Asymptotics
Recal that w=t^3. Expression of w in terms of the series S
> | wS; |
> | algSv:=collect(numer(w-wS),w,factor); |
> |
Expansion of S in terms of w
> | factor(op(2,algeqtoseries(subs(vnu,algS),w,S,5,true))); |
A better series, for the analysis, is U such that S=v(1-2U). Indeed, its coefficients are non negative
> | algUv:=collect(op(3,factor(subs(S=v*(1-2*U),algSv))),U,factor); |
> | algU:=collect(numer(factor(subs(vnu,algUv))),U,factor); |
> | op(3,algeqtoseries(algU,w,U,5,true)); |
> | factor(discrim(algU,U)); |
> | algr2:=collect(op(5,%),w,factor); |
> | algr1:=collect(op(6,%%),w,factor); |
The radius is found among the roots of these two polynomials
> | coeff(algU,U,5); |
> | with(plots): |
The general picture is similar to the case of general planar maps
> | implicitplot([algr1,algr2],nu=0..4,w=0..0.1,numpoints=50000); |
Let us zoom between 2.5 and 4
> | implicitplot([algr1,algr2],nu=2.5..4,w=-0.02..0.02,numpoints=100000); |
Where do the two curves meet?
> | factor(resultant(algr1,algr2,w)); |
We are only interested in positive values of nu
> | res1:=(67482-31038*nu+69811*nu^2-54292*nu^3+13573*nu^4); res2:= (6-14*nu+7*nu^2); |
> | {solve(res1)}: evalf(%); |
> | plot(res1,nu=0..3); |
> | plot(res2,nu=0..4); |
> | {solve(res2)};evalf(%); |
When the sign is - , the common root of algr1 and algr2 is negative and irrelevant for us
> | factor(subs(nu=1-1/7*7^(1/2),algr1));factor(subs(nu=1-1/7*7^(1/2),algr2)); |
> | factor(subs(nu=1+1/7*7^(1/2),algr1));factor(subs(nu=1+1/7*7^(1/2),algr2)); |
> | nuc:=1+1/7*7^(1/2);evalf(nuc); |
> | rhoc:=solve((-864*w-55+25*7^(1/2))); |
> |
Expansions of the roots of algr1 and algr2 before nu_c
> | R1ser:=map(factor,map(expand,map(rationalize,op(algeqtoseries(subs(nu=nuc-x,w=rhoc+x*s,algr1),x,s,3,true))))); |
> | R2ser:=map(factor,map(expand,map(rationalize,op(algeqtoseries(subs(nu=nuc-x,w=rhoc+x*s,algr2),x,s,3,true))))); |
> | factor(series(R2ser-R1ser,x)); |
> | evalf(%); |
Hence before nu_c, rho_2 dominates rho_1 (we denote by rho_i the root of P_i or algr_i)
After nu_c
> | R1ser:=map(factor,map(expand,map(rationalize,op(algeqtoseries(subs(nu=nuc+x,w=rhoc+x*s,algr1),x,s,3,true))))); |
> | R2ser:=map(factor,map(expand,map(rationalize,op(algeqtoseries(subs(nu=nuc+x,w=rhoc+x*s,algr2),x,s,3,true))))); |
> | factor(series(R2ser-R1ser,x)); |
> | evalf(%); |
The two curves have crossed and rho_2 < rho_1
Some plots of U
> | implicitplot(subs(nu=0.8,algU),w=0..0.05,U=0..0.4); |
The radius is rho_2
> | fsolve(subs(nu=0.8,algr1));fsolve(subs(nu=0.8,algr2)); |
> | implicitplot(subs(nu=1.2,algU),w=0..0.03,U=0..0.4); |
The radius is again rho_2
> | fsolve(subs(nu=1.2,algr1));fsolve(subs(nu=1.2,algr2)); |
> | implicitplot(subs(nu=nuc,algU),w=0..0.04,U=0..0.5); |
The two curves meet
> | fsolve(subs(nu=nuc,algr1));fsolve(subs(nu=nuc,algr2)); |
Beyond nu_c
> | implicitplot(subs(nu=2,algU),w=0..0.02,U=0..0.4); |
The radius is now rho_1
> | fsolve(subs(nu=2,algr1));fsolve(subs(nu=2,algr2)); |
The type of singularity of U
Before nu_c
> | subs(nu=4/5,algr2); |
A square root singularity
> | op(2,algeqtoseries(subs(nu=4/5,w=RootOf(-14148/625+7077888/625*w^2+169344/625*w)*(1-x),algU),x,U,2,true)); |
> | evalf(%); |
At the critical point, singularity in 1/3
> | op(2,algeqtoseries(subs(nu=nuc,w=rhoc*(1-x),algU),x,U,2,true)); |
Beyond nu_c, a square root again
> | subs(nu=4,algr1); |
> | op(2,algeqtoseries(subs(nu=4,w=RootOf(-40108032*w^2-27648*w+27+34359738368*w^3)*(1-x),algU),x,U,2,true)); |
> | evalf(%); |
> |
The form of Q1 in terms of S shows that it will inherit the singularities of S
Equation for Q1t=t*Q1
> | algQ1t:=collect(op(5,factor(resultant(numer(Q1t-Q1tsol),algS,S))),Q1t,factor); |
> | algQ1tnu:=collect(numer(factor(subs(vnu,algQ1t))),Q1t,factor); |
Quel est le type de singularité de Q1 ?
Before nu_c
> | subs(nu=4/5,algr2); |
singularité en 3/2
> | op(2,algeqtoseries(subs(nu=4/5,w=RootOf(-14148/625+7077888/625*w^2+169344/625*w)*(1-x),algQ1tnu),x,Q1t,3,true)); |
> | evalf(%); |
Au point critique, singularité en 4/3
> | op(2,algeqtoseries(subs(nu=nuc,w=rhoc*(1-x),algQ1tnu),x,Q1t,3,true)); |
Au dela, on retrouve une racine
> | subs(nu=4,algr1); |
> | op(2,algeqtoseries(subs(nu=4,w=RootOf(-40108032*w^2-27648*w+27+34359738368*w^3)*(1-x),algQ1tnu),x,Q1t,3,true)); |
> | evalf(%); |
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