Homoclinic trajectories and chaotic behaviour in a piecewise linear oscillator (Q2475521)

Let \(F\) be the Poincaré map (shift at time \(\sqrt{2}\pi\)) of the differential equation \[ x''+x=\sin(\sqrt{2}t)+s(x), \] where \(s(x)\) is a continuous, piecewise-linear function given by \(\min(5x,1)\) for \(x\geq 0\) and \(\max(5x,-1)\) for \(x<0\). The authors show that \(F\) exhibits chaotic behavior (in particular, it is shown that \((0,-\sqrt{2}/6)\) is a hyperbolic fixed point having a transverse homoclinic point).