Compute the monodromy matrix of a periodic orbit

Question

I need some help to solve this exercise:

Let (r, θ ) be a point in the phase space $R+ × S$ that obeys the system

$r' = r(1 + a cos θ − r^2), ̇ θ'= 1,$

where $|a| < 1$.

Compute the monodromy matrix M = T(2π,0) for the circle r = 0 and show that its Floquet multipliers are μ = 1 and $e^{2π}$. $S$ is the circle in $R^2$.

I have tried to solve it and I have showed that the circle $r=0$ is a periodic orbit but I don't know how to continue. There is a hint that says I should start computing the solutions of the linear system... Thank you in advance!

Answer 1

You have to explore the solution for initial value $r(0)=εu_0$, $θ(0)=εv_0$ up to first order in $ε$. This solution will have the form $r(t)=εu(t)$, $θ(t)=t+εv(t)$ with equation $$ u'(t)=u(t)(1-a\cos t),\,v'(t)=0 \implies \ln (u(t)/u_0)=t-a\sin t,\,v(t)=v_0 $$ so that $u(2\pi)=u_0e^{2\pi}$, $v(2\pi)=v_0$.