Dynamics of Systems with a Discontinuous Hysteresis Operator and Interval Translation Maps

TL;DR

This paper investigates the properties of interval translation maps (ITMs), focusing on their invariant measures and parameter dependence, with applications demonstrated in a risk management context.

Contribution

It introduces the maximal invariant measure for ITMs and analyzes how these measures vary with system parameters, providing new insights into their dynamics.

Findings

Defined the maximal invariant measure for ITMs

Analyzed the dependence of invariant measures on parameters

Applied results to a risk management model

Abstract

We studied topological and metric properties of the so-called interval translation maps (ITMs). For these maps, we introduced the maximal invariant measure and study its properties. Further, we study how the invariant measures depend on the parameters of the system. These results were illustrated by a simple example or a risk management model where interval translation maps appear~naturally.