Improved Matlab code for Lyapunov exponents of fractional order systems
Abstract
This paper presents an improved Matlab routine, FOLE, 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 FOLyapunov and FONCLyapunov 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, FOLE 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 FOLE are provided, while a fast implementation of LIL for commensurate and non-commensurate orders, LILnc, 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.
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