On Mixed Dynamics in the Duffing Equation with Periodic Perturbations

We study chaotic dynamics and bifurcations in reversible and time-periodic perturbations of the classical Duffing equation in the case when this equation has a homoclinic figure-8 to the saddle zero equilibrium. We consider the perturbation to be nonconservative and show that the phenomenon of mixed dynamics occurs in the corresponding Poincaré map $T$ over the period. However, for small perturbations, the dynamics is predominantly dissipative: here, almost all orbits of $T$ from the interior of the homoclinic-8 (perhaps, except for orbits from small resonant zones) tend to a stable fixed point (sink) at forward iterations and to an unstable fixed point (source) at backward iterations. We show that, when the amplitude of perturbation increases, mixed dynamics, as the phenomenon of the intersection of the attractor with the repeller, becomes quite noticeable and even prevalent. In the paper, we propose two different bifurcation scenarios that lead to such mixed dynamics. In the first scenario, it emerges as a result of the phenomenon known as the attractor-repeller collision. In the second scenario, mixed dynamics arises immediately after a symmetric pair of simple saddle-node bifurcations that, due to reversibility, occur simultaneously with the sink and the source.