Shilnikov Chaos in Oscillators with Huygens Coupling

We consider van der Pol-Duffing oscillators coupled via a two-dimensional linear system, representing Huygens coupling. We demonstrate that two identical synchronized oscillators can have chaotic movements and exhibit Shilnikov chaos. We rigorously prove the existence of a homoclinic orbit of a saddle-focus and a symmetrical heteroclinic contour of two saddle-foci, leading to the appearance of chaos via the Shilnikov saddle-focus bifurcation. We also support our analytical results with numerical simulations, revealing the main signature of Shilnikov chaos, the coexistence of chaotic and regular attractors with riddled basins of attraction, including wild attractors.