A Unified Framework for Optimization and Analysis of Fractional-Order Chaotic Systems

Abstract

Maximizing the dominant Lyapunov exponent λ1 of an incommensurate fractional-order chaotic system, while respecting the dynamical conditions for a strange attractor, is a non-convex, gradient-free problem on a history-dependent landscape. Existing metaheuristic studies typically use hard-cutoff penalties that distort the fitness landscape and integer-order Lyapunov estimators that can be biased for strongly fractional regimes. This paper presents a constraint-faithful optimization framework combining (i) subtractive-hinge penalties that vanish on the feasible set, (ii) a memory-consistent Grünwald–Letnikov variational Lyapunov estimator with adaptive tail-sum truncation, (iii) joint search over parameters and incommensurate orders by the Marine Predators Algorithm, and (iv) a fractional conditional Lyapunov exponent (FCLE) that recovers the integer-order limit. Applied with a fixed configuration to the fractional-order Lorenz, Ma–Chen financial, Iqbal–Wang, and Hyper–Chen systems, the framework converges to feasible attractors with enlarged Lyapunov spectra. Dissipativity is rigorously verified; all selected optima have strictly negative Lyapunov trace at the reported precision. FCLE analysis on the optimized Lorenz attractor recovers the integer-order identity cmin=λ1 under full-state coupling, and shows that single-state x-coupling raises the threshold to ≈9λ1*. The optimized fractional-order Lorenz attractor is employed as the random-number generator of a recent chaos-based image-encryption scheme, where it yields strong statistical results across standard benchmarks.