Lyapunov exponents of a class of piecewise continuous systems of fractional order

TL;DR

This paper establishes the existence of Lyapunov exponents for a class of fractional order piecewise continuous systems and demonstrates their numerical computation through examples of chaotic systems.

Contribution

It introduces a method to define and compute Lyapunov exponents for fractional order discontinuous systems, extending existing theory and numerical techniques.

Findings

Lyapunov exponents are well-defined for the studied systems.

Numerical methods successfully compute exponents for chaotic fractional systems.

Chaotic behavior confirmed in multiple fractional order systems.

Abstract

In this paper, we prove that a class of autonomous piecewise continuous systems of fractional order has well-defined Lyapunov exponents. For this purpose, based on some known results from differential inclusions of integer and fractional order and differential equations with discontinuous right-hand side, the associated discontinuous initial value problem is approximated with a continuous one of fractional order. Then, the Lyapunov exponents are numerically determined using, for example, the known Wolf's algorithm. Three examples of piecewise continuous chaotic systems of fractional order are simulated and analyzed: Sprott's system, Chen's system and Shimizu-Morioka's system.