Constructing a Poincare map for dynamical system

Question

I'm trying to construct a Poincare map for the system:

$$\dot{x} = y$$ $$\dot{y} = -a^2x + b\cos(\theta)$$ $$\dot{\theta} = a$$

I have always thought of the Poincare map as more of a theoretical tool and have never had to calculate one before.

My first difficulty here is deciding a surface on which to find the intersections. My initial thought was to use $\theta = 0$ and then take snapshots of the system every time $\theta$ is a multiple of $2\pi$ but I'm having trouble finding the actual map.

Thanks for any help

Answer 1

Yes, $\theta = 0$ is a good place to start. Thus take $\theta = at$, and then solve the second-order linear DE $\ddot{x} + a^2 x = b \cos(at)$ with $x(0) = x_0$, $y(0) = \dot{x}(0) = y_0$.
Your Poincare map takes $(x_0, y_0)$ to $(x(2\pi), y(2\pi))$.