It is mathematically challenging to analytically show that complex dynamical phenomena observed in simulations and experiments are truly chaotic. Extensive numerical experiments demonstrate the chaotic behavior of the harmonically perturbed hybrid inverted pendulum.
Moreover, some general properties of the hybrid inverted pendulum are characterized through its topological equivalence to the classical pendulum.
As a matter of fact, one can observe many, possibly even infinitely many, distinct chaotic attractors for a single system when the forcing amplitude does not exceed the mentioned threshold. Nevertheless, such an observation holds only for constant external forcing and even arbitrary small time-varying external forcing may destabilize the origin. It is actually a trivial example of the so-called pseudo-equilibrium. The latter is due to the hybrid nature of the equilibrium at the origin, which exists only in the Filippov sense. Secondly, the constant lateral forcing less than a certain threshold does not affect the periodic behavior of the hybrid inverted pendulum and preserves its equilibrium at the origin. Firstly, the unperturbed undamped hybrid inverted pendulum behaves inside a certain stability region periodically and its respective frequencies range from zero (close to the boundary of that stability region) to infinity (close to its double support equilibrium). Two rather exceptional features are analyzed. The chaotic behavior, when being externally harmonically perturbed, is demonstrated. A detailed mathematical analysis of the two-dimensional hybrid model for the lateral dynamics of walking-like mechanical systems (the so-called hybrid inverted pendulum) is presented in this article.
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