Matlab code for Lyapunov exponents of fractional order systems

TL;DR

This paper presents a Matlab implementation of the Benettin-Wolf algorithm to compute Lyapunov exponents for fractional-order systems, including numerical methods, code, and examples, facilitating stability analysis of such systems.

Contribution

It introduces a Matlab code for Lyapunov exponents of fractional systems, extending existing integer-order algorithms with efficient numerical methods and practical examples.

Findings

The code accurately computes Lyapunov exponents for fractional systems.

The predictor-corrector Adams-Bashforth-Moulton method is effective for fractional differential equations.

Examples demonstrate the code's capability to analyze bifurcations and fractional order effects.

Abstract

In this paper the Benettin-Wolf algorithm to determine all Lyapunov exponents for a class of fractional-order systems modeled by Caputo's derivative and the corresponding Matlab code are presented. First it is proved that the considered class of fractional-order systems admits the necessary variational system necessary to find the Lyapunov exponents. The underlying numerical method to solve the extended system of fractional order, composed of the initial value problem and the variational system, is the predictor-corrector Adams-Bashforth-Moulton for fractional differential equations. The Matlab program prints and plots the Lyapunov exponents as function of time. Also, the programs to obtain Lyapunov exponents as function of the bifurcation parameter and as function of the fractional order are described. The Matlab program for Lyapunov exponents is developed from an existing Matlab program for Lyapunov exponents of integer order. To decrease the computing time, a fast Matlab program which implements the Adams-Bashforth-Moulton method, is utilized. Four representative examples are considered.