Dynamical Systems: Discrete, Continuous and Hybrid

TL;DR

This paper unifies the study of discrete, continuous, and hybrid dynamical systems by extending relation dynamics to analyze attractor structures, chain recurrence, and Lyapunov functions across different system types.

Contribution

It introduces a unified framework for analyzing various dynamical systems using relation dynamics, covering semiflows and hybrid systems.

Findings

Unified description of attractor-repeller structures

Extension of Conley's chain recurrence to hybrid systems

Construction of Lyapunov functions for all systems

Abstract

The dynamics by iteration of a function on a compact metric space, sometimes called a cascade, can be extended to the dynamics of a closed relation on such a space. Here we apply this relation dynamics to study semiflows (and their relation extension) as well as hybrid dynamical systems which combine both continuous time and discrete time dynamics. In a unified way we describe the attractor-repeller structure, Conley's chain recurrence relation and the construction of Lyapunov functions for all of these systems.