Parallel Bifurcation and Lyapunov Analysis of Commensurate and Incommensurate Fractional-Order Lorenz Systems

Abstract

The numerical investigation of fractional-order chaotic systems is computationally demanding due to the long-memory effect of fractional derivatives and the need for small integration step sizes. In particular, computing bifurcation diagrams and Lyapunov exponent spectra over wide parameter ranges becomes time-consuming when conventional serial for-loop implementations are used. In this study, a parallel computational approach is proposed for the simultaneous extraction of bifurcation diagrams and Lyapunov exponents of fractional-order chaotic systems. For each parameter value, the system equations and the corresponding variational equations are integrated simultaneously, while periodic Gram–Schmidt reorthonormalization is applied for Lyapunov exponent estimation. MATLAB’s parfor structure is employed to distribute independent parameter realizations across multiple workers. The proposed approach is validated using the commensurate fractional-order Lorenz system with q1 = q2 = q3 = 0.99, where the bifurcation parameter β is varied over [1, 10]. The obtained bifurcation structures are consistent with the sign variation of the largest Lyapunov exponent, confirming the transition between chaotic and regular regimes. The serial and parallel implementations produce numerically identical Lyapunov spectra within the reported precision, indicating that the proposed parallelization strategy preserves the dynamical outputs of the corresponding serial computation. In terms of computational performance, the serial implementation required 6617.57 s, whereas the parallel implementation completed the same analysis in 1473.20 s, corresponding to a speed-up of approximately 4.49× with a parallel efficiency of 74.87%. In addition, an incommensurate fractional-order case with q1 = 0.99, q2 = 0.98, and q3 = 0.97 was tested, where the parallel implementation produced the same largest Lyapunov exponent curve as the serial implementation while reducing the computation time from 6843.45 s to 1974.79 s. These results show that the proposed parfor-based approach is a numerically consistent and computationally efficient tool for high-resolution bifurcation and Lyapunov exponent analyses of fractional-order chaotic systems.