Kicked Mechanical Models and Differential Equations with Periodic Switch

Abstract

Some physical systems admit very natural description in discrete time. For example, it relates to mechanical or electronic systems under action of periodic pulses (kicks). Instead considering the motion in continuous time, one can derive a map, which expresses the state variables just before (or after) the next kick via the state just before (or after) the previous kick. In papers and textbooks on nonlinear dynamics, maps of Hénon, Ikeda, and Chirikov-Taylor-Zaslavsky are derived and studied in this conlexl (Heagy, 1992; Kuznetsov el al., 2008; Sagdeev et al., 1988). Is it possible for the uniformly hyperbolic attractors to occur for diffeomorphisms arising in description of periodically kicked systems? In this chapter we introduce some simple physically motivated toy models with dynamics described by maps possessing attractors of Smale-Williams type.

One more approach to design of non-autonomous systems with uniformly hyperbolic attractors is to organize evolution in time as certain periodically repeated distinct stages, in such a way that the right-hand parts in the differential equations are defined in a special form for each of the successive stages. In oilier words, the right-hand parts may be thought of as being represented by functions piecewise continuous and periodic in time. It may be referred to as differential equations with periodic switch. One example of such kind will be suggested with attractor of Smale-Williams type.

Earlier, the approach based on the periodically switching equations was developed by Hunt (Hunt, 2000) in the context of the problem of suspending attractor of Plykin type, as mentioned in Chap. 2. We introduce a simpler model with analogous attractor governed by equations with right-hand parts represented by periodic piece-wise continuous functions of time. Then, exploiting the structural stability, we modify the model to obtain differential equations with smooth coefficients possessing attractor of the same type.