Dynamics and bifurcations of nonsmooth systems: A survey

Abstract

In this survey we discuss current directions of research in the dynamics of nonsmooth systems, with emphasis on bifurcation theory. An introduction to the state-of-the-art (also for non-specialists) is complemented by a presentation of main open problems. We illustrate the theory by means of elementary examples. The main focus is on piecewise smooth systems, that have recently attracted a lot of attention, but we also briefly discuss other important classes of nonsmooth systems such as nowhere differentiable ones and differential variational inequalities. This extended framework allows us to put the diverse range of papers and surveys in this special issue in a common context. A dedicated section is devoted to concrete applications that stimulate the development of the field. This survey is concluded by an extensive bibliography.

Piecewise linear differential systems without equilibria produce limit cycles?

“…See, for instance, the books of di Bernardo, Budd, Champneys and Kowalczyk [5] and Simpson [28], the survey of Makarenkov and Lamb [26], and the hundreds of references which appear in these last three cited works.…”

On limit cycles bifurcating from the infinity in discontinuous piecewise linear differential systems

“…Interest stems particularly from discontinuous dynamical models in control theory [3], nonlinear oscillations [2,15], impact and friction mechanics [5], economics [8,10], biology [4], and others; recent reviews appear in [18,14].…”

A Note on Forced Oscillations in Differential Equations with Jumping Nonlinearities

“…where f is a force applied to the mass in the vertical direction, see [1,15,11]. The case where equation (1) takes the form where c ε → 0 as ε → 0, the bifurcation of asymptotically stable periodic solutions is studied in Glover-Lazer-McKenna [8].…”

“…A degree theoretic approach is developed in [12]. See our survey [11] for a broad analysis of the research around equations of type (5). Extending the range of conclusions about the dynamics of (5) is important as this equation occurs in a variety of applications, e.g.…”