Let $f(x) \in C^1(R^n,R^n)$ have a non-constant, $T>0$ periodic trajectory $\gamma(t)$ satisfying $\dot \gamma=f(\gamma(t))$.
a) Find a periodic solution $\gamma(t)$ to $$\dot x=x-y-x(x^2+y^2)$$ $$\dot y=x+y-y(x^2+y^2)$$
b) Determine the orbital stability of $\gamma(t)$.
I am trying to understand an alternate way (instead of converting to polar and explicitly computing the Poincare map) that my professor mentioned. He said we could use a variation principle that given $\phi(t,x_0)$ as a solution to $\dot x=f(t,x), x(t_0)=x_0$ we have $$\frac{\partial}{\partial t}(\frac{\partial x(t)}{\partial x_0})=D_2f(t,\phi(t,x_0))\frac{\partial x(t)}{\partial x_0}$$ $$\frac{\partial x(t_0)}{\partial x_0}=I$$
He also mentioned floquet theory and knowing that since we have a 2x2 system, one of the floquet multipliers is -1 and the other gives the stability of the poincare map (or something like that...) I was wondering if this rings a bell for anyone? Thanks!
Solution:
a) $\gamma(t)=(\cos(t),\sin(t))$
b)