1. Fundamentals of piecewise-smooth, continuous systems. 1.1. Applications. 1.2. A
framework for local behavior. 1.3. Existence of equilibria and fixed points. 1.4. The observer …

Our objective is to investigate the innovative dynamics of piecewise smooth systems with
multiple discontinuous switching manifolds. This paper establishes the coexistence of …

Ordinary differential equations (ODEs) are the preferred language for the investigation and
understanding of various natural phenomena. Employed extensively in natural sciences …

We first study the problem of chaotic behaviour in time-perturbed discontinuous systems
whose unperturbed part has a piecewise C1 homoclinic solution transversally crossing the …

The existence of horseshoes is proved in a class of 3-dim piecewise linear systems, in which
a homoclinic orbit connecting the origin to itself is explicitly given. Based on these results, a …

Investigating homoclinic trajectories and chaotic behaviors, is conducive to better
understanding the fourteenth problem listed by Smale in his response to Arnold. This paper …

We study the chaotic behaviour of a time dependent perturbation of a discontinuous
differential equation whose unperturbed part has a sliding homoclinic orbit that is a solution …

It is a great challenge to detect singular cycles and chaos in dynamical systems with multiple
discontinuous boundaries. This paper takes the challenge to investigate the coexistence of …

Homoclinic Trajectories in Discontinuous Systems Page 1 DOI: 10.1007/s10884-007-9087-9
Journal of Dynamics and Differential Equations, Vol. 20, No. 2, June 2008 (© 2007) Homoclinic …

Systems that are not smooth can undergo bifurcations that are forbidden in smooth systems.
We review some of the phenomena that can occur for piecewise-smooth, continuous maps …