I'm trying to plot the boundaries of Arnold tongues (the regions were periodic solutions exist) for the circle map, f(x) = 2x + a + b*sin(2*pi*x)/pi mod 1. These are defined when f^n(x)=x and d/dx(f^n(x)) = 1, where f^n(x) represents iterating the function n times, i.e. f^2(x) = f(f(x)), n is the period of the periodic point.
I would like to able to take the two equations and write an equation for the boundary of the Arnold tongues in terms of b, so I will get x = g(b) and a = h(b) which satisfies the equation. I then want to plot a against b.
Analytically I can solve this in this way for the period 1 boundary by rearrange d/dx(f(x)) = 1 for x which gives x in terms of b then substituting this value into f(x) = x to give a in terms of b. I've also managed to do this in MATLAB using symbolic equations in the following way.
clear;
syms x a b
f = 2*x + a + (b/pi)*sin(2*pi*x);
g = diff(f,x);
solx = solve(g==1,x);
fnox = subs(f,x,solx);
solb(1) = solve(fnox(1)==solx(1), a);
solb(2) = solve(fnox(2)==solx(2),a);
[xval1,yval1] = fplot(matlabFunction(solb(1)),[0 1]);
[xval2,yval2] = fplot(matlabFunction(solb(2)),[0 1]);
A1 = [xval1,yval1];
A2 = [xval2,yval2];
A1 = A1(imag(A1(:,2))==0,:);
A2 = A2(imag(A2(:,2))==0,:);
figure(1)
hold on;
plot(A1(:,2),A1(:,1),'b')
plot(A2(:,2),A2(:,1),'b')
hold off;
The question is this, is there a way for me to solve for period 2 or higher boundary? I've tried the following,
f2 = subs(f,x,f)
g2 = diff(f2,x)
solx2 = solve(g2==1,x);
However I get a 'cannot find explicit solution' warning. I think perhaps the equation is too complicated for MATLAB to solve symbolically. Is there a way I can get it to work using symbolic equations? If not is there a suitable numeric method to perform the above?
Any help is much appreciated, thanks in advance.