Chaotic and strange nonchaotic attractors induced by homoclinic bifurcations in a Duffing-type inverted pendulum under periodic and quasi-periodic excitations

By the rank one chaos theory on homoclinic tangles in periodically forced second order smooth systems for dissipative saddles, when the perturbed stable and unstable manifolds intersect transversally, as the magnitude \(\varepsilon >0\) $\epsilon >0$ of the perturbation varies, there exists complicated dynamical objects, such as attractive quasi-periodic torus, Newhouse sinks, Hénon-like attractors, rank one attractors with positive Sinai–Ruelle–Bowen (SRB) measures and full stochastic behavior. Stimulated by those results, in this paper we investigate the dynamics induced by homoclinic bifurcations in a Duffing-type inverted pendulum between two rigid walls under periodic and quasi-periodic excitations. By the method of numerical simulations, we show that when the perturbed stable and unstable manifolds intersect transversally, as the magnitude \(\varepsilon >0\) $\epsilon >0$ of the perturbation varies, the system also exhibits rich dynamical behaviors. For the periodic excitation case, as \(\varepsilon \rightarrow 0\) $\epsilon \to 0$ , there is an alternating series of chaotic motions and complicated periodic cascades, such as period-doubling, period-incrementing and period-adding cascades. Moreover, hysteresis effects are also observed. For the quasi-periodic case, as \(\varepsilon \rightarrow 0\) $\epsilon \to 0$ , strange nonchaotic attractors (SNAs) and chaotic attractors appear alternatively. To our knowledge, SNAs arising from homoclinic bifurcations of piecewise smooth systems have not been reported previously.