Improved Matlab Code for Lyapunov Exponents of Fractional-Order Systems

This paper presents an improved Matlab code, FO_LE, for numerical computation of the Lyapunov Exponents (LEs) of Fractional-Order (FO) systems modeled by Caputo’s derivative, conceived as an enhanced version of the former FO_Lyapunov and FO_NC_Lyapunov codes for commensurate and non-commensurate orders, respectively [ Danca & Kuznetsov , 2018 ; Danca , 2021 ]. 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 code benefits from the higher-order of the fractional integrator, LIL, and applies to both commensurate and non-commensurate models. As 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 LEs FO_LE is provided, while a fast implementation of LIL, for commensurate and non-commensurate orders, LIL_nc, is available on MathWorks File Exchange ( https://www.mathworks.com/matlabcentral/fileexchange/183478-lil_nc ). A benchmark problem with exact solution is used to compare the LIL-based solver against ABM-type methods, whereas the Rabinovich–Fabrikant system illustrates the computation of LEs 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 FO systems.