In this session, we examine models with an infinite group, and establish that for 56-9=47 of them, there is no decoupling function. In each section, the order of the models is the same as in the table of the paper
<Text-field style="Heading 1" layout="Heading 1">Some tools on singular pairs (X,Y) (means that a(X)d(X) at(Y)dt(Y)=0 =K(X,Y))</Text-field>
Valeurs u telles que a(u)d(u)=0 (x-singuli\303\250res)
singu:=proc(model) local K,ax,ay,dx,dy;
K:=expand(t*x*y*model-x*y);
dx:=factor(discrim(K,y)); #print(dx);
ax:=factor(coeff(K,y,2)); #print(ax);
map(Z->if degree(Z[1],x)=1 then factor(solve (Z[1],x)): fi,op(2,factors(ax*dx,{sqrt(3),I}))):
end:
Valeur de Y associ\303\251e \303\240 une valeur x-singuli\303\250re
Ysolu:=proc(u,model) local rac,K:
K:=expand(t*x*y*model-x*y);
rac:=map(factor,[solve(subs(x=u,K),y)]):
seq([u,rac[i]],i=1..nops(rac)): end:
Idem en v
singv:=proc(model) local K,ax,ay,dx,dy;
K:=expand(t*x*y*model-x*y);
dy:=factor(discrim(K,x));
ay:=factor(coeff(K,x,2));
map(Z->if degree(Z[1],y)=1 then factor(solve (Z[1],y)): fi,op(2,factors(ay*dy,{sqrt(3),I}))):
end:
Xsolv:=proc(v,model) local rac,K:
K:=expand(t*x*y*model-x*y);
rac:=map(factor,[solve(subs(y=v,K),x)]):
seq([rac[i],v],i=1..nops(rac)): end:
La liste des paires (u,Y) x-singuli\303\250res
uYpairs:=proc(liste)
{op(map(Ysolu,singu(liste),liste))};
end:
Xvpairs:=proc(liste)
{op(map(Xsolv,singv(liste),liste))};
end:
<Text-field style="Heading 1" layout="Heading 1">Case 1 : a(x)=t(1+x^2) : 17 models (2 of which are singular)</Text-field>
In all these examples, the key point is to prove that the x-orbit of I (and -I) is infinite.
<Text-field style="Heading 2" layout="Heading 2">NE,NW,SE (singular, detailed in the paper)</Text-field>
liste:=x*y+y/x+x/y;
K:=expand(t*x*y*liste-x*y);
uYpairs(liste);
Xvpairs(liste);
Xsol:=factor([solve(K,x)]);
Ysol:=factor([solve(K,y)]);
The discriminants
dx:=factor(discrim(K,y));
dy:=factor(discrim(K,x));
The branch points x_i and y_i (note that GFUN misses the zero solutions...)
with(gfun): map(allvalues,algeqtoseries(dx,t,x,3));
map(allvalues,algeqtoseries(dy,t,y,3));
Let's have a look at the curve K(x,y)=0
with(plots):
implicitplot(subs(t=0.2,K),x=-8..8,y=-8..8,numpoints=10000);
L'orbite de x=I
k:=0: xx[0]:=I:
factor(subs(x=I,K));
yy[0]:=solve(%,y);
factor(subs(y=%,K));
for i from 1 to 10 do
k:=k+1: factor([solve(%%,x)]):xx[k]:=%[2]:numer(factor(subs(x=xx[k],K))):factor([solve(%,y)]):yy[k]:=%[2]: numer(factor(subs(y=%,K))):od:
k;
We observe how the valuation in t increases
seq(xx[k],k=0..5);
seq(yy[k],k=0..5);
L'orbite de x=0 : (0,0)
k:=0: xx[0]:=0:
factor(subs(x=0,K));
yy[0]:=0;
factor(subs(y=%,K));
<Text-field style="Heading 2" layout="Heading 2">NE,NW,SE,E (singular, differs from the model of the paper by an x/y flip)</Text-field>
liste:=x*y+y/x+x/y+x;
K:=expand(t*x*y*liste-x*y);
factor(subs(x=0,K));factor(subs(y=0,K));
uYpairs(liste);
Xvpairs(liste);
Xsol:=factor([solve(K,x)]);
Ysol:=factor([solve(K,y)]);
dx:=factor(discrim(K,y));
dy:=factor(discrim(K,x));
with(gfun): map(allvalues,algeqtoseries(dx,t,x,3));
map(allvalues,algeqtoseries(dy,t,y,3));
with(plots):
implicitplot(subs(t=0.2,K),x=-8..8,y=-8..8,numpoints=10000);
L'orbite de x=I
xx[0]:=I:
factor(subs(x=xx[0],K));
yy[0]:=solve(%,y);
factor(subs(y=%,K));
for i from 1 to 10 do
k:=k+1: factor([solve(%%,x)]):xx[k]:=%[2]:numer(factor(subs(x=xx[k],K))):factor([solve(%,y)]):yy[k]:=%[2]: numer(factor(subs(y=%,K))):od:
k;
Observe the increasing valuation in t
xx[0];
seq(xx[k],k=0..4);
seq(yy[k],k=0..4);
<Text-field style="Heading 2" layout="Heading 2">NE,NW,SW,E (flipped) (period 3)</Text-field>
liste:=x*y+y/x+1/x/y+x;
K:=expand(t*x*y*liste-x*y);
uYpairs(liste);
Xvpairs(liste);
Xsol:=factor([solve(K,x)]);
Ysol:=factor([solve(K,y)]);
dx:=factor(discrim(K,y));
dy:=factor(discrim(K,x));
with(gfun): map(allvalues,algeqtoseries(dx,t,x,3));
map(allvalues,algeqtoseries(dy,t,y,3));
with(plots):
implicitplot(subs(t=0.2,K),x=-8..8,y=-8..8,numpoints=10000);
factor(discrim(dx,x));
fsolve(%,complex);
L'orbite de x=I
k:=0:xx[0]:= I: factor(subs(x=xx[0],K));
yy[0]:=solve(%,y);
factor(subs(y=yy[0],K));
Version en boucle
for i from 1 to 10 do
k:=k+1: factor([solve(%%,x)]):xx[k]:=%[2]:numer(factor(subs(x=xx[k],K))):factor([solve(%,y)]):yy[k]:=%[2]: numer(factor(subs(y=%,K))):od:
k;
period 3 ?
map(series,[seq(xx[i],i=0..k)],t,4);
map(series,[seq(yy[i],i=0..k)],t,3);
map(series,[seq(xx[3*i],i=0..floor(k/3))],t,4);
Let us prove this
ax:=coeff(K,y,2);
ay:=coeff(K,x,2);
cx:=coeff(K,y,0);
cy:=coeff(K,x,0);
phi:=proc(paire) local u,v:
u:=paire[1]: v:=paire[2]:
[factor(1/u*subs(y=v,cy)/subs(y=v,ay)),v]: end:
psi:=proc(paire) local u,v:
u:=paire[1]: v:=paire[2]:
[u,factor(1/v*subs(x=u,cx)/subs(x=u,ax))]: end:
We start from X0=Xser
Xser:=I+a*t^2+ t^3*R(t);
The corresponding values of Y
normal(subs(csgn(I*a^4*t^10+I*R(t)^4*t^14+I*R(t)^2*t^6+I*a^2*t^4+(6*I)*t^4*R(t)-(2*I)*R(t)^3*t^10-(2*I)*a^3*t^7-12*R(t)^2*a*t^10-12*R(t)*a^2*t^9-2*t^3*R(t)-2*a*t^2-4*R(t)^3*t^11-4*a^3*t^8+12*R(t)*t^5+12*a*t^4+(6*I)*R(t)^2*a^2*t^12+(4*I)*R(t)*a^3*t^11-(6*I)*R(t)^2*a*t^9-(20*I)*R(t)*a*t^7-(6*I)*R(t)*a^2*t^8+(2*I)*R(t)*a*t^5+(4*I)*R(t)^3*a*t^13+6*a^2*t^5+12*a*t^6*R(t)-(10*I)*a^2*t^6+(6*I)*a*t^3-(10*I)*R(t)^2*t^8-2*t+6*t^7*R(t)^2+I*t^2-I)=1,map(series,[solve(subs(x=Xser,K),y)],t,3)));;
The root we want is, like yy[0], a formal power series
yy[0];
op(1,normal(subs(csgn(I*a^4*t^10+I*R(t)^4*t^14+I*R(t)^2*t^6+I*a^2*t^4+(6*I)*t^4*R(t)-(2*I)*R(t)^3*t^10-(2*I)*a^3*t^7-12*R(t)^2*a*t^10-12*R(t)*a^2*t^9-2*t^3*R(t)-2*a*t^2-4*R(t)^3*t^11-4*a^3*t^8+12*R(t)*t^5+12*a*t^4+(6*I)*R(t)^2*a^2*t^12+(4*I)*R(t)*a^3*t^11-(6*I)*R(t)^2*a*t^9-(20*I)*R(t)*a*t^7-(6*I)*R(t)*a^2*t^8+(2*I)*R(t)*a*t^5+(4*I)*R(t)^3*a*t^13+6*a^2*t^5+12*a*t^6*R(t)-(10*I)*a^2*t^6+(6*I)*a*t^3-(10*I)*R(t)^2*t^8-2*t+6*t^7*R(t)^2+I*t^2-I)=1,map(series,[solve(subs(x=Xser,K),y)],t,8))));;
Yser:=series(%,t,4);
[u,v]: phi(%):
factor(map(series,subs(u=Xser,v=Yser,%),t,3));
[u,v]: phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser,%),t,3));
[u,v]: phi(%):psi(%):phi(%):
factor(map(series,subs(u=Xser,v=Yser,%),t,3));
[u,v]: phi(%):psi(%):phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser,%),t,3));
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):
factor(map(series,subs(u=Xser,v=Yser,%),t,3));
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser,%),t,3));
uYpairs(liste);
Xvpairs(liste);
y orbit of -1: whether finite or not, cannot reach non real coefficients. Let us still have a look
factor(subs(y=-1,K));
solve(%,x);
factor(subs(x=%,K));
From this point on, one can iterate the next lines
factor([solve(%,y)]);
numer(factor(subs(y=%[2],K))):
factor([solve(%,x)]);
numer(factor(subs(x=%[2],K))):
<Text-field style="Heading 2" layout="Heading 2">NE,NW,SW,S (detailed in the paper, period 3)</Text-field>
liste:=x*y+y/x+1/x/y+1/y;
K:=expand(t*x*y*liste-x*y);
uYpairs(liste);
Xvpairs(liste);
Xsol:=factor([solve(K,x)]);
Ysol:=factor([solve(K,y)]);
dx:=factor(discrim(K,y));
dy:=factor(discrim(K,x));
with(gfun): map(allvalues,algeqtoseries(dx,t,x,3));
map(allvalues,algeqtoseries(dy,t,y,3));
with(plots):
implicitplot(subs(t=0.2,K),x=-8..8,y=-8..8,numpoints=10000);
L'orbite de x=I
k:=0: xx[0]:=I:
factor(subs(x=I,K));
yy[0]:=solve(%,y);
factor(subs(y=%,K));
for i from 1 to 10 do
k:=k+1: factor([solve(%%,x)]):xx[k]:=%[2]:numer(factor(subs(x=xx[k],K))):factor([solve(%,y)]):yy[k]:=%[2]: numer(factor(subs(y=%,K))):od:
k;
period 3 ?
map(series,[seq(xx[i],i=0..k)],t,4);
map(series,[seq(yy[i],i=0..k)],t,3);
map(series,[seq(xx[3*i],i=0..floor(k/3))],t,4);
ax:=coeff(K,y,2);
ay:=coeff(K,x,2);
cx:=coeff(K,y,0);
cy:=coeff(K,x,0);
phi:=proc(paire) local u,v:
u:=paire[1]: v:=paire[2]:
[factor(1/u*subs(y=v,cy)/subs(y=v,ay)),v]: end:
psi:=proc(paire) local u,v:
u:=paire[1]: v:=paire[2]:
[u,factor(1/v*subs(x=u,cx)/subs(x=u,ax))]: end:
When X=I+at^2+ O(t^3), one of the roots Y reads as follows
Xser:=I+a*t^2+(t^3)*R(t);
normal(subs(csgn((4*I)*a^3*t^8+(4*I)*R(t)^2*t^8+(12*I)*R(t)^2*a*t^10+2*a*t^2+(12*I)*R(t)*a^2*t^9+(4*I)*a^2*t^6-12*R(t)^2*t^8+(4*I)*R(t)^3*t^11-12*a^2*t^6+(8*I)*R(t)*a*t^7-(8*I)*a*t^4-I*a^2*t^4-24*R(t)*a*t^7-I*R(t)^2*t^6-(2*I)*R(t)*a*t^5-8*a*t^4-(8*I)*R(t)*t^5-8*R(t)*t^5+2*t^3*R(t)+I)=1,map(series,[solve(subs(x=Xser,K),y)],t,1)));;
We want the root that is finite at t=0
yy[0];
op(2,normal(subs(csgn((4*I)*a^3*t^8+(4*I)*R(t)^2*t^8+(12*I)*R(t)^2*a*t^10+2*a*t^2+(12*I)*R(t)*a^2*t^9+(4*I)*a^2*t^6-12*R(t)^2*t^8+(4*I)*R(t)^3*t^11-12*a^2*t^6+(8*I)*R(t)*a*t^7-(8*I)*a*t^4-I*a^2*t^4-24*R(t)*a*t^7-I*R(t)^2*t^6-(2*I)*R(t)*a*t^5-8*a*t^4-(8*I)*R(t)*t^5-8*R(t)*t^5+2*t^3*R(t)+I)=1,map(series,[solve(subs(x=Xser,K),y)],t,5))));
Yser:=series(%,t,4);
[u,v]: phi(%): psi(%):phi(%): psi(%):phi(%): psi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,3));
Step by step
[u,v]: phi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,5));
[u,v]: phi(%): psi(%);
factor(map(series,subs(u=Xser,v=Yser ,%),t,5));
[u,v]: phi(%): psi(%):phi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,5));
[u,v]: phi(%): psi(%):phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,5));
[u,v]: phi(%): psi(%):phi(%):psi(%):phi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,4));
[u,v]: phi(%): psi(%):phi(%):psi(%):phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,4));
What if a2=-4*I ? The valuation of Y2 decreases and that's it.
Xser;Yser;
[u,v]: phi(%): psi(%):phi(%):psi(%):
factor(map(series,subs(u=subs(a=-4*I,Xser),v=subs(a=-4*I,Yser) ,%),t,5));
<Text-field style="Heading 2" layout="Heading 2">NE,NW,S,SE (period 3)</Text-field>
liste:=x*y+y/x+1/y+x/y;
K:=expand(t*x*y*liste-x*y);
uYpairs(liste);
Xvpairs(liste);
Xsol:=factor([solve(K,x)]);
Ysol:=factor([solve(K,y)]);
dx:=factor(discrim(K,y));
dy:=factor(discrim(K,x));
with(gfun): map(allvalues,algeqtoseries(dx,t,x,3));
map(allvalues,algeqtoseries(dy,t,y,3));
with(plots):
implicitplot(subs(t=0.2,K),x=-8..8,y=-8..8,numpoints=10000);
L'orbite de x=I
k:=0:xx[0]:= I: factor(subs(x=xx[0],K));
yy[0]:=solve(%,y);
factor(subs(y=yy[0],K));
Version en boucle
for i from 1 to 10 do
k:=k+1: factor([solve(%%,x)]):xx[k]:=%[2]:numer(factor(subs(x=xx[k],K))):factor([solve(%,y)]):yy[k]:=%[2]: numer(factor(subs(y=%,K))):od:
The xx are even in t, while the yy are odd
xx[1];yy[1];
k;
period 3 ?
map(series,[seq(xx[i],i=0..k)],t,4);
map(series,[seq(yy[i],i=0..k)],t,3);
map(series,[seq(xx[3*i],i=0..floor(k/3))],t,4);
ax:=coeff(K,y,2);
ay:=coeff(K,x,2);
cx:=coeff(K,y,0);
cy:=coeff(K,x,0);
phi:=proc(paire) local u,v:
u:=paire[1]: v:=paire[2]:
[factor(1/u*subs(y=v,cy)/subs(y=v,ay)),v]: end:
psi:=proc(paire) local u,v:
u:=paire[1]: v:=paire[2]:
[u,factor(1/v*subs(x=u,cx)/subs(x=u,ax))]: end:
Xser:=I+a*t^2+ t^4*R(t);
normal(subs(csgn(2*t^4*R(t)-16*a^3*t^8+8*R(t)*t^6-48*R(t)^2*a*t^12-48*R(t)*a^2*t^10-12*R(t)^2*t^10-16*R(t)^3*t^14-24*R(t)*a*t^8+(16*I)*R(t)*a^3*t^12-12*a^2*t^6-(40*I)*R(t)*a*t^8-I*R(t)^2*t^8+8*a*t^4-(2*I)*R(t)*a*t^6+(16*I)*R(t)^3*a*t^16+(24*I)*R(t)^2*a^2*t^14+2*a*t^2-I*a^2*t^4-(8*I)*a*t^4-(8*I)*R(t)*t^6-(20*I)*a^2*t^6-(20*I)*R(t)^2*t^10+(4*I)*a^4*t^10+(4*I)*R(t)^4*t^18+(4*I)*R(t)^3*t^14+(4*I)*a^3*t^8+(12*I)*R(t)*a^2*t^10+(12*I)*R(t)^2*a*t^12+I)=1,map(series,[solve(subs(x=Xser,K),y)],t,2)));
op(2,normal(subs(csgn(2*t^4*R(t)-16*a^3*t^8+8*R(t)*t^6-48*R(t)^2*a*t^12-48*R(t)*a^2*t^10-12*R(t)^2*t^10-16*R(t)^3*t^14-24*R(t)*a*t^8+(16*I)*R(t)*a^3*t^12-12*a^2*t^6-(40*I)*R(t)*a*t^8-I*R(t)^2*t^8+8*a*t^4-(2*I)*R(t)*a*t^6+(16*I)*R(t)^3*a*t^16+(24*I)*R(t)^2*a^2*t^14+2*a*t^2-I*a^2*t^4-(8*I)*a*t^4-(8*I)*R(t)*t^6-(20*I)*a^2*t^6-(20*I)*R(t)^2*t^10+(4*I)*a^4*t^10+(4*I)*R(t)^4*t^18+(4*I)*R(t)^3*t^14+(4*I)*a^3*t^8+(12*I)*R(t)*a^2*t^10+(12*I)*R(t)^2*a*t^12+I)=1,map(series,[solve(subs(x=Xser,K),y)],t,9))));;
Yser:=series(%,t,5);
[u,v]: phi(%):
factor(map(series,subs(u=Xser,v=Yser,%),t,3));
[u,v]: phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser,%),t,3));
[u,v]: phi(%):psi(%):phi(%):
factor(map(series,subs(u=Xser,v=Yser,%),t,3));
[u,v]: phi(%):psi(%):phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser,%),t,3));
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):
factor(map(series,subs(u=Xser,v=Yser,%),t,3));
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser,%),t,3));
<Text-field style="Heading 2" layout="Heading 2">NE,NW,S,E (flipped) (period 7)</Text-field>
liste:=x*y+y/x+1/y+x;
K:=expand(t*x*y*liste-x*y);
uYpairs(liste);
Xvpairs(liste);
Xsol:=factor([solve(K,x)]);
Ysol:=factor([solve(K,y)]);
dx:=factor(discrim(K,y));
dy:=factor(discrim(K,x));
with(gfun): map(allvalues,algeqtoseries(dx,t,x,3));
map(allvalues,algeqtoseries(dy,t,y,3));
with(plots):
implicitplot(subs(t=0.2,K),x=-8..8,y=-8..8,numpoints=10000);
L'orbite de x=I
k:=0:xx[0]:= I: factor(subs(x=xx[0],K));
yy[0]:=solve(%,y);
factor(subs(y=yy[0],K));
Version en boucle
for i from 1 to 10 do
k:=k+1: factor([solve(%%,x)]):xx[k]:=%[2]:numer(factor(subs(x=xx[k],K))):factor([solve(%,y)]):yy[k]:=%[2]: numer(factor(subs(y=%,K))):od:
k;
period 7 ?
map(series,[seq(xx[i],i=0..k)],t,4);
map(series,[seq(yy[i],i=0..k)],t,3);
map(series,[seq(xx[7*i],i=0..2)],t,4);
ax:=coeff(K,y,2);
ay:=coeff(K,x,2);
cx:=coeff(K,y,0);
cy:=coeff(K,x,0);
phi:=proc(paire) local u,v:
u:=paire[1]: v:=paire[2]:
[factor(1/u*subs(y=v,cy)/subs(y=v,ay)),v]: end:
psi:=proc(paire) local u,v:
u:=paire[1]: v:=paire[2]:
[u,factor(1/v*subs(x=u,cx)/subs(x=u,ax))]: end:
Xser:=I+a*t+ t^2*R(t);
op(1,normal(subs(csgn(4*t^4*R(t)+12*R(t)^2*t^6+12*a^2*t^4-(6*I)*R(t)^2*t^6+(8*I)*t^4*R(t)+12*R(t)*a*t^4-(12*I)*R(t)*a^2*t^6-(6*I)*R(t)*a^2*t^5-(12*I)*R(t)*a*t^5-(4*I)*R(t)^3*t^8-(6*I)*a^2*t^4-(2*I)*a^3*t^4+I*t^2+(2*I)*R(t)*a*t^3+4*a*t^3+24*R(t)*a*t^5+(6*I)*R(t)^2*a^2*t^8+(4*I)*R(t)*a^3*t^7-(12*I)*R(t)^2*a*t^7-(6*I)*R(t)^2*a*t^6+(6*I)*a*t^2+(8*I)*a*t^3+(6*I)*t^3*R(t)+(4*I)*R(t)^3*a*t^9+I*R(t)^2*t^4+I*a^4*t^6+I*R(t)^4*t^10+I*a^2*t^2-12*R(t)^2*a*t^7-12*R(t)*a^2*t^6-(2*I)*R(t)^3*t^7-(4*I)*a^3*t^5-4*a^3*t^5-4*R(t)^3*t^8+6*t^5*R(t)^2+6*a^2*t^3-2*t^2*R(t)-2*a*t-I-2*t)=1,map(series,[solve(subs(x=Xser,K),y)],t,6))));;
Yser:=series(%,t,4);
[u,v]: phi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,2));
[u,v]: phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,2));
[u,v]: phi(%):psi(%):phi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,2));
[u,v]: phi(%):psi(%):phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,2));
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,2));
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,2));
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,2));
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,2));
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,2));
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,2));
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,2));
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,2));
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,2));
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,3));
Xser; Yser;
<Text-field style="Heading 2" layout="Heading 2">NE,NW,W,S (period 7)</Text-field>
liste:=x*y+y/x+1/x+1/y;
K:=expand(t*x*y*liste-x*y);
uYpairs(liste);
Xvpairs(liste);
Xsol:=factor([solve(K,x)]);
Ysol:=factor([solve(K,y)]);
dx:=factor(discrim(K,y));
dy:=factor(discrim(K,x));
with(gfun): map(allvalues,algeqtoseries(dx,t,x,3));
map(allvalues,algeqtoseries(dy,t,y,3));
with(plots):
implicitplot(subs(t=0.2,K),x=-8..8,y=-8..8,numpoints=10000);
L'orbite de x=I
k:=0:xx[0]:= I: factor(subs(x=xx[0],K));
yy[0]:=solve(%,y);
factor(subs(y=yy[0],K));
Version en boucle
for i from 1 to 10 do
k:=k+1: factor([solve(%%,x)]):xx[k]:=%[2]:numer(factor(subs(x=xx[k],K))):factor([solve(%,y)]):yy[k]:=%[2]: numer(factor(subs(y=%,K))):od:
k;
period 7 ?
map(series,[seq(xx[i],i=0..k)],t,4);
map(series,[seq(yy[i],i=0..k)],t,3);
map(series,[seq(xx[7*i],i=0..2)],t,4);
ax:=coeff(K,y,2);
ay:=coeff(K,x,2);
cx:=coeff(K,y,0);
cy:=coeff(K,x,0);
phi:=proc(paire) local u,v:
u:=paire[1]: v:=paire[2]:
[factor(1/u*subs(y=v,cy)/subs(y=v,ay)),v]: end:
psi:=proc(paire) local u,v:
u:=paire[1]: v:=paire[2]:
[u,factor(1/v*subs(x=u,cx)/subs(x=u,ax))]: end:
Xser:=I+a*t+t^2*R(t);
op(2,normal(subs(csgn(-I*t^2+(4*I)*a^3*t^5-12*a^2*t^4+(12*I)*R(t)^2*a*t^7-2*t+(4*I)*R(t)^3*t^8-(2*I)*R(t)*a*t^3+2*a*t+(2*I)*a*t^2+(2*I)*t^3*R(t)-I*R(t)^2*t^4-24*R(t)*a*t^5-(8*I)*a*t^3-(8*I)*t^4*R(t)+2*t^2*R(t)+(12*I)*R(t)*a^2*t^6+I-12*R(t)^2*t^6-I*a^2*t^2)=1,map(series,[solve(subs(x=Xser,K),y)],t,7))));;
Yser:=series(%,t,4);
[u,v]: phi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,2));
[u,v]: phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,2));
[u,v]: phi(%):psi(%):phi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,2));
[u,v]: phi(%):psi(%):phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,2));
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,2));
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,2));
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,2));
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,2));
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,2));
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,2));
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,2));
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,2));
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,2));
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,3));
Xser; Yser;
<Text-field style="Heading 2" layout="Heading 2">NE,NW,S,SE,E (flipped) (period 3)</Text-field>
liste:=x*y+y/x+1/y+x/y+x;
K:=expand(t*x*y*liste-x*y);
Xsol:=factor([solve(K,x)]);
Ysol:=factor([solve(K,y)]);
uYpairs(liste);
Xvpairs(liste);
dx:=factor(discrim(K,y));
dy:=factor(discrim(K,x));
with(gfun): map(allvalues,algeqtoseries(dx,t,x,3));
map(allvalues,algeqtoseries(dy,t,y,3));
with(plots):
implicitplot(subs(t=0.2,K),x=-8..8,y=-8..8,numpoints=10000);
L'orbite de x=I
k:=0:xx[0]:= I: factor(subs(x=xx[0],K));
yy[0]:=solve(%,y);
factor(subs(y=yy[0],K));
Version en boucle
for i from 1 to 10 do
k:=k+1: factor([solve(%%,x)]):xx[k]:=%[2]:numer(factor(subs(x=xx[k],K))):factor([solve(%,y)]):yy[k]:=%[2]: numer(factor(subs(y=%,K))):od:
k;
period 3 ?
map(series,[seq(xx[i],i=0..k)],t,4);
map(series,[seq(yy[i],i=0..k)],t,3);
map(series,[seq(xx[3*i],i=0..4)],t,4);
ax:=coeff(K,y,2);
ay:=coeff(K,x,2);
cx:=coeff(K,y,0);
cy:=coeff(K,x,0);
phi:=proc(paire) local u,v:
u:=paire[1]: v:=paire[2]:
[factor(1/u*subs(y=v,cy)/subs(y=v,ay)),v]: end:
psi:=proc(paire) local u,v:
u:=paire[1]: v:=paire[2]:
[u,factor(1/v*subs(x=u,cx)/subs(x=u,ax))]: end:
Xser:=I+a*t+t^2*R(t);
op(2,normal(subs(csgn((2*I)*a^3*t^4-(6*I)*t^3*R(t)+(3*I)*R(t)^4*t^10+(2*I)*R(t)^3*t^7+(4*I)*R(t)^3*t^8+(4*I)*a^3*t^5-(8*I)*a*t^3-(8*I)*t^4*R(t)+(6*I)*R(t)*a^2*t^5-(2*I)*R(t)*a*t^3-I*t^2-(14*I)*a^2*t^4+(6*I)*R(t)^2*a*t^6+(12*I)*R(t)^3*a*t^9+(3*I)*a^4*t^6-(28*I)*R(t)*a*t^5+(12*I)*R(t)*a^3*t^7+(18*I)*R(t)^2*a^2*t^8+(12*I)*R(t)*a^2*t^6+(12*I)*R(t)^2*a*t^7-I*R(t)^2*t^4-I*a^2*t^2-36*R(t)*a^2*t^6-12*R(t)*a*t^4-24*R(t)*a*t^5-36*R(t)^2*a*t^7+4*t^4*R(t)-12*R(t)^2*t^6-12*a^2*t^4+4*a*t^3-12*a^3*t^5-12*R(t)^3*t^8-6*t^5*R(t)^2-6*a^2*t^3+2*t^2*R(t)+2*a*t-(14*I)*R(t)^2*t^6-(6*I)*a*t^2+2*t+I)=1,map(series,[solve(subs(x=Xser,K),y)],t,5))));;
Yser:=series(%,t,3);
[u,v]: phi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,2));
[u,v]: phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,2));
[u,v]: phi(%):psi(%):phi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,2));
[u,v]: phi(%):psi(%):phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,2));
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,2));
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,2));
Xser; Yser;;
<Text-field style="Heading 2" layout="Heading 2">NE,NW,W,SW,S (period 3)</Text-field>
liste:=x*y+y/x+1/x+1/x/y+1/y;
K:=expand(t*x*y*liste-x*y);
uYpairs(liste);
Xvpairs(liste);
Xsol:=factor([solve(K,x)]);
Ysol:=factor([solve(K,y)]);
dx:=factor(discrim(K,y));
dy:=factor(discrim(K,x));
with(gfun): map(allvalues,algeqtoseries(dx,t,x,3));
map(allvalues,algeqtoseries(dy,t,y,3));
with(plots):
implicitplot(subs(t=0.2,K),x=-8..8,y=-8..8,numpoints=10000);
L'orbite de x=I
k:=0:xx[0]:= I: factor(subs(x=xx[0],K));
yy[0]:=solve(%,y);
factor(subs(y=yy[0],K));
Version en boucle
for i from 1 to 10 do
k:=k+1: factor([solve(%%,x)]):xx[k]:=%[2]:numer(factor(subs(x=xx[k],K))):factor([solve(%,y)]):yy[k]:=%[2]: numer(factor(subs(y=%,K))):od:
k;
period 3 ?
map(series,[seq(xx[i],i=0..k)],t,4);
map(series,[seq(yy[i],i=0..k)],t,3);
ax:=coeff(K,y,2);
ay:=coeff(K,x,2);
cx:=coeff(K,y,0);
cy:=coeff(K,x,0);
phi:=proc(paire) local u,v:
u:=paire[1]: v:=paire[2]:
[factor(1/u*subs(y=v,cy)/subs(y=v,ay)),v]: end:
psi:=proc(paire) local u,v:
u:=paire[1]: v:=paire[2]:
[u,factor(1/v*subs(x=u,cx)/subs(x=u,ax))]: end:
Xser:=I+a*t+ t^2*R(t);
op(2,normal(subs(csgn((4*I)*a^2*t^4-I*t^2-12*a^2*t^4+(4*I)*a^3*t^5+(12*I)*R(t)^2*a*t^7-2*t+(4*I)*R(t)^3*t^8-(2*I)*R(t)*a*t^3+(2*I)*a*t^2+2*a*t+(8*I)*R(t)*a*t^5-24*R(t)*a*t^5+(2*I)*t^3*R(t)-I*R(t)^2*t^4-(8*I)*a*t^3+2*t^2*R(t)-8*a*t^3-(8*I)*t^4*R(t)+(12*I)*R(t)*a^2*t^6-8*t^4*R(t)-I*a^2*t^2+I-12*R(t)^2*t^6+(4*I)*R(t)^2*t^6)=1,map(series,[solve(subs(x=Xser,K),y)],t,5))));;
Yser:=series(%,t,4);
[u,v]: phi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,2));
[u,v]: phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,2));
[u,v]: phi(%):psi(%):phi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,2));
[u,v]: phi(%):psi(%):phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,2));
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,2));
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,2));
Xser; Yser;
<Text-field style="Heading 2" layout="Heading 2">NE,NW,W,SW,E (flipped) (period 3)</Text-field>
liste:=x*y+y/x+1/x+1/x/y+x;
K:=expand(t*x*y*liste-x*y);
Xsol:=factor([solve(K,x)]);
Ysol:=factor([solve(K,y)]);
dx:=factor(discrim(K,y));
dy:=factor(discrim(K,x));
with(gfun): map(allvalues,algeqtoseries(dx,t,x,3));
map(allvalues,algeqtoseries(dy,t,y,3));
with(plots):
implicitplot(subs(t=0.2,K),x=-8..8,y=-8..8,numpoints=10000);
L'orbite de x=I
k:=0:xx[0]:= I: factor(subs(x=xx[0],K));
yy[0]:=solve(%,y);
factor(subs(y=yy[0],K));
Version en boucle
for i from 1 to 10 do
k:=k+1: factor([solve(%%,x)]):xx[k]:=%[2]:numer(factor(subs(x=xx[k],K))):factor([solve(%,y)]):yy[k]:=%[2]: numer(factor(subs(y=%,K))):od:
k;
period 3 ?
map(series,[seq(xx[i],i=0..k)],t,4);
map(series,[seq(yy[i],i=0..k)],t,3);
map(series,[seq(xx[3*i],i=0..4)],t,4);
ax:=coeff(K,y,2);
ay:=coeff(K,x,2);
cx:=coeff(K,y,0);
cy:=coeff(K,x,0);
phi:=proc(paire) local u,v:
u:=paire[1]: v:=paire[2]:
[factor(1/u*subs(y=v,cy)/subs(y=v,ay)),v]: end:
psi:=proc(paire) local u,v:
u:=paire[1]: v:=paire[2]:
[u,factor(1/v*subs(x=u,cx)/subs(x=u,ax))]: end:
Xser:=I+a*t^2+ t^4*R(t);
op(1,normal(subs(csgn(-4*R(t)^3*t^14-4*a^3*t^8+I*a^4*t^10+I*R(t)^4*t^18+I*R(t)^2*t^8+I*a^2*t^4+6*a^2*t^5+(4*I)*R(t)^3*a*t^16+(6*I)*R(t)^2*a^2*t^14+(4*I)*R(t)*a^3*t^12-(6*I)*R(t)^2*a*t^11-(16*I)*R(t)*a*t^8-(6*I)*R(t)*a^2*t^9+(2*I)*R(t)*a*t^6+(4*I)*a*t^3+(4*I)*t^5*R(t)-(2*I)*a^3*t^7-(8*I)*a^2*t^6+8*R(t)*t^6-2*t^4*R(t)-2*a*t^2+12*a*t^7*R(t)-(2*I)*R(t)^3*t^13-(8*I)*R(t)^2*t^10-12*R(t)^2*a*t^12-12*R(t)*a^2*t^10+8*a*t^4+6*t^9*R(t)^2-I)=1,map(series,[solve(subs(x=Xser,K),y)],t,10))));;
Yser:=series(%,t,5);
[u,v]: phi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,4));
[u,v]: phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,4));
[u,v]: phi(%):psi(%):phi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,4));
[u,v]: phi(%):psi(%):phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,4));
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,4));
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,4));
Xser; Yser;
<Text-field style="Heading 2" layout="Heading 2">NE,NW,W,SE,E (flipped) (period 3)</Text-field>
liste:=x*y+y/x+1/x+x/y+x;
K:=expand(t*x*y*liste-x*y);
Xsol:=factor([solve(K,x)]);
Ysol:=factor([solve(K,y)]);
dx:=factor(discrim(K,y));
dy:=factor(discrim(K,x));
with(gfun): map(allvalues,algeqtoseries(dx,t,x,3));
map(allvalues,algeqtoseries(dy,t,y,3));
with(plots):
implicitplot(subs(t=0.2,K),x=-8..8,y=-8..8,numpoints=10000);
L'orbite de x=I
k:=0:xx[0]:= I: factor(subs(x=xx[0],K));
yy[0]:=solve(%,y);
factor(subs(y=yy[0],K));
Version en boucle
for i from 1 to 10 do
k:=k+1: factor([solve(%%,x)]):xx[k]:=%[2]:numer(factor(subs(x=xx[k],K))):factor([solve(%,y)]):yy[k]:=%[2]: numer(factor(subs(y=%,K))):od:
k;
period 3 ?
map(series,[seq(xx[i],i=0..k)],t,4);
map(series,[seq(yy[i],i=0..k)],t,3);
map(series,[seq(xx[3*i],i=0..4)],t,4);
ax:=coeff(K,y,2);
ay:=coeff(K,x,2);
cx:=coeff(K,y,0);
cy:=coeff(K,x,0);
phi:=proc(paire) local u,v:
u:=paire[1]: v:=paire[2]:
[factor(1/u*subs(y=v,cy)/subs(y=v,ay)),v]: end:
psi:=proc(paire) local u,v:
u:=paire[1]: v:=paire[2]:
[u,factor(1/v*subs(x=u,cx)/subs(x=u,ax))]: end:
Xser:=I+a*t^2+ t^4*R(t);
op(2,normal(subs(csgn(-12*R(t)^3*t^14-12*a^3*t^8-I*R(t)^2*t^8-6*a^2*t^5-(16*I)*a^2*t^6+(3*I)*R(t)^4*t^18-(4*I)*a*t^3+(2*I)*a^3*t^7+(2*I)*R(t)^3*t^13-I*a^2*t^4+8*R(t)*t^6+(3*I)*a^4*t^10-(16*I)*R(t)^2*t^10-(4*I)*t^5*R(t)+(12*I)*R(t)^3*a*t^16+(18*I)*R(t)^2*a^2*t^14+(12*I)*R(t)*a^3*t^12+(6*I)*R(t)^2*a*t^11+(6*I)*R(t)*a^2*t^9+2*t^4*R(t)+2*a*t^2-(32*I)*R(t)*a*t^8-(2*I)*R(t)*a*t^6-12*a*t^7*R(t)-36*R(t)^2*a*t^12-36*R(t)*a^2*t^10+8*a*t^4-6*t^9*R(t)^2+I)=1,map(series,[solve(subs(x=Xser,K),y)],t,10))));;
Yser:=series(%,t,5);
[u,v]: phi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,4));
[u,v]: phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,4));
[u,v]: phi(%):psi(%):phi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,4));
[u,v]: phi(%):psi(%):phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,4));
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,4));
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,4));
Xser; Yser;
<Text-field style="Heading 2" layout="Heading 2">NE,NW,SW,S,E (flipped) (period 5)</Text-field>
liste:=x*y+y/x+1/x/y+1/y+x;
K:=expand(t*x*y*liste-x*y);
Xsol:=factor([solve(K,x)]);
Ysol:=factor([solve(K,y)]);
dx:=factor(discrim(K,y));
dy:=factor(discrim(K,x));
with(gfun): map(allvalues,algeqtoseries(dx,t,x,3));
map(allvalues,algeqtoseries(dy,t,y,3));
with(plots):
implicitplot(subs(t=0.2,K),x=-8..8,y=-8..8,numpoints=10000);
L'orbite de x=I
k:=0:xx[0]:= I: factor(subs(x=xx[0],K));
yy[0]:=solve(%,y);
factor(subs(y=yy[0],K));
Version en boucle
for i from 1 to 10 do
k:=k+1: factor([solve(%%,x)]):xx[k]:=%[2]:numer(factor(subs(x=xx[k],K))):factor([solve(%,y)]):yy[k]:=%[2]: numer(factor(subs(y=%,K))):od:
k;
period 5 ?
map(series,[seq(xx[i],i=0..k)],t,4);
map(series,[seq(yy[i],i=0..k)],t,3);
map(series,[seq(xx[5*i],i=0..2)],t,4);
ax:=coeff(K,y,2);
ay:=coeff(K,x,2);
cx:=coeff(K,y,0);
cy:=coeff(K,x,0);
phi:=proc(paire) local u,v:
u:=paire[1]: v:=paire[2]:
[factor(1/u*subs(y=v,cy)/subs(y=v,ay)),v]: end:
psi:=proc(paire) local u,v:
u:=paire[1]: v:=paire[2]:
[u,factor(1/v*subs(x=u,cx)/subs(x=u,ax))]: end:
Xser:=I+a*t+t^2*R(t);
op(1,normal(subs(csgn((2*I)*R(t)*a*t^3+(6*I)*R(t)^2*a^2*t^8+(6*I)*t^3*R(t)+(8*I)*t^4*R(t)-(2*I)*a^3*t^4-(12*I)*R(t)*a^2*t^6-(10*I)*R(t)^2*t^6-(4*I)*R(t)^3*t^8-(10*I)*a^2*t^4-(4*I)*a^3*t^5+(4*I)*R(t)*a^3*t^7+(4*I)*R(t)^3*a*t^9-(20*I)*R(t)*a*t^5+(8*I)*a*t^3-(2*I)*R(t)^3*t^7-12*R(t)*a^2*t^6+12*R(t)*a*t^4+24*R(t)*a*t^5-12*R(t)^2*a*t^7+(6*I)*a*t^2+12*t^4*R(t)+12*R(t)^2*t^6+12*a^2*t^4+12*a*t^3-4*a^3*t^5-4*R(t)^3*t^8+6*t^5*R(t)^2+6*a^2*t^3-2*t^2*R(t)-2*a*t+I*R(t)^2*t^4+I*a^4*t^6+I*R(t)^4*t^10+I*a^2*t^2-(6*I)*R(t)*a^2*t^5-(12*I)*R(t)^2*a*t^7-(6*I)*R(t)^2*a*t^6+I*t^2-2*t-I)=1,map(series,[solve(subs(x=Xser,K),y)],t,5))));;
Yser:=series(%,t,3);
[u,v]: phi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,2));
[u,v]: phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,2));
[u,v]: phi(%):psi(%):phi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,2));
[u,v]: phi(%):psi(%):phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,2));
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,2));
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,2));
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,2));
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,2));
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,2));
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,2));
Xser; Yser;
<Text-field style="Heading 2" layout="Heading 2">NE,NW,W,S,SE (period 5)</Text-field>
liste:=x*y+y/x+1/x+1/y+x/y;
K:=expand(t*x*y*liste-x*y);
Xsol:=factor([solve(K,x)]);
Ysol:=factor([solve(K,y)]);
dx:=factor(discrim(K,y));
dy:=factor(discrim(K,x));
with(gfun): map(allvalues,algeqtoseries(dx,t,x,3));
map(allvalues,algeqtoseries(dy,t,y,3));
with(plots):
implicitplot(subs(t=0.2,K),x=-8..8,y=-8..8,numpoints=10000);
L'orbite de x=I
k:=0:xx[0]:= I: factor(subs(x=xx[0],K));
yy[0]:=solve(%,y);
factor(subs(y=yy[0],K));
for i from 1 to 10 do
k:=k+1: factor([solve(%%,x)]):xx[k]:=%[2]:numer(factor(subs(x=xx[k],K))):factor([solve(%,y)]):yy[k]:=%[2]: numer(factor(subs(y=%,K))):od:
k;
period 5 ?
map(series,[seq(xx[i],i=0..k)],t,4);
map(series,[seq(yy[i],i=0..k)],t,3);
map(series,[seq(xx[5*i],i=0..2)],t,4);
ax:=coeff(K,y,2);
ay:=coeff(K,x,2);
cx:=coeff(K,y,0);
cy:=coeff(K,x,0);
phi:=proc(paire) local u,v:
u:=paire[1]: v:=paire[2]:
[factor(1/u*subs(y=v,cy)/subs(y=v,ay)),v]: end:
psi:=proc(paire) local u,v:
u:=paire[1]: v:=paire[2]:
[u,factor(1/v*subs(x=u,cx)/subs(x=u,ax))]: end:
Xser:=I+a*t+ t^2*R(t);
op(2,normal(subs(csgn((12*I)*R(t)^2*a*t^7+(4*I)*a^3*t^5+(4*I)*R(t)^3*t^8-(8*I)*t^4*R(t)+(2*I)*a*t^2-(2*I)*R(t)*a*t^3-(20*I)*a^2*t^4+(2*I)*t^3*R(t)-(8*I)*a*t^3+(4*I)*R(t)^4*t^10+(12*I)*R(t)*a^2*t^6-(40*I)*R(t)*a*t^5+(16*I)*R(t)*a^3*t^7+(24*I)*R(t)^2*a^2*t^8+(16*I)*R(t)^3*a*t^9-I*t^2-(20*I)*R(t)^2*t^6-I*R(t)^2*t^4-I*a^2*t^2-48*R(t)*a^2*t^6-24*R(t)*a*t^5-48*R(t)^2*a*t^7+8*t^4*R(t)-12*R(t)^2*t^6-12*a^2*t^4+8*a*t^3-16*a^3*t^5-16*R(t)^3*t^8+2*t^2*R(t)+2*a*t+(4*I)*a^4*t^6-2*t+I)=1,map(series,[solve(subs(x=Xser,K),y)],t,5))));;
Yser:=series(%,t,3);
[u,v]: phi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,2));
[u,v]: phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,2));
[u,v]: phi(%):psi(%):phi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,2));
[u,v]: phi(%):psi(%):phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,2));
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,2));
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,2));
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,2));
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,2));
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,2));
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,2));
Xser; Yser;
<Text-field style="Heading 2" layout="Heading 2">NE,NW,W,SW,S,E (flipped) (period 5)</Text-field>
liste:=x*y+y/x+1/x+1/x/y+1/y+x;
K:=expand(t*x*y*liste-x*y);
Xsol:=factor([solve(K,x)]);
Ysol:=factor([solve(K,y)]);
dx:=factor(discrim(K,y));
dy:=factor(discrim(K,x));
with(gfun): map(allvalues,algeqtoseries(dx,t,x,3));
map(allvalues,algeqtoseries(dy,t,y,3));
with(plots):
implicitplot(subs(t=0.2,K),x=-8..18,y=-8..18,numpoints=10000);
L'orbite de x=I
k:=0:xx[0]:= I: factor(subs(x=xx[0],K));
yy[0]:=solve(%,y);
factor(subs(y=yy[0],K));
Version en boucle
for i from 1 to 10 do
k:=k+1: factor([solve(%%,x)]):xx[k]:=%[2]:numer(factor(subs(x=xx[k],K))):factor([solve(%,y)]):yy[k]:=%[2]: numer(factor(subs(y=%,K))):od:
k;
period 5 ?
map(series,[seq(xx[i],i=0..k)],t,4);
map(series,[seq(yy[i],i=0..k)],t,3);
map(series,[seq(xx[5*i],i=0..2)],t,4);
ax:=coeff(K,y,2);
ay:=coeff(K,x,2);
cx:=coeff(K,y,0);
cy:=coeff(K,x,0);
phi:=proc(paire) local u,v:
u:=paire[1]: v:=paire[2]:
[factor(1/u*subs(y=v,cy)/subs(y=v,ay)),v]: end:
psi:=proc(paire) local u,v:
u:=paire[1]: v:=paire[2]:
[u,factor(1/v*subs(x=u,cx)/subs(x=u,ax))]: end:
Xser:=I+a*t+ t^2*R(t);
op(1,normal(subs(csgn(-(2*I)*R(t)^3*t^7+(4*I)*R(t)^3*a*t^9-(12*I)*R(t)^2*a*t^7-(6*I)*R(t)^2*a*t^6-(2*I)*a^3*t^4-(8*I)*R(t)^2*t^6-(8*I)*a^2*t^4+(4*I)*a*t^2+(4*I)*t^3*R(t)-(4*I)*R(t)^3*t^8-(4*I)*a^3*t^5-(12*I)*R(t)*a^2*t^6-(6*I)*R(t)*a^2*t^5+(8*I)*t^4*R(t)+(6*I)*R(t)^2*a^2*t^8+(4*I)*R(t)*a^3*t^7-12*R(t)*a^2*t^6+12*R(t)*a*t^4+24*R(t)*a*t^5-12*R(t)^2*a*t^7+(2*I)*R(t)*a*t^3+8*t^4*R(t)+12*R(t)^2*t^6+12*a^2*t^4+8*a*t^3-4*a^3*t^5-4*R(t)^3*t^8+6*t^5*R(t)^2+6*a^2*t^3-2*t^2*R(t)-2*a*t+I*R(t)^2*t^4+I*a^4*t^6+I*R(t)^4*t^10+I*a^2*t^2+(8*I)*a*t^3-(16*I)*R(t)*a*t^5-I)=1,map(series,[solve(subs(x=Xser,K),y)],t,5))));;
Yser:=series(%,t,3);
[u,v]: phi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,2));
[u,v]: phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,2));
[u,v]: phi(%):psi(%):phi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,2));
[u,v]: phi(%):psi(%):phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,2));
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,2));
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,2));
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,2));
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,2));
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,2));
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,2));
Xser; Yser;
<Text-field style="Heading 2" layout="Heading 2">NE,NW,W,S,SE,E (flipped) (period 5)</Text-field>
liste:=x*y+y/x+1/x+1/y+x/y+x;
K:=expand(t*x*y*liste-x*y);
Xsol:=factor([solve(K,x)]);
Ysol:=factor([solve(K,y)]);
dx:=factor(discrim(K,y));
dy:=factor(discrim(K,x));
with(gfun): map(allvalues,algeqtoseries(dx,t,x,3));
map(allvalues,algeqtoseries(dy,t,y,3));
with(plots):
implicitplot(subs(t=0.2,K),x=-8..8,y=-8..8,numpoints=10000);
L'orbite de x=I
k:=0:xx[0]:= I: factor(subs(x=xx[0],K));
yy[0]:=solve(%,y);
factor(subs(y=yy[0],K));
Version en boucle
for i from 1 to 10 do
k:=k+1: factor([solve(%%,x)]):xx[k]:=%[2]:numer(factor(subs(x=xx[k],K))):factor([solve(%,y)]):yy[k]:=%[2]: numer(factor(subs(y=%,K))):od:
k;
period 5 ?
map(series,[seq(xx[i],i=0..k)],t,4);
map(series,[seq(yy[i],i=0..k)],t,3);
ax:=coeff(K,y,2);
ay:=coeff(K,x,2);
cx:=coeff(K,y,0);
cy:=coeff(K,x,0);
phi:=proc(paire) local u,v:
u:=paire[1]: v:=paire[2]:
[factor(1/u*subs(y=v,cy)/subs(y=v,ay)),v]: end:
psi:=proc(paire) local u,v:
u:=paire[1]: v:=paire[2]:
[u,factor(1/v*subs(x=u,cx)/subs(x=u,ax))]: end:
Xser:=I+a*t+t^2*R(t);
op(2,normal(subs(csgn(-(32*I)*R(t)*a*t^5+(12*I)*R(t)*a^2*t^6-I*R(t)^2*t^4-I*a^2*t^2-36*R(t)*a^2*t^6-12*R(t)*a*t^4-24*R(t)*a*t^5-36*R(t)^2*a*t^7+8*t^4*R(t)-12*R(t)^2*t^6-12*a^2*t^4+8*a*t^3-12*a^3*t^5-12*R(t)^3*t^8-6*t^5*R(t)^2-6*a^2*t^3+2*t^2*R(t)+2*a*t+(4*I)*R(t)^3*t^8+(4*I)*a^3*t^5+I-(2*I)*R(t)*a*t^3+(6*I)*R(t)*a^2*t^5+(6*I)*R(t)^2*a*t^6+(12*I)*R(t)^3*a*t^9+(12*I)*R(t)*a^3*t^7+(18*I)*R(t)^2*a^2*t^8+(12*I)*R(t)^2*a*t^7-(16*I)*a^2*t^4-(16*I)*R(t)^2*t^6-(4*I)*a*t^2-(4*I)*t^3*R(t)-(8*I)*a*t^3-(8*I)*t^4*R(t)+(3*I)*a^4*t^6+(2*I)*a^3*t^4+(3*I)*R(t)^4*t^10+(2*I)*R(t)^3*t^7)=1,map(series,[solve(subs(x=Xser,K),y)],t,5))));;
Yser:=series(%,t,3);
[u,v]: phi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,2));
[u,v]: phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,2));
[u,v]: phi(%):psi(%):phi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,2));
[u,v]: phi(%):psi(%):phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,2));
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,2));
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,2));
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,2));
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,2));
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,2));
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,2));
Xser; Yser;
<Text-field style="Heading 2" layout="Heading 2">NE,NW,SW,S,SE,E (flipped) (period 2)</Text-field>
liste:=x*y+y/x+1/x/y+1/y+x/y+x;
K:=expand(t*x*y*liste-x*y);
Xsol:=factor([solve(K,x)]);
Ysol:=factor([solve(K,y)]);
dx:=factor(discrim(K,y));
dy:=factor(discrim(K,x));
with(gfun): map(allvalues,algeqtoseries(dx,t,x,3));
map(allvalues,algeqtoseries(dy,t,y,3));
with(plots):
implicitplot(subs(t=0.2,K),x=-8..8,y=-8..8,numpoints=10000);
L'orbite de x=I
k:=0:xx[0]:= I: factor(subs(x=xx[0],K));
yy[0]:=solve(%,y);
factor(subs(y=yy[0],K));
Version en boucle
for i from 1 to 10 do
k:=k+1: factor([solve(%%,x)]):xx[k]:=%[2]:numer(factor(subs(x=xx[k],K))):factor([solve(%,y)]):yy[k]:=%[2]: numer(factor(subs(y=%,K))):od:
k;
period 2 ?
map(series,[seq(xx[i],i=0..k)],t,4);
map(series,[seq(yy[i],i=0..k)],t,3);
map(series,[seq(xx[2*i],i=0..k/2)],t,4);
ax:=coeff(K,y,2);
ay:=coeff(K,x,2);
cx:=coeff(K,y,0);
cy:=coeff(K,x,0);
phi:=proc(paire) local u,v:
u:=paire[1]: v:=paire[2]:
[factor(1/u*subs(y=v,cy)/subs(y=v,ay)),v]: end:
psi:=proc(paire) local u,v:
u:=paire[1]: v:=paire[2]:
[u,factor(1/v*subs(x=u,cx)/subs(x=u,ax))]: end:
Xser:=I+a*t+ t^2*R(t);
op(2,normal(subs(csgn((6*I)*R(t)*a^2*t^5+(6*I)*R(t)^2*a*t^6+(12*I)*R(t)^3*a*t^9+(12*I)*R(t)*a^3*t^7+(18*I)*R(t)^2*a^2*t^8+(12*I)*R(t)^2*a*t^7+(12*I)*R(t)*a^2*t^6-I*t^2-(20*I)*R(t)*a*t^5-I*R(t)^2*t^4-(2*I)*R(t)*a*t^3-I*a^2*t^2-36*R(t)*a^2*t^6-12*R(t)*a*t^4-24*R(t)*a*t^5-36*R(t)^2*a*t^7-4*t^4*R(t)-12*R(t)^2*t^6-12*a^2*t^4-4*a*t^3-12*a^3*t^5-12*R(t)^3*t^8-6*t^5*R(t)^2-6*a^2*t^3+2*t^2*R(t)+2*a*t+(4*I)*R(t)^3*t^8+(4*I)*a^3*t^5+(3*I)*R(t)^4*t^10-(6*I)*t^3*R(t)+(3*I)*a^4*t^6+(2*I)*R(t)^3*t^7-(6*I)*a*t^2+(2*I)*a^3*t^4-(10*I)*R(t)^2*t^6-(8*I)*t^4*R(t)-(8*I)*a*t^3-(10*I)*a^2*t^4+2*t+I)=1,map(series,[solve(subs(x=Xser,K),y)],t,5))));;
Yser:=%;
[u,v]: phi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,2));
[u,v]: phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,2));
[u,v]: phi(%):psi(%):phi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,2));
[u,v]: phi(%):psi(%):phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,2));
Xser; Yser;
<Text-field style="Heading 2" layout="Heading 2">NE,NW,W,SW,S,SE (period 2)</Text-field>
liste:=x*y+y/x+1/x+1/x/y+1/y+x/y;
K:=expand(t*x*y*liste-x*y);
Xsol:=factor([solve(K,x)]);
Ysol:=factor([solve(K,y)]);
dx:=factor(discrim(K,y));
dy:=factor(discrim(K,x));
with(gfun): map(allvalues,algeqtoseries(dx,t,x,3));
map(allvalues,algeqtoseries(dy,t,y,3));
with(plots):
implicitplot(subs(t=0.2,K),x=-8..8,y=-8..8,numpoints=10000);
L'orbite de x=I
k:=0:xx[0]:= I: factor(subs(x=xx[0],K));
yy[0]:=solve(%,y);
factor(subs(y=yy[0],K));
Version en boucle
for i from 1 to 10 do
k:=k+1: factor([solve(%%,x)]):xx[k]:=%[2]:numer(factor(subs(x=xx[k],K))):factor([solve(%,y)]):yy[k]:=%[2]: numer(factor(subs(y=%,K))):od:
k;
period 2 ?
map(series,[seq(xx[i],i=0..k)],t,4);
map(series,[seq(yy[i],i=0..k)],t,3);
ax:=coeff(K,y,2);
ay:=coeff(K,x,2);
cx:=coeff(K,y,0);
cy:=coeff(K,x,0);
phi:=proc(paire) local u,v:
u:=paire[1]: v:=paire[2]:
[factor(1/u*subs(y=v,cy)/subs(y=v,ay)),v]: end:
psi:=proc(paire) local u,v:
u:=paire[1]: v:=paire[2]:
[u,factor(1/v*subs(x=u,cx)/subs(x=u,ax))]: end:
Xser:=I+a*t+t^2*R(t);
op(2,normal(subs(csgn((4*I)*R(t)^4*t^10-(16*I)*R(t)^2*t^6-(16*I)*a^2*t^4+(4*I)*a^3*t^5-(8*I)*a*t^3+(2*I)*t^3*R(t)-(8*I)*t^4*R(t)+(12*I)*R(t)*a^2*t^6+(12*I)*R(t)^2*a*t^7-(32*I)*R(t)*a*t^5+(16*I)*R(t)*a^3*t^7+(24*I)*R(t)^2*a^2*t^8+(16*I)*R(t)^3*a*t^9+(4*I)*R(t)^3*t^8+(2*I)*a*t^2-I*t^2+(4*I)*a^4*t^6-I*R(t)^2*t^4-I*a^2*t^2-48*R(t)*a^2*t^6-24*R(t)*a*t^5-48*R(t)^2*a*t^7-(2*I)*R(t)*a*t^3-12*R(t)^2*t^6-12*a^2*t^4-16*a^3*t^5-16*R(t)^3*t^8+2*t^2*R(t)+2*a*t-2*t+I)=1,map(series,[solve(subs(x=Xser,K),y)],t,5))));;
Yser:=%;
[u,v]: phi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,2));
[u,v]: phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,2));
[u,v]: phi(%):psi(%):phi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,2));
[u,v]: phi(%):psi(%):phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,2));
Xser; Yser;
<Text-field style="Heading 1" layout="Heading 1">Case 2 : a(x)=t(1+x+x^2) : 12 more models (1 is singular)</Text-field>
In all these examples, the key point is to prove that the x-orbit of j=exp(2i pi/3) (and the x-orbit of its conjugate) are infinite.
<Text-field style="Heading 2" layout="Heading 2">NE,N,NW,S,E (period 7)</Text-field>
liste:=x*y+y+y/x+1/y+x;
K:=expand(t*x*y*liste-x*y);
Xsol:=factor([solve(K,x)]);
Ysol:=factor([solve(K,y)]);
dx:=factor(discrim(K,y));
dy:=factor(discrim(K,x));
with(gfun): map(allvalues,algeqtoseries(dx,t,x,3));
map(allvalues,algeqtoseries(dy,t,y,3));
with(plots):
implicitplot(subs(t=0.2,K),x=-8..8,y=-8..8,numpoints=10000);
L'orbite de x=J
algJ:=1+x+x^2;
solve(algJ,x);
J:=-1/2+(1/2*I)*sqrt(3);
L'orbite de x=J
k:=0:xx[0]:= J: factor(subs(x=xx[0],K));
yy[0]:=solve(%,y);
factor(expand(factor(subs(y=yy[0],K))));
Version en boucle
for i from 1 to 10 do
k:=k+1: factor(expand(factor([solve(%%,x)]))):xx[k]:=%[2]:numer(factor(subs(x=xx[k],K))):factor(expand(factor([solve(%,y)]))):yy[k]:=%[2]: numer(factor(subs(y=%,K))):od:
k;
period 7 ?
map(factor,map(rationalize,map(normal,map(series,[seq(xx[i],i=0..k)],t,4))));
map(factor,map(rationalize,map(normal,map(series,[seq(yy[i],i=0..k)],t,4))));
map(factor,map(rationalize,map(normal,map(series,[seq(xx[7*i],i=0..2)],t,4))));
ax:=coeff(K,y,2);
ay:=coeff(K,x,2);
cx:=coeff(K,y,0);
cy:=coeff(K,x,0);
phi:=proc(paire) local u,v:
u:=paire[1]: v:=paire[2]:
[factor(1/u*subs(y=v,cy)/subs(y=v,ay)),v]: end:
psi:=proc(paire) local u,v:
u:=paire[1]: v:=paire[2]:
[u,factor(1/v*subs(x=u,cx)/subs(x=u,ax))]: end:
Xser:=J+a*t+t^2*R(t);
op(1,normal(subs(csgn((4*I)*a^2*t^4-I*t^2-12*a^2*t^4+(4*I)*a^3*t^5+(12*I)*R(t)^2*a*t^7-2*t+(4*I)*R(t)^3*t^8-(2*I)*R(t)*a*t^3+(2*I)*a*t^2+2*a*t+(8*I)*R(t)*a*t^5-24*R(t)*a*t^5+(2*I)*t^3*R(t)-I*R(t)^2*t^4-(8*I)*a*t^3+2*t^2*R(t)-8*a*t^3-(8*I)*t^4*R(t)+(12*I)*R(t)*a^2*t^6-8*t^4*R(t)-I*a^2*t^2+I-12*R(t)^2*t^6+(4*I)*R(t)^2*t^6)=1,map(simplify,map(series,[solve(subs(x=Xser,K),y)],t,6)))));;
Yser:=series(%,t,4);
[u,v]: phi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,3));
[u,v]: phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,2));
[u,v]: phi(%):psi(%):phi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,2));
[u,v]: phi(%):psi(%):phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,3));
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,3));
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,3));
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,3));
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,3));
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,3));
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,3));
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,2));
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,2));
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,2));
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,3));
Xser,Yser;
<Text-field style="Heading 2" layout="Heading 2">NE,N,NW,W,S (period 7)</Text-field>
liste:=x*y+y+y/x+1/x+1/y;
K:=expand(t*x*y*liste-x*y);
Xsol:=factor([solve(K,x)]);
Ysol:=factor([solve(K,y)]);
dx:=factor(discrim(K,y));
dy:=factor(discrim(K,x));
with(gfun): map(allvalues,algeqtoseries(dx,t,x,3));
map(allvalues,algeqtoseries(dy,t,y,3));
with(plots):
implicitplot(subs(t=0.2,K),x=-8..8,y=-8..8,numpoints=10000);
L'orbite de x=J
algJ:=1+x+x^2;
solve(algJ,x);
J:=-1/2+(1/2*I)*sqrt(3);
L'orbite de x=J
k:=0:xx[0]:= J: factor(subs(x=xx[0],K));
yy[0]:=solve(%,y);
factor(expand(factor(subs(y=yy[0],K))));
Version en boucle
for i from 1 to 10 do
k:=k+1: factor(expand(factor([solve(%%,x)]))):xx[k]:=%[2]:numer(factor(subs(x=xx[k],K))):factor(expand(factor([solve(%,y)]))):yy[k]:=%[2]: numer(factor(subs(y=%,K))):od:
k;
period 7 ?
map(factor,map(rationalize,map(normal,map(series,[seq(xx[i],i=0..k)],t,4))));
map(factor,map(rationalize,map(normal,map(series,[seq(yy[i],i=0..k)],t,4))));
map(factor,map(rationalize,map(normal,map(series,[seq(xx[7*i],i=0..1)],t,4))));
ax:=coeff(K,y,2);
ay:=coeff(K,x,2);
cx:=coeff(K,y,0);
cy:=coeff(K,x,0);
phi:=proc(paire) local u,v:
u:=paire[1]: v:=paire[2]:
[factor(1/u*subs(y=v,cy)/subs(y=v,ay)),v]: end:
psi:=proc(paire) local u,v:
u:=paire[1]: v:=paire[2]:
[u,factor(1/v*subs(x=u,cx)/subs(x=u,ax))]: end:
Xser:=J+a*t+t^2*R(t);
op(1,normal(map(simplify,map(series,[solve(subs(x=Xser,K),y)],t,6))));;
Yser:=series(%,t,4);
[u,v]: phi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,3));
[u,v]: phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,2));
[u,v]: phi(%):psi(%):phi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,2));
[u,v]: phi(%):psi(%):phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,3));
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,2));
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,3));
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,3));
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,3));
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,3));
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,3));
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,2));
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,2));
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,2));
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,3));
Xser,Yser;
<Text-field style="Heading 2" layout="Heading 2">NE,N,NW,SW,E (period 3)</Text-field>
liste:=x*y+y+y/x+1/x/y+x;
K:=expand(t*x*y*liste-x*y);
Xsol:=factor([solve(K,x)]);
Ysol:=factor([solve(K,y)]);
dx:=factor(discrim(K,y));
dy:=factor(discrim(K,x));
with(gfun): map(allvalues,algeqtoseries(dx,t,x,3));
map(allvalues,algeqtoseries(dy,t,y,3));
with(plots):
implicitplot(subs(t=0.2,K),x=-8..8,y=-8..8,numpoints=10000);
L'orbite de x=J
algJ:=1+x+x^2;
solve(algJ,x);
J:=-1/2+(1/2*I)*sqrt(3);
L'orbite de x=J
k:=0:xx[0]:= J: factor(subs(x=xx[0],K));
yy[0]:=solve(%,y);
factor(expand(factor(subs(y=yy[0],K))));
Version en boucle
for i from 1 to 10 do
k:=k+1: factor(expand(factor([solve(%%,x)]))):xx[k]:=%[2]:numer(factor(subs(x=xx[k],K))):factor(expand(factor([solve(%,y)]))):yy[k]:=%[2]: numer(factor(subs(y=%,K))):od:
k;
period 3 ?
map(factor,map(rationalize,map(normal,map(series,[seq(xx[i],i=0..k)],t,4))));
map(factor,map(rationalize,map(normal,map(series,[seq(yy[i],i=0..k)],t,4))));
map(factor,map(rationalize,map(normal,map(series,[seq(xx[3*i],i=0..4)],t,2))));
ax:=coeff(K,y,2);
ay:=coeff(K,x,2);
cx:=coeff(K,y,0);
cy:=coeff(K,x,0);
phi:=proc(paire) local u,v:
u:=paire[1]: v:=paire[2]:
[factor(1/u*subs(y=v,cy)/subs(y=v,ay)),v]: end:
psi:=proc(paire) local u,v:
u:=paire[1]: v:=paire[2]:
[u,factor(1/v*subs(x=u,cx)/subs(x=u,ax))]: end:
Xser:=J+a*t+t^2*R(t);
op(1,normal(map(simplify,map(series,[solve(subs(x=Xser,K),y)],t,6))));;
Yser:=series(%,t,3);
[u,v]: phi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,3));
[u,v]: phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,2));
[u,v]: phi(%):psi(%):phi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,2));
[u,v]: phi(%):psi(%):phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,2));
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,2));
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,2));
Xser,Yser;
<Text-field style="Heading 2" layout="Heading 2">NE,N,NW,SW,S (period 3)</Text-field>
liste:=x*y+y+y/x+1/x/y+1/y;
K:=expand(t*x*y*liste-x*y);
Xsol:=factor([solve(K,x)]);
Ysol:=factor([solve(K,y)]);
dx:=factor(discrim(K,y));
dy:=factor(discrim(K,x));
with(gfun): map(allvalues,algeqtoseries(dx,t,x,3));
map(allvalues,algeqtoseries(dy,t,y,3));
with(plots):
implicitplot(subs(t=0.2,K),x=-8..8,y=-8..8,numpoints=10000);
L'orbite de x=J
algJ:=1+x+x^2;
solve(algJ,x);
J:=-1/2+(1/2*I)*sqrt(3);
L'orbite de x=J
k:=0:xx[0]:= J: factor(subs(x=xx[0],K));
yy[0]:=solve(%,y);
factor(expand(factor(subs(y=yy[0],K))));
Version en boucle
for i from 1 to 10 do
k:=k+1: factor(expand(factor([solve(%%,x)]))):xx[k]:=%[2]:numer(factor(subs(x=xx[k],K))):factor(expand(factor([solve(%,y)]))):yy[k]:=%[2]: numer(factor(subs(y=%,K))):od:
k;
period 3 ?
map(factor,map(rationalize,map(normal,map(series,[seq(xx[i],i=0..k)],t,4))));
map(factor,map(rationalize,map(normal,map(series,[seq(yy[i],i=0..k)],t,4))));
map(factor,map(rationalize,map(normal,map(series,[seq(xx[3*i],i=0..4)],t,2))));
ax:=coeff(K,y,2);
ay:=coeff(K,x,2);
cx:=coeff(K,y,0);
cy:=coeff(K,x,0);
phi:=proc(paire) local u,v:
u:=paire[1]: v:=paire[2]:
[factor(1/u*subs(y=v,cy)/subs(y=v,ay)),v]: end:
psi:=proc(paire) local u,v:
u:=paire[1]: v:=paire[2]:
[u,factor(1/v*subs(x=u,cx)/subs(x=u,ax))]: end:
Xser:=J+a*t^2+t^4*R(t);
op(1,normal(map(simplify,map(series,[solve(subs(x=Xser,K),y)],t,9))));;
Yser:=series(%,t,5);
[u,v]: phi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,3));
[u,v]: phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,2));
[u,v]: phi(%):psi(%):phi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,2));
[u,v]: phi(%):psi(%):phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,3));
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,3));
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,3));
Xser,Yser;
<Text-field style="Heading 2" layout="Heading 2">NE,N,NW,SE,E (singular)</Text-field>
liste:=x*y+y+y/x+x/y+x;
K:=expand(t*x*y*liste-x*y);
Xsol:=factor([solve(K,x)]);
Ysol:=factor([solve(K,y)]);
dx:=factor(discrim(K,y));
dy:=factor(discrim(K,x));
with(gfun): map(allvalues,algeqtoseries(dx,t,x,3));
map(allvalues,algeqtoseries(dy,t,y,3));
with(plots):
implicitplot(subs(t=0.2,K),x=-8..8,y=-8..8,numpoints=10000);
L'orbite de x=J
algJ:=1+x+x^2;
solve(algJ,x);
J:=-1/2+(1/2*I)*sqrt(3);
L'orbite de x=J
k:=0:xx[0]:= J: factor(subs(x=xx[0],K));
yy[0]:=solve(%,y);
factor(expand(factor(subs(y=yy[0],K))));
Version en boucle
for i from 1 to 5 do
k:=k+1: factor(expand(factor([solve(%%,x)]))):xx[k]:=%[2]:numer(factor(subs(x=xx[k],K))):factor(expand(factor([solve(%,y)]))):yy[k]:=%[2]: numer(factor(subs(y=%,K))):od:
k;
seq(numer(xx[k]),k=0..5);
seq(numer(yy[k]),k=0..5);
<Text-field style="Heading 2" layout="Heading 2">NE,N,NW,W,SE,E (period 3)</Text-field>
liste:=x*y+y+y/x+1/x+x/y+x;
K:=expand(t*x*y*liste-x*y);
Xsol:=factor([solve(K,x)]);
Ysol:=factor([solve(K,y)]);
dx:=factor(discrim(K,y));
dy:=factor(discrim(K,x));
with(gfun): map(allvalues,algeqtoseries(dx,t,x,3));
map(allvalues,algeqtoseries(dy,t,y,3));
with(plots):
implicitplot(subs(t=0.2,K),x=-8..8,y=-8..8,numpoints=10000);
L'orbite de x=J
algJ:=1+x+x^2;
solve(algJ,x);
J:=-1/2+(1/2*I)*sqrt(3);
L'orbite de x=J
k:=0:xx[0]:= J: factor(subs(x=xx[0],K));
yy[0]:=solve(%,y);
factor(expand(factor(subs(y=yy[0],K))));
Version en boucle
for i from 1 to 10 do
k:=k+1: factor(expand(factor([solve(%%,x)]))):xx[k]:=%[2]:numer(factor(subs(x=xx[k],K))):factor(expand(factor([solve(%,y)]))):yy[k]:=%[2]: numer(factor(subs(y=%,K))):od:
k;
period 3 ?
map(factor,map(rationalize,map(normal,map(series,[seq(xx[i],i=0..k)],t,2))));
map(factor,map(rationalize,map(normal,map(series,[seq(yy[i],i=0..k)],t,2))));
map(factor,map(rationalize,map(normal,map(series,[seq(xx[3*i],i=0..4)],t,2))));
ax:=coeff(K,y,2);
ay:=coeff(K,x,2);
cx:=coeff(K,y,0);
cy:=coeff(K,x,0);
phi:=proc(paire) local u,v:
u:=paire[1]: v:=paire[2]:
[factor(1/u*subs(y=v,cy)/subs(y=v,ay)),v]: end:
psi:=proc(paire) local u,v:
u:=paire[1]: v:=paire[2]:
[u,factor(1/v*subs(x=u,cx)/subs(x=u,ax))]: end:
Xser:=J+a*t+t^2*R(t);
op(1,normal(map(simplify,map(series,[solve(subs(x=Xser,K),y)],t,2))));;
Yser:=series(%,t,2);
[u,v]: phi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,3));
[u,v]: phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,2));
[u,v]: phi(%):psi(%):phi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,2));
[u,v]: phi(%):psi(%):phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,2));
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,3));
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,3));
Xser,Yser;
<Text-field style="Heading 2" layout="Heading 2">NE,N,NW,W,SW,S (period 3)</Text-field>
liste:=x*y+y+y/x+1/x+1/x/y+1/y;
K:=expand(t*x*y*liste-x*y);
Xsol:=factor([solve(K,x)]);
Ysol:=factor([solve(K,y)]);
dx:=factor(discrim(K,y));
dy:=factor(discrim(K,x));
with(gfun): map(allvalues,algeqtoseries(dx,t,x,3));
map(allvalues,algeqtoseries(dy,t,y,3));
with(plots):
implicitplot(subs(t=0.2,K),x=-8..8,y=-8..8,numpoints=10000);
L'orbite de x=J
algJ:=1+x+x^2;
solve(algJ,x);
J:=-1/2+(1/2*I)*sqrt(3);
L'orbite de x=J
k:=0:xx[0]:= J: factor(subs(x=xx[0],K));
yy[0]:=solve(%,y);
factor(expand(factor(subs(y=yy[0],K))));
Version en boucle
for i from 1 to 10 do
k:=k+1: factor(expand(factor([solve(%%,x)]))):xx[k]:=%[2]:numer(factor(subs(x=xx[k],K))):factor(expand(factor([solve(%,y)]))):yy[k]:=%[2]: numer(factor(subs(y=%,K))):od:
k;
period 3 ?
map(factor,map(rationalize,map(normal,map(series,[seq(xx[i],i=0..k)],t,2))));
map(factor,map(rationalize,map(normal,map(series,[seq(yy[i],i=0..k)],t,2))));
map(factor,map(rationalize,map(normal,map(series,[seq(xx[3*i],i=0..4)],t,2))));
ax:=coeff(K,y,2);
ay:=coeff(K,x,2);
cx:=coeff(K,y,0);
cy:=coeff(K,x,0);
phi:=proc(paire) local u,v:
u:=paire[1]: v:=paire[2]:
[factor(1/u*subs(y=v,cy)/subs(y=v,ay)),v]: end:
psi:=proc(paire) local u,v:
u:=paire[1]: v:=paire[2]:
[u,factor(1/v*subs(x=u,cx)/subs(x=u,ax))]: end:
Xser:=J+a*t+t^2*R(t);
op(1,normal(map(simplify,map(series,[solve(subs(x=Xser,K),y)],t,6))));;
Yser:=series(%,t,3);
[u,v]: phi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,3));
[u,v]: phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,2));
[u,v]: phi(%):psi(%):phi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,2));
[u,v]: phi(%):psi(%):phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,2));
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,2));
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,2));
Xser,Yser;
<Text-field style="Heading 2" layout="Heading 2">NE,N,NW,W,SW,E (flipped) (period 3)</Text-field>
liste:=x*y+y+y/x+1/x+1/x/y+x;
K:=expand(t*x*y*liste-x*y);
Xsol:=factor([solve(K,x)]);
Ysol:=factor([solve(K,y)]);
dx:=factor(discrim(K,y));
dy:=factor(discrim(K,x));
with(gfun): map(allvalues,algeqtoseries(dx,t,x,3));
map(allvalues,algeqtoseries(dy,t,y,3));
with(plots):
implicitplot(subs(t=0.2,K),x=-8..8,y=-8..8,numpoints=10000);
L'orbite de x=J
algJ:=1+x+x^2;
solve(algJ,x);
J:=-1/2+(1/2*I)*sqrt(3);
L'orbite de x=J
k:=0:xx[0]:= J: factor(subs(x=xx[0],K));
yy[0]:=solve(%,y);
factor(expand(factor(subs(y=yy[0],K))));
Version en boucle
for i from 1 to 10 do
k:=k+1: factor(expand(factor([solve(%%,x)]))):xx[k]:=%[2]:numer(factor(subs(x=xx[k],K))):factor(expand(factor([solve(%,y)]))):yy[k]:=%[2]: numer(factor(subs(y=%,K))):od:
k;
period 3 ?
map(factor,map(rationalize,map(normal,map(series,[seq(xx[i],i=0..k)],t,2))));
map(factor,map(rationalize,map(normal,map(series,[seq(yy[i],i=0..k)],t,2))));
map(factor,map(rationalize,map(normal,map(series,[seq(xx[3*i],i=0..4)],t,2))));
ax:=coeff(K,y,2);
ay:=coeff(K,x,2);
cx:=coeff(K,y,0);
cy:=coeff(K,x,0);
phi:=proc(paire) local u,v:
u:=paire[1]: v:=paire[2]:
[factor(1/u*subs(y=v,cy)/subs(y=v,ay)),v]: end:
psi:=proc(paire) local u,v:
u:=paire[1]: v:=paire[2]:
[u,factor(1/v*subs(x=u,cx)/subs(x=u,ax))]: end:
Xser:=J+a*t+t^2*R(t);
op(1,normal(map(simplify,map(series,[solve(subs(x=Xser,K),y)],t,6))));;
Yser:=series(%,t,3);
[u,v]: phi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,3));
[u,v]: phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,2));
[u,v]: phi(%):psi(%):phi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,2));
[u,v]: phi(%):psi(%):phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,2));
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,2));
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,2));
Xser,Yser;
<Text-field style="Heading 2" layout="Heading 2">NE,N,NW,SW,S,E (flipped) (period 5)</Text-field>
liste:=x*y+y+y/x+1/x/y+1/y+x;
K:=expand(t*x*y*liste-x*y);
Xsol:=factor([solve(K,x)]);
Ysol:=factor([solve(K,y)]);
dx:=factor(discrim(K,y));
dy:=factor(discrim(K,x));
with(gfun): map(allvalues,algeqtoseries(dx,t,x,3));
map(allvalues,algeqtoseries(dy,t,y,3));
with(plots):
implicitplot(subs(t=0.2,K),x=-8..8,y=-8..8,numpoints=10000);
L'orbite de x=J
algJ:=1+x+x^2;
solve(algJ,x);
J:=-1/2+(1/2*I)*sqrt(3);
L'orbite de x=J
k:=0:xx[0]:= J: factor(subs(x=xx[0],K));
yy[0]:=solve(%,y);
factor(expand(factor(subs(y=yy[0],K))));
Version en boucle
for i from 1 to 10 do
k:=k+1: factor(expand(factor([solve(%%,x)]))):xx[k]:=%[2]:numer(factor(subs(x=xx[k],K))):factor(expand(factor([solve(%,y)]))):yy[k]:=%[2]: numer(factor(subs(y=%,K))):od:
k;
period 5 ?
map(factor,map(rationalize,map(normal,map(series,[seq(xx[i],i=0..k)],t,2))));
map(factor,map(rationalize,map(normal,map(series,[seq(yy[i],i=0..k)],t,2))));
map(factor,map(rationalize,map(normal,map(series,[seq(xx[5*i],i=0..2)],t,2))));
ax:=coeff(K,y,2);
ay:=coeff(K,x,2);
cx:=coeff(K,y,0);
cy:=coeff(K,x,0);
phi:=proc(paire) local u,v:
u:=paire[1]: v:=paire[2]:
[factor(1/u*subs(y=v,cy)/subs(y=v,ay)),v]: end:
psi:=proc(paire) local u,v:
u:=paire[1]: v:=paire[2]:
[u,factor(1/v*subs(x=u,cx)/subs(x=u,ax))]: end:
Xser:=J+a*t+t^2*R(t);
op(1,normal(map(simplify,map(series,[solve(subs(x=Xser,K),y)],t,6))));;
Yser:=series(%,t,3);
[u,v]: phi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,3));
[u,v]: phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,2));
[u,v]: phi(%):psi(%):phi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,2));
[u,v]: phi(%):psi(%):phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,2));
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,2));
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,2));
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,2));
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,2));
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,2));
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,2));
Xser,Yser;
<Text-field style="Heading 2" layout="Heading 2">NE,N,NW,SW,SE,E (period 2)</Text-field>
liste:=x*y+y+y/x+1/x/y+x/y+x;
K:=expand(t*x*y*liste-x*y);
Xsol:=factor([solve(K,x)]);
Ysol:=factor([solve(K,y)]);
dx:=factor(discrim(K,y));
dy:=factor(discrim(K,x));
with(gfun): map(allvalues,algeqtoseries(dx,t,x,3));
map(allvalues,algeqtoseries(dy,t,y,3));
with(plots):
implicitplot(subs(t=0.2,K),x=-8..8,y=-8..8,numpoints=10000);
L'orbite de x=J
algJ:=1+x+x^2;
solve(algJ,x);
J:=-1/2+(1/2*I)*sqrt(3);
L'orbite de x=J
k:=0:xx[0]:= J: factor(subs(x=xx[0],K));
yy[0]:=solve(%,y);
factor(expand(factor(subs(y=yy[0],K))));
Version en boucle
for i from 1 to 8 do
k:=k+1: factor(expand(factor([solve(%%,x)]))):xx[k]:=%[2]:numer(factor(subs(x=xx[k],K))):factor(expand(factor([solve(%,y)]))):yy[k]:=%[2]: numer(factor(subs(y=%,K))):od:
k;
period 2 ?
map(factor,map(rationalize,map(normal,map(series,[seq(xx[i],i=0..k)],t,2))));
map(factor,map(rationalize,map(normal,map(series,[seq(yy[i],i=0..k)],t,2))));
map(factor,map(rationalize,map(normal,map(series,[seq(xx[2*i],i=0..4)],t,2))));
ax:=coeff(K,y,2);
ay:=coeff(K,x,2);
cx:=coeff(K,y,0);
cy:=coeff(K,x,0);
phi:=proc(paire) local u,v:
u:=paire[1]: v:=paire[2]:
[factor(1/u*subs(y=v,cy)/subs(y=v,ay)),v]: end:
psi:=proc(paire) local u,v:
u:=paire[1]: v:=paire[2]:
[u,factor(1/v*subs(x=u,cx)/subs(x=u,ax))]: end:
Xser:=J+a*t+t^2*R(t);
op(1,normal(map(simplify,map(series,[solve(subs(x=Xser,K),y)],t,6))));;
Yser:=series(%,t,3);
[u,v]: phi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,2));
[u,v]: phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,2));
[u,v]: phi(%):psi(%):phi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,2));
[u,v]: phi(%):psi(%):phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,2));
Xser,Yser;
<Text-field style="Heading 2" layout="Heading 2">NE,N,NW,W,S,SE,E (period 5)</Text-field>
liste:=x*y+y+y/x+1/x+1/y+x/y+x;
K:=expand(t*x*y*liste-x*y);
Xsol:=factor([solve(K,x)]);
Ysol:=factor([solve(K,y)]);
dx:=factor(discrim(K,y));
dy:=factor(discrim(K,x));
with(gfun): map(allvalues,algeqtoseries(dx,t,x,3));
map(allvalues,algeqtoseries(dy,t,y,3));
with(plots):
implicitplot(subs(t=0.2,K),x=-8..8,y=-8..8,numpoints=10000);
L'orbite de x=J
algJ:=1+x+x^2;
solve(algJ,x);
J:=-1/2+(1/2*I)*sqrt(3);
L'orbite de x=J
k:=0:xx[0]:= J: factor(subs(x=xx[0],K));
yy[0]:=solve(%,y);
factor(expand(factor(subs(y=yy[0],K))));
Version en boucle
for i from 1 to 10 do
k:=k+1: factor(expand(factor([solve(%%,x)]))):xx[k]:=%[2]:numer(factor(subs(x=xx[k],K))):factor(expand(factor([solve(%,y)]))):yy[k]:=%[2]: numer(factor(subs(y=%,K))):od:
k;
period 5 ?
map(factor,map(rationalize,map(normal,map(series,[seq(xx[i],i=0..k)],t,2))));
map(factor,map(rationalize,map(normal,map(series,[seq(yy[i],i=0..k)],t,2))));
map(factor,map(rationalize,map(normal,map(series,[seq(xx[5*i],i=0..3)],t,3))));
ax:=coeff(K,y,2);
ay:=coeff(K,x,2);
cx:=coeff(K,y,0);
cy:=coeff(K,x,0);
phi:=proc(paire) local u,v:
u:=paire[1]: v:=paire[2]:
[factor(1/u*subs(y=v,cy)/subs(y=v,ay)),v]: end:
psi:=proc(paire) local u,v:
u:=paire[1]: v:=paire[2]:
[u,factor(1/v*subs(x=u,cx)/subs(x=u,ax))]: end:
Xser:=J+a*t^2+t^3*R(t);
map(simplify,op(1,normal(map(simplify,map(series,[solve(subs(x=Xser,K),y)],t,6)))));;
Yser:=series(%,t,3);
[u,v]: phi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,3));
[u,v]: phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,2));
[u,v]: phi(%):psi(%):phi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,3));
[u,v]: phi(%):psi(%):phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,3));
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,3));
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,3));
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,3));
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,3));
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,3));
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,3));
Xser,Yser;
<Text-field style="Heading 2" layout="Heading 2">NE,N,NW,W,SW,S,E (period 5)</Text-field>
liste:=x*y+y+y/x+1/x+1/x/y+1/y+x;
K:=expand(t*x*y*liste-x*y);
Xsol:=factor([solve(K,x)]);
Ysol:=factor([solve(K,y)]);
dx:=factor(discrim(K,y));
dy:=factor(discrim(K,x));
with(gfun): map(allvalues,algeqtoseries(dx,t,x,3));
map(allvalues,algeqtoseries(dy,t,y,3));
with(plots):
implicitplot(subs(t=0.2,K),x=-8..8,y=-8..8,numpoints=10000);
L'orbite de x=J
algJ:=1+x+x^2;
solve(algJ,x);
J:=-1/2+(1/2*I)*sqrt(3);
L'orbite de x=J
k:=0:xx[0]:= J: factor(subs(x=xx[0],K));
yy[0]:=solve(%,y);
factor(expand(factor(subs(y=yy[0],K))));
Version en boucle
for i from 1 to 10 do
k:=k+1: factor(expand(factor([solve(%%,x)]))):xx[k]:=%[2]:numer(factor(subs(x=xx[k],K))):factor(expand(factor([solve(%,y)]))):yy[k]:=%[2]: numer(factor(subs(y=%,K))):od:
k;
period 5 ?
map(factor,map(rationalize,map(normal,map(series,[seq(xx[i],i=0..k)],t,2))));
map(factor,map(rationalize,map(normal,map(series,[seq(yy[i],i=0..k)],t,2))));
map(factor,map(rationalize,map(normal,map(series,[seq(xx[5*i],i=0..3)],t,3))));
ax:=coeff(K,y,2);
ay:=coeff(K,x,2);
cx:=coeff(K,y,0);
cy:=coeff(K,x,0);
phi:=proc(paire) local u,v:
u:=paire[1]: v:=paire[2]:
[factor(1/u*subs(y=v,cy)/subs(y=v,ay)),v]: end:
psi:=proc(paire) local u,v:
u:=paire[1]: v:=paire[2]:
[u,factor(1/v*subs(x=u,cx)/subs(x=u,ax))]: end:
Xser:=J+a*t^2+t^3*R(t);
map(simplify,op(1,normal(map(simplify,map(series,[solve(subs(x=Xser,K),y)],t,7)))));;
Yser:=series(%,t,4);
[u,v]: phi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,3));
[u,v]: phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,2));
[u,v]: phi(%):psi(%):phi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,2));
[u,v]: phi(%):psi(%):phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,3));
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,3));
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,3));
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,3));
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,3));
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,3));
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,3));
Xser,Yser;
<Text-field style="Heading 1" layout="Heading 1">Case 3 : a(x)=t(1+x) (12 more models, among which 2 are singular; two variants of the same method, 3.1 and 3.2)</Text-field>
Here, the x-orbit of -1 is infiniteIn 5 cases, one also has to look at the y-orbit of y=-1 (or y=0) to conclude.
<Text-field style="Heading 2" layout="Heading 2">N, NW, SE (singular) (method 1)</Text-field>
liste:=y+y/x+x/y;
K:=expand(t*x*y*liste-x*y);
degree(coeff(K,y,1),x);
Xsol:=factor([solve(K,x)]);
Ysol:=factor([solve(K,y)]);
dx:=factor(discrim(K,y));
dy:=factor(discrim(K,x));
with(gfun): map(allvalues,algeqtoseries(dx,t,x,3));
map(allvalues,algeqtoseries(dy,t,y,3));
with(plots):
implicitplot(subs(t=0.2,K),x=-18..12,y=-18..18,numpoints=10000);
L'orbite de x=-1
k:=0:xx[0]:= -1: factor(subs(x=xx[0],K));
yy[0]:=solve(%,y);
factor(expand(factor(subs(y=yy[0],K))));
Version en boucle
for i from 1 to 5 do
k:=k+1: factor(expand(factor([solve(%%,x)]))):xx[k]:=%[2]:numer(factor(subs(x=xx[k],K))):factor(expand(factor([solve(%,y)]))):yy[k]:=%[2]: numer(factor(subs(y=%,K))):od:
k;
seq(numer(xx[k]),k=0..5);
seq(numer(yy[k]),k=0..5);
En x=-infini, the two branches diverge like sqrt(-x)
map(allvalues,algeqtoseries(numer(subs(x=-1/epsilon,K)),epsilon,y,3));;
<Text-field style="Heading 2" layout="Heading 2">N, NW, SW, SE (1 singular pair) (period 3) (method 1)</Text-field>
liste:=y+y/x+1/x/y+x/y;
K:=expand(t*x*y*liste-x*y);
degree(coeff(K,y,1),x);
uYpairs(liste);
Xvpairs(liste);
Xsol:=factor([solve(K,x)]);
Ysol:=factor([solve(K,y)]);
dx:=factor(discrim(K,y));
dy:=factor(discrim(K,x));
with(gfun): map(allvalues,algeqtoseries(dx,t,x,3));
map(allvalues,algeqtoseries(dy,t,y,3));
with(plots):
implicitplot(subs(t=0.2,K),x=-18..12,y=-18..18,numpoints=10000);
implicitplot(subs(t=0.2,K),x=-5..1,y=-3..10,numpoints=10000);
L'orbite de x=-1
k:=0:xx[0]:= -1: factor(subs(x=xx[0],K));
yy[0]:=solve(%,y);
factor(expand(factor(subs(y=yy[0],K))));
Version en boucle
for i from 1 to 10 do
k:=k+1: factor(expand(factor([solve(%%,x)]))):xx[k]:=%[2]:numer(factor(subs(x=xx[k],K))):factor(expand(factor([solve(%,y)]))):yy[k]:=%[2]: numer(factor(subs(y=%,K))):od:
k;
period 3 ?
map(factor,map(rationalize,map(normal,map(series,[seq(xx[i],i=0..k)],t,3))));
map(factor,map(rationalize,map(normal,map(series,[seq(yy[i],i=0..k)],t,4))));
map(factor,map(rationalize,map(normal,map(series,[seq(xx[3*i],i=0..4)],t,4))));
ax:=coeff(K,y,2);
ay:=coeff(K,x,2);
cx:=coeff(K,y,0);
cy:=coeff(K,x,0);
phi:=proc(paire) local u,v:
u:=paire[1]: v:=paire[2]:
[factor(1/u*subs(y=v,cy)/subs(y=v,ay)),v]: end:
psi:=proc(paire) local u,v:
u:=paire[1]: v:=paire[2]:
[u,factor(1/v*subs(x=u,cx)/subs(x=u,ax))]: end:
Xser:=-1+a*t^2+t^4*R(t);
map(simplify,op(1,normal(map(simplify,map(series,[solve(subs(x=Xser,K),y)],t,12)))));;
Yser:=series(%,t,7);
[u,v]: phi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,3));
[u,v]: phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,3));
[u,v]: phi(%):psi(%):phi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,3));
[u,v]: phi(%):psi(%):phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,3));
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,3));
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,4));
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,3));
Xser,Yser;
<Text-field style="Heading 2" layout="Heading 2">N, NW, W, SE (symm 2nd diagonal, two singular pairs) (period 4) (method 1)</Text-field>
liste:=y+y/x+1/x+x/y;
K:=expand(t*x*y*liste-x*y);
degree(coeff(K,y,1),x);
uYpairs(liste);
Xvpairs(liste);
factor(subs(y=1/x,K));
Xsol:=factor([solve(K,x)]);
Ysol:=factor([solve(K,y)]);
dx:=factor(discrim(K,y));
dy:=factor(discrim(K,x));
with(gfun): map(allvalues,algeqtoseries(dx,t,x,3));
map(allvalues,algeqtoseries(dy,t,y,3));
with(plots):
implicitplot(subs(t=0.2,K),x=-18..12,y=-18..18,numpoints=10000);
L'orbite de x=-1
k:=0:xx[0]:= -1: factor(subs(x=xx[0],K));
yy[0]:=solve(%,y);
factor(expand(factor(subs(y=yy[0],K))));
Version en boucle
for i from 1 to 10 do
k:=k+1: factor(expand(factor([solve(%%,x)]))):xx[k]:=%[2]:numer(factor(subs(x=xx[k],K))):factor(expand(factor([solve(%,y)]))):yy[k]:=%[2]: numer(factor(subs(y=%,K))):od:
k;
period 4 ?
map(factor,map(rationalize,map(normal,map(series,[seq(xx[i],i=0..k)],t,2))));
map(factor,map(rationalize,map(normal,map(series,[seq(yy[i],i=0..k)],t,2))));
map(factor,map(rationalize,map(normal,map(series,[seq(xx[4*i],i=0..3)],t,2))));
ax:=coeff(K,y,2);
ay:=coeff(K,x,2);
cx:=coeff(K,y,0);
cy:=coeff(K,x,0);
phi:=proc(paire) local u,v:
u:=paire[1]: v:=paire[2]:
[factor(1/u*subs(y=v,cy)/subs(y=v,ay)),v]: end:
psi:=proc(paire) local u,v:
u:=paire[1]: v:=paire[2]:
[u,factor(1/v*subs(x=u,cx)/subs(x=u,ax))]: end:
Xser:=-1+a*t+t^2*R(t);
map(simplify,op(1,normal(map(simplify,map(series,[solve(subs(x=Xser,K),y)],t,6)))));;
Yser:=series(%,t,3);
[u,v]: phi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,3));
[u,v]: phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,2));
[u,v]: phi(%):psi(%):phi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,3));
[u,v]: phi(%):psi(%):phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,3));
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,3));
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,3));
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,3));
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,3));
Xser,Yser;
Xser:=-1+a*t+O(t^2):
N:=10:[Xser,Yser];
simplify(normal(map(series,phi(%),t,N)));simplify(normal(map(series,psi(%),t,N)));
simplify(normal(map(series,phi(%),t,N)));simplify(normal(map(series,psi(%),t,N)));
simplify(normal(map(series,phi(%),t,N)));simplify(normal(map(series,psi(%),t,N)));
simplify(normal(map(series,phi(%),t,N)));simplify(normal(map(series,psi(%),t,N)));
Xser; Yser;
[u,v]: phi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,10));
[u,v]: phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,10));
[u,v]: phi(%):psi(%):phi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,10));
[u,v]: phi(%):psi(%):phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,10));
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,10));
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,10));
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,10));
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,10));
<Text-field style="Heading 2" layout="Heading 2">N, NW, W, S, SE (1 singular pair) (period 7) (method 1)</Text-field>
liste:=y+y/x+1/x+1/y+x/y;
K:=expand(t*x*y*liste-x*y);
uYpairs(liste);
Xvpairs(liste);
degree(coeff(K,y,1),x);
Xsol:=factor([solve(K,x)]);
Ysol:=factor([solve(K,y)]);
dx:=factor(discrim(K,y));
dy:=factor(discrim(K,x));
with(gfun): map(allvalues,algeqtoseries(dx,t,x,3));
map(allvalues,algeqtoseries(dy,t,y,3));
with(plots):
implicitplot(subs(t=0.2,K),x=-18..12,y=-18..18,numpoints=10000);
L'orbite de x=-1
k:=0:xx[0]:= -1: factor(subs(x=xx[0],K));
yy[0]:=solve(%,y);
factor(expand(factor(subs(y=yy[0],K))));
k:=k+1: factor(expand(factor([solve(%%,x)])));
xx[k]:=%[2];
numer(factor(subs(x=xx[k],K))):factor(expand(factor([solve(%,y)])));
yy[k]:=%[1];
numer(factor(subs(y=%,K)));
Version en boucle
for i from 1 to 15 do
k:=k+1: factor(expand(factor([solve(%%,x)]))):xx[k]:=%[2]:numer(factor(subs(x=xx[k],K))):factor(expand(factor([solve(%,y)]))):yy[k]:=%[2]: numer(factor(subs(y=%,K))):od:
k;
period 7 ?
map(factor,map(rationalize,map(normal,map(series,[seq(xx[i],i=0..k)],t,2))));
map(factor,map(rationalize,map(normal,map(series,[seq(yy[i],i=0..k)],t,2))));
map(factor,map(rationalize,map(normal,map(series,[seq(xx[7*i],i=0..2)],t,2))));
ax:=coeff(K,y,2);
ay:=coeff(K,x,2);
cx:=coeff(K,y,0);
cy:=coeff(K,x,0);
phi:=proc(paire) local u,v:
u:=paire[1]: v:=paire[2]:
[factor(1/u*subs(y=v,cy)/subs(y=v,ay)),v]: end:
psi:=proc(paire) local u,v:
u:=paire[1]: v:=paire[2]:
[u,factor(1/v*subs(x=u,cx)/subs(x=u,ax))]: end:
Xser:=-1+a*t+b*t^2+c*t^3+t^4*R(t);
map(simplify,op(1,normal(map(simplify,map(series,[solve(subs(x=Xser,K),y)],t,8)))));;
Yser:=series(%,t,5);
[u,v]: phi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,3));
[u,v]: phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,3));
[u,v]: phi(%):psi(%):phi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,3));
[u,v]: phi(%):psi(%):phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,4));
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,4));
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,3));
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,4));
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,5));
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,3));
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,3));
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,5));
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,5));
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,5));
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,8));
Xser,Yser;
At -infinity
map(allvalues,algeqtoseries(numer(subs(x=-1/epsilon,K)),epsilon,y,3));;
<Text-field style="Heading 2" layout="Heading 2">N, NW, W, SW, SE (1 singular pair) (period 3) (method 1)</Text-field>
liste:=y+y/x+1/x+1/x/y+x/y;
K:=expand(t*x*y*liste-x*y);
uYpairs(liste);
Xvpairs(liste);
degree(coeff(K,y,1),x);
Xsol:=factor([solve(K,x)]);
Ysol:=factor([solve(K,y)]);
dx:=factor(discrim(K,y));
dy:=factor(discrim(K,x));
with(gfun): map(allvalues,algeqtoseries(dx,t,x,3));
map(allvalues,algeqtoseries(dy,t,y,3));
with(plots):
implicitplot(subs(t=0.2,K),x=-18..12,y=-18..18,numpoints=10000);
L'orbite de x=-1
k:=0:xx[0]:= -1: factor(subs(x=xx[0],K));
yy[0]:=solve(%,y);
factor(expand(factor(subs(y=yy[0],K))));
Version en boucle
for i from 1 to 10 do
k:=k+1: factor(expand(factor([solve(%%,x)]))):xx[k]:=%[2]:numer(factor(subs(x=xx[k],K))):factor(expand(factor([solve(%,y)]))):yy[k]:=%[2]: numer(factor(subs(y=%,K))):od:
k;
period 3 ?
map(factor,map(rationalize,map(normal,map(series,[seq(xx[i],i=0..k)],t,2))));
map(factor,map(rationalize,map(normal,map(series,[seq(yy[i],i=0..k)],t,2))));
map(factor,map(rationalize,map(normal,map(series,[seq(xx[i*3],i=0..4)],t,3))));
ax:=coeff(K,y,2);
ay:=coeff(K,x,2);
cx:=coeff(K,y,0);
cy:=coeff(K,x,0);
phi:=proc(paire) local u,v:
u:=paire[1]: v:=paire[2]:
[factor(1/u*subs(y=v,cy)/subs(y=v,ay)),v]: end:
psi:=proc(paire) local u,v:
u:=paire[1]: v:=paire[2]:
[u,factor(1/v*subs(x=u,cx)/subs(x=u,ax))]: end:
Xser:=-1+a*t+t^2*R(t);
map(simplify,op(1,normal(map(simplify,map(series,[solve(subs(x=-Xser,K),y)],t,6)))));;
Yser:=series(%,t,4);
[u,v]: phi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,3));
[u,v]: phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,3));
[u,v]: phi(%):psi(%):phi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,2));
[u,v]: phi(%):psi(%):phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,3));
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,3));
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,3));
Xser,Yser;
At infinity
map(allvalues,algeqtoseries(numer(subs(x=-1/epsilon,K)),epsilon,y,3));;
<Text-field style="Heading 2" layout="Heading 2">N, NW, SW, S, SE (1 singular pair) (period 3) (method 1)</Text-field>
liste:=y+y/x+1/x/y+1/y+x/y;
K:=expand(t*x*y*liste-x*y);
uYpairs(liste);
Xvpairs(liste);
degree(coeff(K,y,1),x);
Xsol:=factor([solve(K,x)]);
Ysol:=factor([solve(K,y)]);
dx:=factor(discrim(K,y));
dy:=factor(discrim(K,x));
with(gfun): map(allvalues,algeqtoseries(dx,t,x,3));
map(allvalues,algeqtoseries(dy,t,y,3));
with(plots):
implicitplot(subs(t=0.2,K),x=-18..12,y=-18..18,numpoints=10000);
L'orbite de x=-1
k:=0:xx[0]:= -1: factor(subs(x=xx[0],K));
yy[0]:=solve(%,y);
factor(expand(factor(subs(y=yy[0],K))));
Version en boucle
for i from 1 to 12 do
k:=k+1: factor(expand(factor([solve(%%,x)]))):xx[k]:=%[2]:numer(factor(subs(x=xx[k],K))):factor(expand(factor([solve(%,y)]))):yy[k]:=%[2]: numer(factor(subs(y=%,K))):od:
k;
period 3 ?
map(factor,map(rationalize,map(normal,map(series,[seq(xx[i],i=0..k)],t,4))));
map(factor,map(rationalize,map(normal,map(series,[seq(yy[i],i=0..k)],t,4))));
map(factor,map(rationalize,map(normal,map(series,[seq(xx[i*3],i=0..4)],t,4))));
ax:=coeff(K,y,2);
ay:=coeff(K,x,2);
cx:=coeff(K,y,0);
cy:=coeff(K,x,0);
phi:=proc(paire) local u,v:
u:=paire[1]: v:=paire[2]:
[factor(1/u*subs(y=v,cy)/subs(y=v,ay)),v]: end:
psi:=proc(paire) local u,v:
u:=paire[1]: v:=paire[2]:
[u,factor(1/v*subs(x=u,cx)/subs(x=u,ax))]: end:
Xser:=-1+a*t^2+t^4*R(t);
map(simplify,op(1,normal(map(simplify,map(series,[solve(subs(x=Xser,K),y)],t,7)))));;
Yser:=series(%,t,4);
[u,v]: phi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,3));
[u,v]: phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,3));
[u,v]: phi(%):psi(%):phi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,3));
[u,v]: phi(%):psi(%):phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,3));
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,3));
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,3));
Xser,Yser;
At -infinity
map(allvalues,algeqtoseries(numer(subs(x=-1/epsilon,K)),epsilon,y,3));;
<Text-field style="Heading 2" layout="Heading 2">N, NW, W, SW, S, SE (1 singular pair) (period 3) (method 1)</Text-field>
liste:=y+y/x+1/x+1/x/y+1/y+x/y;
K:=expand(t*x*y*liste-x*y);
uYpairs(liste);
Xvpairs(liste);
degree(coeff(K,y,1),x);
Xsol:=factor([solve(K,x)]);
Ysol:=factor([solve(K,y)]);
dx:=factor(discrim(K,y));
dy:=factor(discrim(K,x));
with(gfun): map(allvalues,algeqtoseries(dx,t,x,3));
map(allvalues,algeqtoseries(dy,t,y,3));
with(plots):
implicitplot(subs(t=0.2,K),x=-18..12,y=-18..18,numpoints=10000);
L'orbite de x=-1
k:=0:xx[0]:= -1: factor(subs(x=xx[0],K));
yy[0]:=solve(%,y);
factor(expand(factor(subs(y=yy[0],K))));
Version en boucle
for i from 1 to 13 do
k:=k+1: factor(expand(factor([solve(%%,x)]))):xx[k]:=%[2]:numer(factor(subs(x=xx[k],K))):factor(expand(factor([solve(%,y)]))):yy[k]:=%[2]: numer(factor(subs(y=%,K))):od:
k;
period 3 ?
map(factor,map(rationalize,map(normal,map(series,[seq(xx[i],i=0..k)],t,4))));
map(factor,map(rationalize,map(normal,map(series,[seq(yy[i],i=0..k)],t,4))));
map(factor,map(rationalize,map(normal,map(series,[seq(xx[3*i],i=0..4)],t,4))));
ax:=coeff(K,y,2);
ay:=coeff(K,x,2);
cx:=coeff(K,y,0);
cy:=coeff(K,x,0);
phi:=proc(paire) local u,v:
u:=paire[1]: v:=paire[2]:
[factor(1/u*subs(y=v,cy)/subs(y=v,ay)),v]: end:
psi:=proc(paire) local u,v:
u:=paire[1]: v:=paire[2]:
[u,factor(1/v*subs(x=u,cx)/subs(x=u,ax))]: end:
Xser:=-1+a*t+t^2*R(t);
map(simplify,op(1,normal(map(simplify,map(series,[solve(subs(x=Xser,K),y)],t,7)))));;
Yser:=series(%,t,4);
[u,v]: phi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,3));
[u,v]: phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,3));
[u,v]: phi(%):psi(%):phi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,3));
[u,v]: phi(%):psi(%):phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,3));
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,3));
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,3));
Xser,Yser;
At -infinity
map(allvalues,algeqtoseries(numer(subs(x=-1/epsilon,K)),epsilon,y,3));;
<Text-field style="Heading 2" layout="Heading 2">N, NW, SE, E (symmetric and singular) (method 2)</Text-field>
liste:=y+y/x+x/y+x;
K:=expand(t*x*y*liste-x*y);
uYpairs(liste);
Xvpairs(liste);
factor(subs(y=x,K));
degree(coeff(K,y,1),x);
Xsol:=factor([solve(K,x)]);
Ysol:=factor([solve(K,y)]);
dx:=factor(discrim(K,y));
dy:=factor(discrim(K,x));
with(gfun): map(allvalues,algeqtoseries(dx,t,x,3));
map(allvalues,algeqtoseries(dy,t,y,3));
with(plots):
implicitplot(subs(t=0.2,K),x=-18..18,y=-18..18,numpoints=10000);
L'orbite de x=-1
k:=0:xx[0]:= -1: factor(subs(x=xx[0],K));
yy[0]:=solve(%,y);
factor(expand(factor(subs(y=yy[0],K))));
for i from 1 to 5 do
k:=k+1: factor(expand(factor([solve(%%,x)]))):xx[k]:=%[2]:numer(factor(subs(x=xx[k],K))):factor(expand(factor([solve(%,y)]))):yy[k]:=%[2]: numer(factor(subs(y=%,K))):od:
k;
seq(numer(xx[k]),k=0..5);
seq(numer(yy[k]),k=0..5);
By symmetry, the y-orbit of y=-1 is also infinite
En x=-infini, one of the branches is infinite
map(allvalues,algeqtoseries(numer(subs(x=-1/epsilon,K)),epsilon,y,3));;
En x=infini, one of the branches is infinite
map(allvalues,algeqtoseries(numer(subs(x=1/epsilon,K)),epsilon,y,3));;
<Text-field style="Heading 2" layout="Heading 2">N, NW, SW, SE, E (symmetric) (2 singular and symmetric pairs) (period 5) (method 2)</Text-field>
liste:=y+y/x+1/x/y+x/y+x;
K:=expand(t*x*y*liste-x*y);
degree(coeff(K,y,1),x);
uYpairs(liste);
Xvpairs(liste);
factor(subs(y=x,K));
Xsol:=factor([solve(K,x)]);
Ysol:=factor([solve(K,y)]);
dx:=factor(discrim(K,y));
dy:=factor(discrim(K,x));
with(gfun): map(allvalues,algeqtoseries(dx,t,x,3));
map(allvalues,algeqtoseries(dy,t,y,3));
with(plots):
implicitplot(subs(t=0.2,K),x=-18..12,y=-18..18,numpoints=10000);
L'orbite de x=-1
k:=0:xx[0]:= -1: factor(subs(x=xx[0],K));
yy[0]:=solve(%,y);
factor(expand(factor(subs(y=yy[0],K))));
Version en boucle
for i from 1 to 13 do
k:=k+1: factor(expand(factor([solve(%%,x)]))):xx[k]:=%[2]:numer(factor(subs(x=xx[k],K))):factor(expand(factor([solve(%,y)]))):yy[k]:=%[2]: numer(factor(subs(y=%,K))):od:
k;
period 5 ?
map(factor,map(rationalize,map(normal,map(series,[seq(xx[i],i=0..k)],t,3))));
map(factor,map(rationalize,map(normal,map(series,[seq(yy[i],i=0..k)],t,3))));
map(factor,map(rationalize,map(normal,map(series,[seq(xx[i*5],i=0..2)],t,3))));
ax:=coeff(K,y,2);
ay:=coeff(K,x,2);
cx:=coeff(K,y,0);
cy:=coeff(K,x,0);
phi:=proc(paire) local u,v:
u:=paire[1]: v:=paire[2]:
[factor(1/u*subs(y=v,cy)/subs(y=v,ay)),v]: end:
psi:=proc(paire) local u,v:
u:=paire[1]: v:=paire[2]:
[u,factor(1/v*subs(x=u,cx)/subs(x=u,ax))]: end:
Xser:=-1+a*t+(t^2)*R(t):
map(simplify,op(1,normal(map(simplify,map(series,[solve(subs(x=Xser,K),y)],t,7)))));;
Yser:=series(%,t,4);
[u,v]: phi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,3));
[u,v]: phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,3));
[u,v]: phi(%):psi(%):phi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,3));
[u,v]: phi(%):psi(%):phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,3));
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,3));
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,3));
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,3));
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,3));
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,3));
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,3));
Xser,Yser;
By symmetry, te y-orbit of y=-1 is also infinite
At -infinity
map(allvalues,algeqtoseries(numer(subs(x=-1/epsilon,K)),epsilon,y,3));;
At +infinity
map(allvalues,algeqtoseries(numer(subs(x=1/epsilon,K)),epsilon,y,3));;
<Text-field style="Heading 2" layout="Heading 2">N, NW, W, SW, SE, E (period 5, for x and y) (method 2)</Text-field>
liste:=y+y/x+1/x+1/x/y+x/y+x;
K:=expand(t*x*y*liste-x*y);
uYpairs(liste);
Xvpairs(liste);
degree(coeff(K,y,1),x);
Xsol:=factor([solve(K,x)]);
Ysol:=factor([solve(K,y)]);
dx:=factor(discrim(K,y));
dy:=factor(discrim(K,x));
with(gfun): map(allvalues,algeqtoseries(dx,t,x,3));
map(allvalues,algeqtoseries(dy,t,y,3));
with(plots):
implicitplot(subs(t=0.1,K),x=-10..20,y=-15..25,numpoints=10000);
L'orbite de x=-1
k:=0:xx[0]:= -1: factor(subs(x=xx[0],K));
yy[0]:=solve(%,y);
factor(expand(factor(subs(y=yy[0],K))));
Version en boucle
for i from 1 to 15 do
k:=k+1: factor(expand(factor([solve(%%,x)]))):xx[k]:=%[2]:numer(factor(subs(x=xx[k],K))):factor(expand(factor([solve(%,y)]))):yy[k]:=%[2]: numer(factor(subs(y=%,K))):od:
k;
period 5 ?
map(factor,map(rationalize,map(normal,map(series,[seq(xx[i],i=0..k)],t,3))));
map(factor,map(rationalize,map(normal,map(series,[seq(yy[i],i=0..k)],t,3))));
map(factor,map(rationalize,map(normal,map(series,[seq(xx[5*i],i=0..3)],t,3))));
ax:=coeff(K,y,2);
ay:=coeff(K,x,2);
cx:=coeff(K,y,0);
cy:=coeff(K,x,0);
phi:=proc(paire) local u,v:
u:=paire[1]: v:=paire[2]:
[factor(1/u*subs(y=v,cy)/subs(y=v,ay)),v]: end:
psi:=proc(paire) local u,v:
u:=paire[1]: v:=paire[2]:
[u,factor(1/v*subs(x=u,cx)/subs(x=u,ax))]: end:
(below we need a<>0 at some point; hence we start with xx[5] rather than xx[0], say
Xser:=-1+a*t+R(t)*(t^2):
map(simplify,op(1,normal(map(simplify,map(series,[solve(subs(x=Xser,K),y)],t,7)))));;
Yser:=series(%,t,4);
[u,v]: phi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,3));
[u,v]: phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,3));
[u,v]: phi(%):psi(%):phi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,3));
[u,v]: phi(%):psi(%):phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,3));
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,3));
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,3));
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,3));
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,3));
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,3));
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,3));
Xser,Yser;
We also need to prove that the y-orbit of y=-1 is infinite
coeff(K,x,2);
L'orbite de y=-1
k:=0:yy[0]:= -1: factor(subs(y=yy[0],K));
xx[0]:=solve(%,x);
factor(expand(factor(subs(x=xx[0],K))));
Version en boucle
for i from 1 to 15 do
k:=k+1: factor(expand(factor([solve(%%,y)]))):yy[k]:=%[2]:
numer(factor(subs(y=yy[k],K))):factor(expand(factor([solve(%,x)]))):xx[k]:=%[2]: numer(factor(subs(x=%,K))):od:
k;
period 5 ?
map(factor,map(rationalize,map(normal,map(series,[seq(xx[i],i=0..k)],t,3))));
map(factor,map(rationalize,map(normal,map(series,[seq(yy[i],i=0..k)],t,3))));
map(factor,map(rationalize,map(normal,map(series,[seq(yy[5*i],i=0..3)],t,3))));
Yser:=-1+a*t+R(t)*(t^2):
map(simplify,op(1,normal(map(simplify,map(series,[solve(subs(y=Yser,K),x)],t,7)))));;
Xser:=series(%,t,4);
[u,v]: psi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,3));
[u,v]: psi(%): phi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,3));
[u,v]: psi(%):phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,3));
[u,v]: psi(%):phi(%):psi(%):phi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,3));
[u,v]: psi(%):phi(%):psi(%):phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,3));
[u,v]: psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,3));
[u,v]: psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,3));
[u,v]:psi(%): phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,3));
[u,v]: psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,3));
[u,v]: psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,3));
Xser,Yser;
At -infinity, one of the branches diverges
map(allvalues,algeqtoseries(numer(subs(x=-1/epsilon,K)),epsilon,y,3));;
map(allvalues,algeqtoseries(numer(subs(x=1/epsilon,K)),epsilon,y,3));;
<Text-field style="Heading 2" layout="Heading 2">N, NW, W, SW, S, SE, E (symmetric) (2 symmetric singular pairs) (period 5) (method 2)</Text-field>
liste:=y+y/x+1/x+1/x/y+1/y+x/y+x;
K:=expand(t*x*y*liste-x*y);
uYpairs(liste);
Xvpairs(liste);
factor(subs(y=x,K));
degree(coeff(K,y,1),x);
Xsol:=factor([solve(K,x)]);
Ysol:=factor([solve(K,y)]);
dx:=factor(discrim(K,y));
dy:=factor(discrim(K,x));
with(gfun): map(allvalues,algeqtoseries(dx,t,x,3));
map(allvalues,algeqtoseries(dy,t,y,3));
with(plots):
implicitplot(subs(t=0.2,K),x=-18..12,y=-18..18,numpoints=10000);
L'orbite de x=-1
k:=0:xx[0]:= -1: factor(subs(x=xx[0],K));
yy[0]:=solve(%,y);
factor(expand(factor(subs(y=yy[0],K))));
Version en boucle
for i from 1 to 10 do
k:=k+1: factor(expand(factor([solve(%%,x)]))):xx[k]:=%[2]:numer(factor(subs(x=xx[k],K))):factor(expand(factor([solve(%,y)]))):yy[k]:=%[2]: numer(factor(subs(y=%,K))):od:
k;
period 5 ?
map(factor,map(rationalize,map(normal,map(series,[seq(xx[i],i=0..k)],t,4))));
map(factor,map(rationalize,map(normal,map(series,[seq(yy[i],i=0..k)],t,4))));
map(factor,map(rationalize,map(normal,map(series,[seq(xx[i*5],i=0..2)],t,4))));
ax:=coeff(K,y,2);
ay:=coeff(K,x,2);
cx:=coeff(K,y,0);
cy:=coeff(K,x,0);
phi:=proc(paire) local u,v:
u:=paire[1]: v:=paire[2]:
[factor(1/u*subs(y=v,cy)/subs(y=v,ay)),v]: end:
psi:=proc(paire) local u,v:
u:=paire[1]: v:=paire[2]:
[u,factor(1/v*subs(x=u,cx)/subs(x=u,ax))]: end:
Xser:=-1+a*t^2+t^3*R(t);
map(simplify,op(1,normal(map(simplify,map(series,[solve(subs(x=Xser,K),y)],t,7)))));;
Yser:=series(%,t,4);
[u,v]: phi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,3));
[u,v]: phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,3));
[u,v]: phi(%):psi(%):phi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,3));
[u,v]: phi(%):psi(%):phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,3));
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,3));
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,3));
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,3));
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,3));
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,3));
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,3));
Xser,Yser;
At infinity
map(allvalues,algeqtoseries(numer(subs(x=-1/epsilon,K)),epsilon,y,3));;
map(allvalues,algeqtoseries(numer(subs(x=1/epsilon,K)),epsilon,y,3));;
<Text-field style="Heading 2" layout="Heading 2">N, NW, W, S, E (flipped) (symm 2nd diagonal, two singular pairs) (period 7, at x=-1 and y=0) (method 2)</Text-field>
liste:=y+y/x+1/x+1/y+x;
K:=expand(t*x*y*liste-x*y);
uYpairs(liste);
Xvpairs(liste);
factor(subs(y=1/x,K));
degree(coeff(K,y,1),x);
Xsol:=factor([solve(K,x)]);
Ysol:=factor([solve(K,y)]);
dx:=factor(discrim(K,y));
dy:=factor(discrim(K,x));
with(gfun): map(allvalues,algeqtoseries(dx,t,x,3));
map(allvalues,algeqtoseries(dy,t,y,3));
with(plots):
implicitplot(subs(t=0.2,K),x=-18..12,y=-18..18,numpoints=10000);
L'orbite de x=-1
k:=0:xx[0]:= -1: factor(subs(x=xx[0],K));
yy[0]:=solve(%,y);
factor(expand(factor(subs(y=yy[0],K))));
Version en boucle
for i from 1 to 15 do
k:=k+1: factor(expand(factor([solve(%%,x)]))):xx[k]:=%[2]:numer(factor(subs(x=xx[k],K))):factor(expand(factor([solve(%,y)]))):yy[k]:=%[2]: numer(factor(subs(y=%,K))):od:
k;
period 7 ?
map(factor,map(rationalize,map(normal,map(series,[seq(xx[i],i=0..k)],t,4))));
map(factor,map(rationalize,map(normal,map(series,[seq(yy[i],i=0..k)],t,4))));
map(factor,map(rationalize,map(normal,map(series,[seq(xx[7*i],i=0..2)],t,3))));
ax:=coeff(K,y,2);
ay:=coeff(K,x,2);
cx:=coeff(K,y,0);
cy:=coeff(K,x,0);
phi:=proc(paire) local u,v:
u:=paire[1]: v:=paire[2]:
[factor(1/u*subs(y=v,cy)/subs(y=v,ay)),v]: end:
psi:=proc(paire) local u,v:
u:=paire[1]: v:=paire[2]:
[u,factor(1/v*subs(x=u,cx)/subs(x=u,ax))]: end:
Xser:=-1+a*t+b*t^2+t^3*R(t);
map(normal,op(1,normal(map(simplify,map(series,[solve(subs(x=Xser,K),y)],t,8)))));;
Yser:=series(%,t,5);
[u,v]: phi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,3));
[u,v]: phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,3));
[u,v]: phi(%):psi(%):phi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,3));
[u,v]: phi(%):psi(%):phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,3));
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,3));
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,3));
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,3));[u,v]:
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,3));
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,3));
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,3));
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,3));
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,3));
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,3));
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,3));
Xser,Yser;
We also need to prove that the y-orbit at y=0 is infinite
coeff(K,x,2);
L'orbite de y=0
k:=0:yy[0]:=0: factor(subs(y=yy[0],K));
xx[0]:=solve(%,x);
factor(expand(factor(subs(x=xx[0],K))));
k:=k+1: factor(expand(factor([solve(%%,y)])));
yy[k]:=%[1];
numer(factor(subs(y=yy[k],K))):factor(expand(factor([solve(%,x)]))):xx[k]:=%[2];
numer(factor(subs(x=%,K))):
Version en boucle
for i from 1 to 15 do
k:=k+1: factor(expand(factor([solve(%%,y)]))):yy[k]:=%[2]:
numer(factor(subs(y=yy[k],K))):factor(expand(factor([solve(%,x)]))):xx[k]:=%[2]: numer(factor(subs(x=%,K))):od:
k;
period 7 ?
map(factor,map(rationalize,map(normal,map(series,[seq(xx[i],i=0..k)],t,3))));
map(factor,map(rationalize,map(normal,map(series,[seq(yy[i],i=0..k)],t,3))));
map(factor,map(rationalize,map(normal,map(series,[seq(yy[7*i],i=0..2)],t,3))));
We start at xx[1], yy[2]
Yser:=-1+a*t+b*t^2+c*t^3+R(t)*(t^4):
map(simplify,op(2,normal(map(simplify,map(series,[solve(subs(y=Yser,K),x)],t,5)))));;
Xser:=series(%,t,5);
[u,v]: psi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,3));
[u,v]: psi(%): phi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,3));
[u,v]: psi(%):phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,3));
[u,v]: psi(%):phi(%):psi(%):phi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,3));
[u,v]: psi(%):phi(%):psi(%):phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,3));
[u,v]: psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,3));
[u,v]: psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,3));
[u,v]:psi(%): phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,3));
[u,v]: psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,2));
[u,v]: psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,2));
[u,v]: psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,4));
[u,v]: psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,3));
[u,v]: psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):
factor(map(series,subs(u=Xser,v=Yser ,%),t,4));
Xser,Yser;
At infinity
map(allvalues,algeqtoseries(numer(subs(x=-1/epsilon,K)),epsilon,y,3));;
map(allvalues,algeqtoseries(numer(subs(x=1/epsilon,K)),epsilon,y,3));;
<Text-field style="Heading 2" layout="Heading 2">A decoupled model : N, NW, S, E (the x-orbit of -1 is still infinite, but the behaviour at x=-infinity is different)</Text-field>
liste:=y+y/x+1/y+x;
K:=expand(t*x*y*liste-x*y);
Xsol:=factor([solve(K,x)]);
Ysol:=factor([solve(K,y)]);
dx:=factor(discrim(K,y));
dy:=factor(discrim(K,x));
with(gfun): map(allvalues,algeqtoseries(dx,t,x,3));
map(allvalues,algeqtoseries(dy,t,y,3));
with(plots):
implicitplot(subs(t=0.2,K),x=-18..12,y=-18..18,numpoints=10000);
L'orbite de x=-1 est infinie
factor(subs(x=-1,K));
solve(%,y);
factor(subs(y=%,K));
factor([solve(%,x)]);
numer(factor(subs(x=%[2],K)));
factor([solve(%,y)]);
numer(factor(subs(y=%[1],K))):
From this point on, one can iterate the next lines
factor([solve(%,x)]);
numer(factor(subs(x=%[2],K)));
factor([solve(%,y)]);
numer(factor(subs(y=%[2],K))):
When w-> -infinity, the behaviour is different
map(allvalues,algeqtoseries(numer(subs(x=-1/epsilon,K)),epsilon,y,3));;
<Text-field style="Heading 2" layout="Heading 2">A decoupled model : N, NW, S, SE, E (the x-orbit of -1 is now finite)</Text-field>
liste:=y+y/x+1/y+x/y+x;
K:=expand(t*x*y*liste-x*y);
uYpairs(liste);
Xvpairs(liste);
Xsol:=factor([solve(K,x)]);
Ysol:=factor([solve(K,y)]);
dx:=factor(discrim(K,y));
dy:=factor(discrim(K,x));
with(gfun): map(allvalues,algeqtoseries(dx,t,x,3));
map(allvalues,algeqtoseries(dy,t,y,3));
with(plots):
implicitplot(subs(t=0.2,K),x=-8..1,y=-18..18,numpoints=10000);
L'orbite de x=-1 est FINIE
factor(subs(x=-1,K));
solve(%,y);
subs(y=0,K);
solve(%,x);
subs(x=0,K);
<Text-field style="Heading 1" layout="Heading 1">Case 4 : a(x)=tx (4 more models)</Text-field>
The key ingredients are that the x-orbit of 0 is infinite, and the square root behaviour of Y(x) as x tends to - infinity (method 4.1)The symmetric case is a bit different (method 4.2)
<Text-field style="Heading 2" layout="Heading 2">N, W, S, SE (1 singular pair) (period 8)</Text-field>
liste:=y+1/x+1/y+x/y;
K:=expand(t*x*y*liste-x*y);
uYpairs(liste);
Xvpairs(liste);
Xsol:=factor([solve(K,x)]);
Ysol:=factor([solve(K,y)]);
dx:=factor(discrim(K,y));
dy:=factor(discrim(K,x));
with(gfun): map(allvalues,algeqtoseries(dx,t,x,3));
map(allvalues,algeqtoseries(dy,t,y,3));
with(plots):
implicitplot(subs(t=0.2,K),x=-8..6,y=-8..10,numpoints=10000);
L'orbite de x=0
k:=0:xx[0]:=0: factor(subs(x=xx[0],K));
yy[0]:=solve(%,y);
factor(expand(factor(subs(y=yy[0],K))));
k:=k+1: factor(expand(factor([solve(%%,x)])));
xx[k]:=%[1]:numer(factor(subs(x=xx[k],K))):factor(expand(factor([solve(%,y)])));
yy[k]:=%[2]: numer(factor(subs(y=%,K))):
k:=k+1: factor(expand(factor([solve(%%,x)])));
xx[k]:=%[2]:numer(factor(subs(x=xx[k],K))):factor(expand(factor([solve(%,y)])));
yy[k]:=%[1]: numer(factor(subs(y=%,K))):
Version en boucle
for i from 1 to 12 do
k:=k+1: factor(expand(factor([solve(%%,x)]))):xx[k]:=%[2]:numer(factor(subs(x=xx[k],K))):factor(expand(factor([solve(%,y)]))):yy[k]:=%[2]: numer(factor(subs(y=%,K))):od:
k;
period 8 ?
map(factor,map(rationalize,map(normal,map(series,[seq(xx[i],i=0..k)],t,4))));
map(factor,map(rationalize,map(normal,map(series,[seq(yy[i],i=0..k)],t,4))));
map(factor,map(rationalize,map(normal,map(series,[seq(xx[8*i+3],i=0..2)],t,4))));
ax:=coeff(K,y,2);
ay:=coeff(K,x,2);
cx:=coeff(K,y,0);
cy:=coeff(K,x,0);
phi:=proc(paire) local u,v:
u:=paire[1]: v:=paire[2]:
[factor(1/u*subs(y=v,cy)/subs(y=v,ay)),v]: end:
psi:=proc(paire) local u,v:
u:=paire[1]: v:=paire[2]:
[u,factor(1/v*subs(x=u,cx)/subs(x=u,ax))]: end:
If we start from a pair (X,Y) with X and Y =O(t^2), we will have to go very far in the expansion of X and Y, due to some idiosyncrasies in the "series" command of Maple. So instead we start from the pair (-1, 1/t+1)
d:=2: map(normal,op(2,normal(map(simplify,map(series,[solve(subs(x=-1+add(aa[i]*t^i,i=1..d)+t^(d+1)*R(t),K),y)],t,d+5))))):
Yser:=series(%,t,d+1);
Xser:=-1+add(aa[i]*t^i,i=1..d)+O(t^(d+1));
The final result
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):paire:=%:
N:=10: map(Z->factor(series(factor(series(subs(u=Xser,v=Yser ,numer(Z)),t,N))/factor(series(subs(u=Xser,v=Yser ,denom(Z)),t,N)),t,40)),paire);
Xser,Yser;
Now step by step
[u,v]: phi(%):paire:=%:
N:=5: map(Z->factor(series(factor(series(subs(u=Xser,v=Yser ,numer(Z)),t,N))/factor(series(subs(u=Xser,v=Yser ,denom(Z)),t,N)),t,5)),paire);
[u,v]: phi(%):psi(%):paire:=%:
map(Z->factor(series(factor(series(subs(u=Xser,v=Yser ,numer(Z)),t,N))/factor(series(subs(u=Xser,v=Yser ,denom(Z)),t,N)),t,5)),paire);
[u,v]: phi(%):psi(%):phi(%):paire:=%:
map(Z->factor(series(factor(series(subs(u=Xser,v=Yser ,numer(Z)),t,N))/factor(series(subs(u=Xser,v=Yser ,denom(Z)),t,N)),t,5)),paire);
[u,v]: phi(%):psi(%):phi(%):psi(%):paire:=%:
map(Z->factor(series(factor(series(subs(u=Xser,v=Yser ,numer(Z)),t,N))/factor(series(subs(u=Xser,v=Yser ,denom(Z)),t,N)),t,7)),paire);
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):paire:=%:
map(Z->factor(series(factor(series(subs(u=Xser,v=Yser ,numer(Z)),t,N))/factor(series(subs(u=Xser,v=Yser ,denom(Z)),t,N)),t,8)),paire);
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):paire:=%:
map(Z->factor(series(factor(series(subs(u=Xser,v=Yser ,numer(Z)),t,N))/factor(series(subs(u=Xser,v=Yser ,denom(Z)),t,N)),t,10)),paire);
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):paire:=%:
map(Z->factor(series(factor(series(subs(u=Xser,v=Yser ,numer(Z)),t,N))/factor(series(subs(u=Xser,v=Yser ,denom(Z)),t,N)),t,12)),paire);
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):paire:=%:
map(Z->factor(series(factor(series(subs(u=Xser,v=Yser ,numer(Z)),t,N))/factor(series(subs(u=Xser,v=Yser ,denom(Z)),t,N)),t,14)),paire);
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):paire:=%:
map(Z->factor(series(factor(series(subs(u=Xser,v=Yser ,numer(Z)),t,N))/factor(series(subs(u=Xser,v=Yser ,denom(Z)),t,N)),t,16)),paire);
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):paire:=%:
map(Z->factor(series(factor(series(subs(u=Xser,v=Yser ,numer(Z)),t,N))/factor(series(subs(u=Xser,v=Yser ,denom(Z)),t,N)),t,18)),paire);
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):paire:=%:
map(Z->factor(series(factor(series(subs(u=Xser,v=Yser ,numer(Z)),t,N))/factor(series(subs(u=Xser,v=Yser ,denom(Z)),t,N)),t,22)),paire);
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):paire:=%:
map(Z->factor(series(factor(series(subs(u=Xser,v=Yser ,numer(Z)),t,N))/factor(series(subs(u=Xser,v=Yser ,denom(Z)),t,N)),t,24)),paire);
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):paire:=%:
map(Z->factor(series(factor(series(subs(u=Xser,v=Yser ,numer(Z)),t,N))/factor(series(subs(u=Xser,v=Yser ,denom(Z)),t,N)),t,30)),paire);
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):paire:=%:
map(Z->factor(series(factor(series(subs(u=Xser,v=Yser ,numer(Z)),t,N))/factor(series(subs(u=Xser,v=Yser ,denom(Z)),t,N)),t,30)),paire);
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):paire:=%:
map(Z->factor(series(factor(series(subs(u=Xser,v=Yser ,numer(Z)),t,N))/factor(series(subs(u=Xser,v=Yser ,denom(Z)),t,N)),t,32)),paire);
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):paire:=%:
map(Z->factor(series(factor(series(subs(u=Xser,v=Yser ,numer(Z)),t,N))/factor(series(subs(u=Xser,v=Yser ,denom(Z)),t,N)),t,40)),paire);
Xser,Yser;
Autour de x = -infini : square root behaviour
with(gfun):map(allvalues,algeqtoseries(numer(subs(x=-1/epsilon,K)),epsilon,y,2));
<Text-field style="Heading 2" layout="Heading 2">N, W, SW, SE (1 singular pair) (period 7)</Text-field>
liste:=y+1/x+1/y/x+x/y;
K:=expand(t*x*y*liste-x*y);
uYpairs(liste);
Xvpairs(liste);
Xsol:=factor([solve(K,x)]);
Ysol:=factor([solve(K,y)]);
dx:=factor(discrim(K,y));
dy:=factor(discrim(K,x));
with(gfun): map(allvalues,algeqtoseries(dx,t,x,3));
map(allvalues,algeqtoseries(dy,t,y,3));
with(plots):
implicitplot(subs(t=0.2,K),x=-8..6,y=-8..10,numpoints=10000);
L'orbite de x=0
k:=0:xx[0]:=0: factor(subs(x=xx[0],K));
yy[0]:=solve(%,y);
factor(expand(factor(subs(y=yy[0],K))));
Version en boucle
for i from 1 to 16 do
k:=k+1: factor(expand(factor([solve(%%,x)]))):xx[k]:=%[2]:numer(factor(subs(x=xx[k],K))):factor(expand(factor([solve(%,y)]))):yy[k]:=%[2]: numer(factor(subs(y=%,K))):od:
k;
period 7 ?
map(factor,map(rationalize,map(normal,map(series,[seq(xx[i],i=0..k)],t,3))));
map(factor,map(rationalize,map(normal,map(series,[seq(yy[i],i=0..k)],t,3))));
map(factor,map(rationalize,map(normal,map(series,[seq(xx[7*i+2],i=0..1)],t,3))));
ax:=coeff(K,y,2);
ay:=coeff(K,x,2);
cx:=coeff(K,y,0);
cy:=coeff(K,x,0);
phi:=proc(paire) local u,v:
u:=paire[1]: v:=paire[2]:
[factor(1/u*subs(y=v,cy)/subs(y=v,ay)),v]: end:
psi:=proc(paire) local u,v:
u:=paire[1]: v:=paire[2]:
[u,factor(1/v*subs(x=u,cx)/subs(x=u,ax))]: end:
If we start from a pair (X,Y) with X and Y =O(t^2), we will have to go very far in the expansion of X and Y, due to some idiosyncrasies in the "series" command of Maple. So instead we start from the pair (-1/t-1, -1)
d:=2: map(normal,op(1,normal(map(simplify,map(series,[solve(subs(x=-1/t+add(aa[i]*t^i,i=0..d)+t^(d+1)*R(t),K),y)],t,d+5))))):
Yser:=series(%,t,d+1);
Xser:=-1/t+add(aa[i]*t^i,i=0..d)+O(t^(d+1));
The final result
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):paire:=%:
N:=10: map(Z->factor(series(factor(series(subs(u=Xser,v=Yser ,numer(Z)),t,N))/factor(series(subs(u=Xser,v=Yser ,denom(Z)),t,N)),t,40)),paire);
Xser,Yser;
Now step by step
[u,v]: phi(%):paire:=%:
N:=5: map(Z->factor(series(factor(series(subs(u=Xser,v=Yser ,numer(Z)),t,N))/factor(series(subs(u=Xser,v=Yser ,denom(Z)),t,N)),t,5)),paire);
[u,v]: phi(%):psi(%):paire:=%:
map(Z->factor(series(factor(series(subs(u=Xser,v=Yser ,numer(Z)),t,N))/factor(series(subs(u=Xser,v=Yser ,denom(Z)),t,N)),t,5)),paire);
[u,v]: phi(%):psi(%):phi(%):paire:=%:
map(Z->factor(series(factor(series(subs(u=Xser,v=Yser ,numer(Z)),t,N))/factor(series(subs(u=Xser,v=Yser ,denom(Z)),t,N)),t,5)),paire);
[u,v]: phi(%):psi(%):phi(%):psi(%):paire:=%:
map(Z->factor(series(factor(series(subs(u=Xser,v=Yser ,numer(Z)),t,N))/factor(series(subs(u=Xser,v=Yser ,denom(Z)),t,N)),t,7)),paire);
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):paire:=%:
map(Z->factor(series(factor(series(subs(u=Xser,v=Yser ,numer(Z)),t,N))/factor(series(subs(u=Xser,v=Yser ,denom(Z)),t,N)),t,8)),paire);
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):paire:=%:
map(Z->factor(series(factor(series(subs(u=Xser,v=Yser ,numer(Z)),t,N))/factor(series(subs(u=Xser,v=Yser ,denom(Z)),t,N)),t,10)),paire);
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):paire:=%:
map(Z->factor(series(factor(series(subs(u=Xser,v=Yser ,numer(Z)),t,N))/factor(series(subs(u=Xser,v=Yser ,denom(Z)),t,N)),t,12)),paire);
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):paire:=%:
map(Z->factor(series(factor(series(subs(u=Xser,v=Yser ,numer(Z)),t,N))/factor(series(subs(u=Xser,v=Yser ,denom(Z)),t,N)),t,14)),paire);
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):paire:=%:
map(Z->factor(series(factor(series(subs(u=Xser,v=Yser ,numer(Z)),t,N))/factor(series(subs(u=Xser,v=Yser ,denom(Z)),t,N)),t,16)),paire);
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):paire:=%:
map(Z->factor(series(factor(series(subs(u=Xser,v=Yser ,numer(Z)),t,N))/factor(series(subs(u=Xser,v=Yser ,denom(Z)),t,N)),t,18)),paire);
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):paire:=%:
map(Z->factor(series(factor(series(subs(u=Xser,v=Yser ,numer(Z)),t,N))/factor(series(subs(u=Xser,v=Yser ,denom(Z)),t,N)),t,22)),paire);
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):paire:=%:
map(Z->factor(series(factor(series(subs(u=Xser,v=Yser ,numer(Z)),t,N))/factor(series(subs(u=Xser,v=Yser ,denom(Z)),t,N)),t,24)),paire);
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):paire:=%:
map(Z->factor(series(factor(series(subs(u=Xser,v=Yser ,numer(Z)),t,N))/factor(series(subs(u=Xser,v=Yser ,denom(Z)),t,N)),t,30)),paire);
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):paire:=%:
map(Z->factor(series(factor(series(subs(u=Xser,v=Yser ,numer(Z)),t,N))/factor(series(subs(u=Xser,v=Yser ,denom(Z)),t,N)),t,30)),paire);
Xser,Yser;
Autour de x = -infini
map(allvalues,algeqtoseries(numer(subs(x=-1/epsilon,K)),epsilon,y,2));
<Text-field style="Heading 2" layout="Heading 2">N, W, SW, S, SE (1 singular pair) (period 7)</Text-field>
liste:=y+1/x+1/y/x+1/y+x/y;
K:=expand(t*x*y*liste-x*y);
uYpairs(liste);
Xvpairs(liste);
Xsol:=factor([solve(K,x)]);
Ysol:=factor([solve(K,y)]);
dx:=factor(discrim(K,y));
dy:=factor(discrim(K,x));
with(gfun): map(allvalues,algeqtoseries(dx,t,x,3));
map(allvalues,algeqtoseries(dy,t,y,3));
with(plots):
implicitplot(subs(t=0.2,K),x=-8..6,y=-8..10,numpoints=10000);
L'orbite de x=0
k:=0:xx[0]:=0: factor(subs(x=xx[0],K));
yy[0]:=solve(%,y);
factor(expand(factor(subs(y=yy[0],K))));
Version en boucle
for i from 1 to 20 do
k:=k+1: factor(expand(factor([solve(%%,x)]))):xx[k]:=%[2]:numer(factor(subs(x=xx[k],K))):factor(expand(factor([solve(%,y)]))):yy[k]:=%[2]: numer(factor(subs(y=%,K))):od:
k;
period 7 ?
map(factor,map(rationalize,map(normal,map(series,[seq(xx[i],i=0..k)],t,3))));
map(factor,map(rationalize,map(normal,map(series,[seq(yy[i],i=0..k)],t,3))));
map(factor,map(rationalize,map(normal,map(series,[seq(xx[i*7+3],i=0..2)],t,3))));
ax:=coeff(K,y,2);
ay:=coeff(K,x,2);
cx:=coeff(K,y,0);
cy:=coeff(K,x,0);
phi:=proc(paire) local u,v:
u:=paire[1]: v:=paire[2]:
[factor(1/u*subs(y=v,cy)/subs(y=v,ay)),v]: end:
psi:=proc(paire) local u,v:
u:=paire[1]: v:=paire[2]:
[u,factor(1/v*subs(x=u,cx)/subs(x=u,ax))]: end:
If we start from a pair (X,Y) with X and Y =O(t^2), we will have to go very far in the expansion of X and Y, due to some idiosyncrasies in the "series" command of Maple. So instead we start from the pair (-1/t-2, -1)
d:=2: map(normal,op(1,normal(map(simplify,map(series,[solve(subs(x=-1/t+add(aa[i]*t^i,i=0..d)+t^(d+1)*R(t),K),y)],t,d+5))))):
Yser:=series(%,t,d+1);
Xser:=-1/t+add(aa[i]*t^i,i=0..d)+O(t^(d+1));
The final result
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):paire:=%:
N:=10: map(Z->factor(series(factor(series(subs(u=Xser,v=Yser ,numer(Z)),t,N))/factor(series(subs(u=Xser,v=Yser ,denom(Z)),t,N)),t,40)),paire);
Xser,Yser;
Now step by step
[u,v]: phi(%):paire:=%:
N:=5: map(Z->factor(series(factor(series(subs(u=Xser,v=Yser ,numer(Z)),t,N))/factor(series(subs(u=Xser,v=Yser ,denom(Z)),t,N)),t,5)),paire);
[u,v]: phi(%):psi(%):paire:=%:
map(Z->factor(series(factor(series(subs(u=Xser,v=Yser ,numer(Z)),t,N))/factor(series(subs(u=Xser,v=Yser ,denom(Z)),t,N)),t,5)),paire);
[u,v]: phi(%):psi(%):phi(%):paire:=%:
map(Z->factor(series(factor(series(subs(u=Xser,v=Yser ,numer(Z)),t,N))/factor(series(subs(u=Xser,v=Yser ,denom(Z)),t,N)),t,5)),paire);
[u,v]: phi(%):psi(%):phi(%):psi(%):paire:=%:
map(Z->factor(series(factor(series(subs(u=Xser,v=Yser ,numer(Z)),t,N))/factor(series(subs(u=Xser,v=Yser ,denom(Z)),t,N)),t,7)),paire);
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):paire:=%:
map(Z->factor(series(factor(series(subs(u=Xser,v=Yser ,numer(Z)),t,N))/factor(series(subs(u=Xser,v=Yser ,denom(Z)),t,N)),t,8)),paire);
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):paire:=%:
map(Z->factor(series(factor(series(subs(u=Xser,v=Yser ,numer(Z)),t,N))/factor(series(subs(u=Xser,v=Yser ,denom(Z)),t,N)),t,10)),paire);
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):paire:=%:
map(Z->factor(series(factor(series(subs(u=Xser,v=Yser ,numer(Z)),t,N))/factor(series(subs(u=Xser,v=Yser ,denom(Z)),t,N)),t,12)),paire);
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):paire:=%:
map(Z->factor(series(factor(series(subs(u=Xser,v=Yser ,numer(Z)),t,N))/factor(series(subs(u=Xser,v=Yser ,denom(Z)),t,N)),t,14)),paire);
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):paire:=%:
map(Z->factor(series(factor(series(subs(u=Xser,v=Yser ,numer(Z)),t,N))/factor(series(subs(u=Xser,v=Yser ,denom(Z)),t,N)),t,16)),paire);
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):paire:=%:
map(Z->factor(series(factor(series(subs(u=Xser,v=Yser ,numer(Z)),t,N))/factor(series(subs(u=Xser,v=Yser ,denom(Z)),t,N)),t,18)),paire);
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):paire:=%:
map(Z->factor(series(factor(series(subs(u=Xser,v=Yser ,numer(Z)),t,N))/factor(series(subs(u=Xser,v=Yser ,denom(Z)),t,N)),t,22)),paire);
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):paire:=%:
map(Z->factor(series(factor(series(subs(u=Xser,v=Yser ,numer(Z)),t,N))/factor(series(subs(u=Xser,v=Yser ,denom(Z)),t,N)),t,24)),paire);
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):paire:=%:
map(Z->factor(series(factor(series(subs(u=Xser,v=Yser ,numer(Z)),t,N))/factor(series(subs(u=Xser,v=Yser ,denom(Z)),t,N)),t,30)),paire);
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):paire:=%:
map(Z->factor(series(factor(series(subs(u=Xser,v=Yser ,numer(Z)),t,N))/factor(series(subs(u=Xser,v=Yser ,denom(Z)),t,N)),t,30)),paire);
Xser,Yser;
Autour de x = -infini
map(allvalues,algeqtoseries(numer(subs(x=-1/epsilon,K)),epsilon,y,2));
<Text-field style="Heading 2" layout="Heading 2">N, W, SW, S, E (symmetric) (two symmetric singular pairs) (period 7)</Text-field>
liste:=y+1/x+1/y/x+1/y+x;
K:=expand(t*x*y*liste-x*y);
uYpairs(liste); Xvpairs(liste);
factor(subs(y=x,K));
Xsol:=factor([solve(K,x)]);
Ysol:=factor([solve(K,y)]);
dx:=factor(discrim(K,y));
dy:=factor(discrim(K,x));
with(gfun): map(allvalues,algeqtoseries(dx,t,x,3));
map(allvalues,algeqtoseries(dy,t,y,3));
with(plots):
implicitplot(subs(t=0.2,K),x=-8..6,y=-8..10,numpoints=10000);
L'orbite de x=0
k:=0:xx[0]:=0: factor(subs(x=xx[0],K));
yy[0]:=solve(%,y);
factor(expand(factor(subs(y=yy[0],K))));
Version en boucle
for i from 1 to 20 do
k:=k+1: factor(expand(factor([solve(%%,x)]))):xx[k]:=%[2]:numer(factor(subs(x=xx[k],K))):factor(expand(factor([solve(%,y)]))):yy[k]:=%[2]: numer(factor(subs(y=%,K))):od:
k;
period 7 ?
map(factor,map(rationalize,map(normal,map(series,[seq(xx[i],i=0..k)],t,3))));
map(factor,map(rationalize,map(normal,map(series,[seq(yy[i],i=0..k)],t,3))));
map(factor,map(rationalize,map(normal,map(series,[seq(xx[7*i+1],i=0..3)],t,3))));
ax:=coeff(K,y,2);
ay:=coeff(K,x,2);
cx:=coeff(K,y,0);
cy:=coeff(K,x,0);
phi:=proc(paire) local u,v:
u:=paire[1]: v:=paire[2]:
[factor(1/u*subs(y=v,cy)/subs(y=v,ay)),v]: end:
psi:=proc(paire) local u,v:
u:=paire[1]: v:=paire[2]:
[u,factor(1/v*subs(x=u,cx)/subs(x=u,ax))]: end:
If we start from a pair (X,Y) with X and Y =O(t^2), we will have to go very far in the expansion of X and Y, due to some idiosyncrasies in the "series" command of Maple. So instead we start from a pair (-1+O(t), 1/t+O(1))
d:=2: map(normal,op(2,normal(map(simplify,map(series,[solve(subs(x=-1+add(aa[i]*t^i,i=1..d)+t^(d+1)*R(t),K),y)],t,d+5))))):
Yser:=series(%,t,d+1);
Xser:=-1+add(aa[i]*t^i,i=1..d)+O(t^(d+1));
The final result
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):paire:=%:
N:=10: map(Z->factor(series(factor(series(subs(u=Xser,v=Yser ,numer(Z)),t,N))/factor(series(subs(u=Xser,v=Yser ,denom(Z)),t,N)),t,40)),paire);
Xser,Yser;
Now step by step
[u,v]: phi(%):paire:=%:
N:=5: map(Z->factor(series(factor(series(subs(u=Xser,v=Yser ,numer(Z)),t,N))/factor(series(subs(u=Xser,v=Yser ,denom(Z)),t,N)),t,5)),paire);
[u,v]: phi(%):psi(%):paire:=%:
map(Z->factor(series(factor(series(subs(u=Xser,v=Yser ,numer(Z)),t,N))/factor(series(subs(u=Xser,v=Yser ,denom(Z)),t,N)),t,5)),paire);
[u,v]: phi(%):psi(%):phi(%):paire:=%:
map(Z->factor(series(factor(series(subs(u=Xser,v=Yser ,numer(Z)),t,N))/factor(series(subs(u=Xser,v=Yser ,denom(Z)),t,N)),t,5)),paire);
[u,v]: phi(%):psi(%):phi(%):psi(%):paire:=%:
map(Z->factor(series(factor(series(subs(u=Xser,v=Yser ,numer(Z)),t,N))/factor(series(subs(u=Xser,v=Yser ,denom(Z)),t,N)),t,7)),paire);
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):paire:=%:
map(Z->factor(series(factor(series(subs(u=Xser,v=Yser ,numer(Z)),t,N))/factor(series(subs(u=Xser,v=Yser ,denom(Z)),t,N)),t,8)),paire);
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):paire:=%:
map(Z->factor(series(factor(series(subs(u=Xser,v=Yser ,numer(Z)),t,N))/factor(series(subs(u=Xser,v=Yser ,denom(Z)),t,N)),t,10)),paire);
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):paire:=%:
map(Z->factor(series(factor(series(subs(u=Xser,v=Yser ,numer(Z)),t,N))/factor(series(subs(u=Xser,v=Yser ,denom(Z)),t,N)),t,12)),paire);
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):paire:=%:
map(Z->factor(series(factor(series(subs(u=Xser,v=Yser ,numer(Z)),t,N))/factor(series(subs(u=Xser,v=Yser ,denom(Z)),t,N)),t,14)),paire);
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):paire:=%:
map(Z->factor(series(factor(series(subs(u=Xser,v=Yser ,numer(Z)),t,N))/factor(series(subs(u=Xser,v=Yser ,denom(Z)),t,N)),t,16)),paire);
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):paire:=%:
map(Z->factor(series(factor(series(subs(u=Xser,v=Yser ,numer(Z)),t,N))/factor(series(subs(u=Xser,v=Yser ,denom(Z)),t,N)),t,18)),paire);
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):paire:=%:
map(Z->factor(series(factor(series(subs(u=Xser,v=Yser ,numer(Z)),t,N))/factor(series(subs(u=Xser,v=Yser ,denom(Z)),t,N)),t,22)),paire);
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):paire:=%:
map(Z->factor(series(factor(series(subs(u=Xser,v=Yser ,numer(Z)),t,N))/factor(series(subs(u=Xser,v=Yser ,denom(Z)),t,N)),t,24)),paire);
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):paire:=%:
map(Z->factor(series(factor(series(subs(u=Xser,v=Yser ,numer(Z)),t,N))/factor(series(subs(u=Xser,v=Yser ,denom(Z)),t,N)),t,30)),paire);
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):paire:=%:
map(Z->factor(series(factor(series(subs(u=Xser,v=Yser ,numer(Z)),t,N))/factor(series(subs(u=Xser,v=Yser ,denom(Z)),t,N)),t,30)),paire);
Xser,Yser;
Autour de x = infini
map(allvalues,algeqtoseries(numer(subs(x=1/epsilon,K)),epsilon,y,2));
<Text-field style="Heading 2" layout="Heading 2">A decoupled case : N, SW, S, E. The behaviour of Y(x) around x=0 is different.</Text-field>
liste:=y+1/x/y+1/y+x;
K:=expand(t*x*y*liste-x*y);
Xsol:=factor([solve(K,x)]);
Ysol:=factor([solve(K,y)]);
dx:=factor(discrim(K,y));
dy:=factor(discrim(K,x));
with(gfun): map(allvalues,algeqtoseries(dx,t,x,3));
map(allvalues,algeqtoseries(dy,t,y,3));
with(plots):
implicitplot(subs(t=0.2,K),x=-8..6,y=-8..10,numpoints=10000);
L'orbite de x=0
map(allvalues,algeqtoseries(K,x,y,2));
factor(subs(x=0,K));
factor(subs(y=0,K));
Autour de y=+infini et x = -infini
map(allvalues,algeqtoseries(numer(subs(x=-1/epsilon,K)),epsilon,y,2));
inconclusive
<Text-field style="Heading 1" layout="Heading 1">Case 5 : a(x)=t(x+x^2) (one model, the NE fork) or a(x)=t x^2 (one model, the SW fork)</Text-field>
<Text-field style="Heading 2" layout="Heading 2">NE, N, SW, E (symmetric, 2 symmetric singular pairs) (period 4)</Text-field>
liste:=x*y+y+1/x/y+x;
K:=expand(t*x*y*liste-x*y);
uYpairs(liste); Xvpairs(liste);
liste;
factor(subs(y=x,K));
coeff(K,y,2);
subs(x=0,K);
Xsol:=factor([solve(K,x)]);
Ysol:=factor([solve(K,y)]);
map(allvalues,algeqtoseries(K,x,y,2));
dx:=factor(discrim(K,y));
dy:=factor(discrim(K,x));
with(gfun): map(allvalues,algeqtoseries(dx,t,x,3));
map(allvalues,algeqtoseries(dy,t,y,3));
with(plots):
implicitplot(subs(t=0.2,K),x=-8..6,y=-8..6,numpoints=10000);
implicitplot(subs(t=0.2,K),x=-1..0,y=-8..16,numpoints=10000);
The x-orbit of -1
k:=0:xx[0]:=-1: factor(subs(x=xx[0],K));
yy[0]:=solve(%,y);
factor(expand(factor(subs(y=yy[0],K))));
Version en boucle
for i from 1 to 10 do
k:=k+1: factor(expand(factor([solve(%%,x)]))):xx[k]:=%[2]:numer(factor(subs(x=xx[k],K))):factor(expand(factor([solve(%,y)]))):yy[k]:=%[2]: numer(factor(subs(y=%,K))):od:
k;
period 4 ?
map(factor,map(rationalize,map(normal,map(series,[seq(xx[i],i=0..k)],t,4))));
map(factor,map(rationalize,map(normal,map(series,[seq(yy[i],i=0..k)],t,4))));
map(factor,map(rationalize,map(normal,map(series,[seq(xx[i*4],i=0..2)],t,4))));
ax:=coeff(K,y,2);
ay:=coeff(K,x,2);
cx:=coeff(K,y,0);
cy:=coeff(K,x,0);
phi:=proc(paire) local u,v:
u:=paire[1]: v:=paire[2]:
[factor(1/u*subs(y=v,cy)/subs(y=v,ay)),v]: end:
psi:=proc(paire) local u,v:
u:=paire[1]: v:=paire[2]:
[u,factor(1/v*subs(x=u,cx)/subs(x=u,ax))]: end:
d:=2: map(normal,op(1,normal(map(simplify,map(series,[solve(subs(x=-1+add(aa[i]*t^i,i=1..d)+t^(d+1)*R(t),K),y)],t,d+5))))):
Yser:=series(%,t,d+1);
Xser:=-1+add(aa[i]*t^i,i=1..d)+O(t^(d+1));
The final result
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):paire:=%:
N:=10: map(Z->factor(series(factor(series(subs(u=Xser,v=Yser ,numer(Z)),t,N))/factor(series(subs(u=Xser,v=Yser ,denom(Z)),t,N)),t,40)),paire);
Xser,Yser;
Now step by step
[u,v]: phi(%):paire:=%:
N:=5: map(Z->factor(series(factor(series(subs(u=Xser,v=Yser ,numer(Z)),t,N))/factor(series(subs(u=Xser,v=Yser ,denom(Z)),t,N)),t,5)),paire);
[u,v]: phi(%):psi(%):paire:=%:
map(Z->factor(series(factor(series(subs(u=Xser,v=Yser ,numer(Z)),t,N))/factor(series(subs(u=Xser,v=Yser ,denom(Z)),t,N)),t,5)),paire);
[u,v]: phi(%):psi(%):phi(%):paire:=%:
map(Z->factor(series(factor(series(subs(u=Xser,v=Yser ,numer(Z)),t,N))/factor(series(subs(u=Xser,v=Yser ,denom(Z)),t,N)),t,5)),paire);
[u,v]: phi(%):psi(%):phi(%):psi(%):paire:=%:
map(Z->factor(series(factor(series(subs(u=Xser,v=Yser ,numer(Z)),t,N))/factor(series(subs(u=Xser,v=Yser ,denom(Z)),t,N)),t,7)),paire);
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):paire:=%:
map(Z->factor(series(factor(series(subs(u=Xser,v=Yser ,numer(Z)),t,N))/factor(series(subs(u=Xser,v=Yser ,denom(Z)),t,N)),t,8)),paire);
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):paire:=%:
map(Z->factor(series(factor(series(subs(u=Xser,v=Yser ,numer(Z)),t,N))/factor(series(subs(u=Xser,v=Yser ,denom(Z)),t,N)),t,10)),paire);
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):paire:=%:
map(Z->factor(series(factor(series(subs(u=Xser,v=Yser ,numer(Z)),t,N))/factor(series(subs(u=Xser,v=Yser ,denom(Z)),t,N)),t,12)),paire);
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):paire:=%:
map(Z->factor(series(factor(series(subs(u=Xser,v=Yser ,numer(Z)),t,N))/factor(series(subs(u=Xser,v=Yser ,denom(Z)),t,N)),t,14)),paire);
Xser,Yser;
L'orbite de x=0.. is empty
K;
factor(subs(x=0,K));
Autour de x = infini
map(allvalues,algeqtoseries(numer(subs(x=1/epsilon,K)),epsilon,y,2));
Around x=0
map(allvalues,algeqtoseries(K,x,y,2));
<Text-field style="Heading 2" layout="Heading 2">NE, W, SW, S (symmetric, 2 symmetric singular pairs) (period 4)</Text-field>
liste:=x*y+1/x/y+1/x+1/y;
K:=expand(t*x*y*liste-x*y);
uYpairs(liste); Xvpairs(liste);
factor(subs(y=x,K));
Xsol:=factor([solve(K,x)]);
Ysol:=factor([solve(K,y)]);
dx:=factor(discrim(K,y));
dy:=factor(discrim(K,x));
with(gfun): map(allvalues,algeqtoseries(dx,t,x,3));
map(allvalues,algeqtoseries(dy,t,y,3));
with(plots):
implicitplot(subs(t=0.2,K),x=-8..6,y=-8..6,numpoints=10000);
The x-orbit of 0
k:=0:xx[0]:=0: factor(subs(x=xx[0],K));
yy[0]:=solve(%,y);
factor(expand(factor(subs(y=yy[0],K))));
Version en boucle
for i from 1 to 16 do
k:=k+1: factor(expand(factor([solve(%%,x)]))):xx[k]:=%[2]:numer(factor(subs(x=xx[k],K))):factor(expand(factor([solve(%,y)]))):yy[k]:=%[2]: numer(factor(subs(y=%,K))):od:
k;
period 4 ?
map(factor,map(rationalize,map(normal,map(series,[seq(xx[i],i=0..k)],t,3))));
map(factor,map(rationalize,map(normal,map(series,[seq(yy[i],i=0..k)],t,3))));
map(factor,map(rationalize,map(normal,map(series,[seq(xx[4*i+2],i=0..3)],t,3))));
ax:=coeff(K,y,2);
ay:=coeff(K,x,2);
cx:=coeff(K,y,0);
cy:=coeff(K,x,0);
phi:=proc(paire) local u,v:
u:=paire[1]: v:=paire[2]:
[factor(1/u*subs(y=v,cy)/subs(y=v,ay)),v]: end:
psi:=proc(paire) local u,v:
u:=paire[1]: v:=paire[2]:
[u,factor(1/v*subs(x=u,cx)/subs(x=u,ax))]: end:
We start from the pair (-1/t-1, -1)
series(xx[1],t); series(xx[5],t);
d:=2: map(normal,op(2,normal(map(simplify,map(series,[solve(subs(x=-1/t+add(aa[i]*t^i,i=0..d)+t^(d+1)*R(t),K),y)],t,d+5))))):
Yser:=series(%,t,d+1);
Xser:=-1/t+add(aa[i]*t^i,i=0..d)+O(t^(d+1));
The final result
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):paire:=%:
N:=10: map(Z->factor(series(factor(series(subs(u=Xser,v=Yser ,numer(Z)),t,N))/factor(series(subs(u=Xser,v=Yser ,denom(Z)),t,N)),t,40)),paire);
Xser,Yser;
Now step by step
[u,v]: phi(%):paire:=%:
N:=5: map(Z->factor(series(factor(series(subs(u=Xser,v=Yser ,numer(Z)),t,N))/factor(series(subs(u=Xser,v=Yser ,denom(Z)),t,N)),t,5)),paire);
[u,v]: phi(%):psi(%):paire:=%:
map(Z->factor(series(factor(series(subs(u=Xser,v=Yser ,numer(Z)),t,N))/factor(series(subs(u=Xser,v=Yser ,denom(Z)),t,N)),t,5)),paire);
[u,v]: phi(%):psi(%):phi(%):paire:=%:
map(Z->factor(series(factor(series(subs(u=Xser,v=Yser ,numer(Z)),t,N))/factor(series(subs(u=Xser,v=Yser ,denom(Z)),t,N)),t,5)),paire);
[u,v]: phi(%):psi(%):phi(%):psi(%):paire:=%:
map(Z->factor(series(factor(series(subs(u=Xser,v=Yser ,numer(Z)),t,N))/factor(series(subs(u=Xser,v=Yser ,denom(Z)),t,N)),t,7)),paire);
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):paire:=%:
map(Z->factor(series(factor(series(subs(u=Xser,v=Yser ,numer(Z)),t,N))/factor(series(subs(u=Xser,v=Yser ,denom(Z)),t,N)),t,8)),paire);
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):paire:=%:
map(Z->factor(series(factor(series(subs(u=Xser,v=Yser ,numer(Z)),t,N))/factor(series(subs(u=Xser,v=Yser ,denom(Z)),t,N)),t,10)),paire);
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):paire:=%:
map(Z->factor(series(factor(series(subs(u=Xser,v=Yser ,numer(Z)),t,N))/factor(series(subs(u=Xser,v=Yser ,denom(Z)),t,N)),t,12)),paire);
[u,v]: phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):phi(%):psi(%):paire:=%:
map(Z->factor(series(factor(series(subs(u=Xser,v=Yser ,numer(Z)),t,N))/factor(series(subs(u=Xser,v=Yser ,denom(Z)),t,N)),t,14)),paire);
Xser,Yser;
Around x=0
map(allvalues,algeqtoseries(K,x,y,2));
Autour de x = -infini
map(allvalues,algeqtoseries(numer(subs(x=-1/epsilon,K)),epsilon,y,2));