Improved Matlab code for Lyapunov exponents of fractional order systems

TL;DR

This paper introduces an improved MATLAB routine, FO_LE, for accurately computing Lyapunov exponents in fractional-order systems, enhancing previous methods with higher-order integration and reorthonormalization techniques.

Contribution

The paper presents a new MATLAB code, FO_LE, that improves the numerical computation of Lyapunov exponents in fractional-order systems using advanced integration and reorthonormalization methods.

Findings

FO_LE provides more accurate Lyapunov exponents than previous codes.

The routine is applicable to both commensurate and non-commensurate fractional orders.

Benchmark and dynamical system examples demonstrate robustness and efficiency.

Abstract

This paper presents an improved Matlab routine, FO_LE, for the numerical computation of Lyapunov exponents of fractional-order systems modeled by Caputo's derivative. It is conceived as an enhanced version of the former FO_Lyapunov and FO_NC_Lyapunov codes for commensurate and non-commensurate orders, respectively. The proposed approach replaces the Gram-Schmidt orthogonalization procedure with QR-based reorthonormalization and uses the new quadratic LIL predictor-corrector scheme for the integration of the extended variational system. Compared with the former implementations, the present routine benefits from the higher order of the fractional integrator LIL and applies to both commensurate and non-commensurate models. Like the previous code, FO_LE retains the full memory structure of the underlying Caputo model. The Matlab code for the LIL solver and for the computation of Lyapunov exponents with FO_LE are provided, while a fast implementation of LIL for commensurate and non-commensurate orders, LIL_nc, is available on MathWorks File Exchange. A benchmark problem with exact solution is used to compare the LIL-based solver with ABM-type methods, whereas the Rabinovich-Fabrikant system illustrates the computation of Lyapunov exponents in different dynamical regimes. The results indicate that the proposed implementation is a compact, robust, and efficient tool for the numerical study of stability and chaos in fractional-order systems.