Ce programme trace les amplitudes a telles qu'il existe r tel que l'équation x''(t)+(1-hcos(2(1+mu)t))x(t)+2mf x'(t) + mv x(t)^3=0 possède une solution périodique "bifurcant" pour m=0 à partir de x0(t)=acos((1+mu)t+r)

> restart:

> with(plots): with(linalg):

> R2:=proc(f,k);# procedure qui resout x"+k^2*x=f
g:= 1/k*int(f*sin(k*(s-t)), t=0..s);
subs(s=t,g); end proc;
 
 

> C:=proc(x,y,z);
> x0:=t-> a*cos((1+m*u)*t+r)+m*x1(t);
> E:=expand(diff(diff(x0(t),t),t)+(1-m*h*cos(2*(1+m*u)*t))*x0(t)+2*m*f*diff(x0(t),t)+m*v*x0(t)^3);
> H1:=simplify(1/m*convert(series(E,m,2), polynom)-x1(t)-diff(diff(x1(t),t),t));
> E1:=convert(series(subs(t=2*Pi/(1+m*u),R2(H1,1)),m,2)-series(subs(t=2*Pi/(1+m*u),R2(H1,1)),m,1),polynom);
> E2:=expand(convert(series(subs(t=2*Pi/(1+m*u), diff(R2(H1,1),t)),m,2)-series(subs(t=2*Pi/(1+m*u), diff(R2(H1,1),t)),m,1),polynom));
> P:=matrix(2,2,[[coeff(E1,cos(r)), coeff(E2,cos(r))],[coeff(E1,sin(r)), coeff(E2,sin(r))]]);
> S:=[solve(det(P),a)];T:=subs(h=x,f=y,v=z,S);u1:=solve(T[3],u); u2:=solve(T[5]-T[3],u);u3:=solve(diff(T[3],u));w:=subs(u=u3,T[3]);
> plot([T[5],T[3]],u=u1-.1..u2+.1, view=[u1-.1..u2+.1,0..w],numpoints=1500,thickness=2);
> end proc;
>

> C(0.65,0.1,0.16);

>

>