We consider a piecewise linear two-dimensional dynamical system that couples a linear equation with the so-called stop operator. Global dynamics and bifurcations of this system are studied depending on two parameters. The system is motivated by modifications to general-equilibrium macroeconomic models that attempt to capture the frictions and memory-dependence of realistic economic agents.

We introduce a new analytical method, which allows to find out chaotic dynamics in non-smooth dynamical systems. A simple mechanical system consisting of a mass and a dry friction element is considered as an example. The corresponding mathematical model is represented. We show that the considered dynamical system is a skew product over a piecewise smooth mapping of a segment (a base map). For this base map we demonstrate there is a domain of parameters where a robust chaotic dynamics can be observed. Namely, we prove existence of an infinite set of periodic points of arbitrarily big period. Moreover, a reduction of the considered map is semi-conjugated to a shift on the set of one-sided infinite boolean sequences. Also, we find conditions, sufficient for existence of a superstable periodic point of this map. The obtained result partially solves a general problem: theoretical confirmation of chaotic and periodic regimes numerically and experimentally observed for models of percussion drilling.

No abstract

We consider dynamics of a scalar piecewise linear "saw map" with infinitely many linear segments. In particular, such maps are generated as a Poincaré map of simple two-dimensional discrete time piecewise linear systems involving a saturation function. Alternatively, these systems can be viewed as a feedback loop with the so-called stop hysteresis operator. We analyze chaotic sets and attractors of the "saw map" depending on its parameters.