Differential Equations with Hysteresis via a Canonical Example

Analysis of closed-loop system is needed and these systems are described by differential equations with hysteresis, and hysteresis terms are to be taken into account in various areas of differential equations, thus leading to numerous distinct branches of study, depending on the subject area, type of hysteresis operators that are used, etc. Operators of hysteresis nonlinearities often admit a simple "picture definition," however their properties are quite different from the properties of more classical operators. The investigation of differential equations with hysteresis nonlinearities requires new mathematical methods. In return, methods that have been originally suggested for the analysis of differential equations with hysteresis appear to be useful in the classical theory of differential-operator equations. This chapter demonstrates the theory of differential equations with hysteresis via a simple canonical example. Essentially, the semi-linear Duffing oscillator is considered with the Preisach non-linearity. The chapter presents various results on existence and uniqueness, on properties of periodic motions, on the convergence of numerical solutions, etc. Moreover, it shows how these fuse with, and complement each other. Apart from results in traditional areas, the chapter also presents a version of the shadowing lemma specifically designed for the analysis of systems with hysteresis.