Chaos Generated by a Class of 3D Three-Zone Piecewise Affine Systems with Coexisting Singular Cycles

It is a significant and challenging task to detect both the coexistence of singular cycles, mainly homoclinic and heteroclinic cycles, and chaos induced by the coexistence in nonsmooth systems. By analyzing the dynamical behaviors on manifolds, this paper proposes some criteria to accurately locate the coexistence of homoclinic cycles and of heteroclinic cycles in a class of three-dimensional (3D) piecewise affine systems (PASs), respectively. It further establishes the existence conditions of chaos arising from such coexistence, and presents a mathematical proof by analyzing the constructed Poincaré map. Finally, the simulations for two numerical examples are provided to validate the established results.

Heteroclinic Chaotic Threshold in a Nonsmooth System with Jump Discontinuities

In smooth systems, the form of the heteroclinic Melnikov chaotic threshold is similar to that of the homoclinic Melnikov chaotic threshold. However, this conclusion may not be valid in nonsmooth systems with jump discontinuities. In this paper, based on a newly constructed nonsmooth pendulum, a kind of impulsive differential system is introduced, whose unperturbed part possesses a nonsmooth heteroclinic solution with multiple jump discontinuities. Using the recursive method and the perturbation principle, the effects of the nonsmooth factors on the behaviors of the nonsmooth dynamical system are converted to the integral items which can be easily calculated. Furthermore, the extended Melnikov function is employed to obtain the nonsmooth heteroclinic Melnikov chaotic threshold, which implies that the existence of the nonsmooth heteroclinic orbits may be due to the breaking of the nonsmooth heteroclinic loops under the perturbation of damping, external forcing and nonsmooth factors. It is worth pointing out that the form of the nonsmooth heteroclinic Melnikov function is different from the one of the nonsmooth homoclinic Melnikov function, which is quite different from the classical Melnikov theory.

Chaotic Dynamics in Four-Dimensional Piecewise Affine Systems with Bifocal Heteroclinic Cycles

The well-known Shil’nikov type theory provides an approach to proving the existence of chaotic invariant sets for some classes of smooth dynamical systems with homoclinic orbits or heteroclinic cycles. However, it cannot be applied to nonsmooth systems directly. Based on the similar ideas, this paper studies the existence of chaotic invariant sets for a class of two-zone four-dimensional piecewise affine systems with bifocal heteroclinic cycles that cross the switching manifold transversally at two points. It turns out that there exist countable infinite chaotic invariant sets in a neighborhood of the bifocal heteroclinic cycle under some eigenvalue conditions. Moreover, the horseshoes of the corresponding Poincaré map are topologically semi-conjugated to a full shift on four symbols.

Editorial

This issue is dedicated to nonsmooth dynamics, particularly the widening applications where dynamics is modelled by nonsmooth differential equations. The modeling of electrical or mechanical switches as nonsmooth can be traced throughout the (at least) 90-year history in which nondifferentiable terms, such as sign or step functions, have been turning up in differential equations. In recent decades, nonsmooth models have found increasing use in areas like contact mechanics, climate modeling, and the life sciences, among others, with a wealth of new theory and novel dynamical phenomena discovered along the way. Our aim here is to give just a partial snapshot of the current landscape of research topics in the field. We open with Paul Glendinning's article extending a classic phenomenon of nonlinear dynamics — a form of chaos introduced by Leonid Pavlovich Shilnikov — to nonsmooth systems. The scenario introduces a fundamental notion of nonsmooth dyamics, that of trajectories (in this case the crucial homoclinic orbit) that can ‘slide’ along a discontinuity. Shilnikov's scenario is moreover shown to occur naturally as an equilibrium hits a discontinuity, helping with a fundamental yet complex problem, namely that of extending the notion of boundary equilibrium bifurcations beyond systems of two variables.