Periodic orbits of planar discontinuous system under discretization

We consider a model planar system with discontinuous right-hand side possessing an attracting periodic orbit, and we investigate what happens to a Euler discretization with stepsize $ τ$ of this system. We show that, in general, the resulting discrete dynamical system does not possess an invariant curve, in sharp contrast to what happens for smooth problems. In our context, we show that the numerical trajectories are forced to remain inside a band, whose width is proportional to the discretization stepsize $ τ$. We further show that if we consider an event-driven discretization of the model problem, whereby the solution is forced to step exactly on the discontinuity line, then there is a discrete periodic solution near the one of the original problem (for sufficiently small $ τ$). Finally, we consider what happens to the Euler discretization of the regularized system rewritten in polar coordinates, and give numerical evidence that the discrete solution now undergoes a period doubling cascade with respect to the regularization parameter $ \epsilon$, for fixed $ τ$.