Matlab code for Lyapunov exponents of fractional-order systems, Part II: The non-commensurate case

TL;DR

This paper presents a Matlab implementation of the Benettin-Wolf algorithm to compute Lyapunov exponents for non-commensurate fractional-order systems, extending previous work on commensurate systems and focusing on the Lorenz system.

Contribution

It introduces a Matlab code for calculating Lyapunov exponents in non-commensurate fractional-order systems using an adapted algorithm and numerical integration method.

Findings

The Matlab code successfully computes Lyapunov exponents for non-commensurate fractional systems.

The program can be adapted to analyze the evolution of exponents with respect to parameters.

Special attention is given to the effects of periodicity in fractional-order systems.

Abstract

In this paper the Benettin-Wolf algorithm to determine all Lyapunov exponents adapted to a class of non-commensurate fractional-order systems modeled by Caputo's derivative and the corresponding Matlab code are presented. The paper continues the work started in [Danca & Kuznetsov, 2018], where the Matlab code of commensurate fractional-order systems is given. To integrate the extended system, the Adams-Bashforth-Moulton for fractional differential equations is utilized. As the the Matlab program for commensurate-order systems, the program presented in this paper prints and plots all Lyapunov exponents as function of time. The program can be simply adapted to plot the evolution of the Lyapunov exponents as function of orders, or a function of a bifurcation parameter. A special attention is payed to the periodicity of fractional-order systems and its influences. The case of the Lorenz system is considered.