Roughness of trajectories of dynamical systems with respect to hysteresis disturbances (Q1363834)

Consider smooth mappings \(f:\mathbb{R}^d\to \mathbb{R}^d\), and systems of the form \[ x_n= f(x_{n- 1}).\tag{1} \] Equation (1) usually describes the dynamics of real systems only approximately, and the problem arises of their roughness with respect to disturbances of different natures. According to classic results (see, e.g., [\textit{J. Guckenheimer} and \textit{P. Holmes}, Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. Applied Mathematical Sciences, 42. New York etc. Springer-Verlag (1983; Zbl 0515.34001)]), sufficiently smooth systems (1) preserve many structural properties for smooth disturbances (small in the metric \(C^1\)). To analyze them, we use the technique that we proposed for an approximate study of chaotic dynamical systems. The main object of analysis is a class of operator disturbances of systems (1) that arises in describing the dynamics of systems with weak hysteresis nonlinearities. Hysteresis nonlinearities are treated in [\textit{M. A. Krasnosel'skij} and \textit{A. V. Pokrovskij}, Systems with hysteresis. Transl. from the Russian, Berlin etc.: Springer-Verlag (1989; Zbl 0665.47038)] and in the paper as continuous but fundamentally nonsmooth dynamical systems \({\mathbf W}\), often with an infinite-dimensional space \(\Omega= \Omega({\mathbf W})\) of internal states \(\omega\). This class contains nonlinearities of the following types: a play, a stop, Ishlinskij-Besseling's, Preisath-Giltay's, and other models.