Unveiling and resolving algorithmic false chaos in fractional-order systems via infinite state representation

Abstract

Lyapunov exponents (LEs) are central to characterizing the long-time dynamics of fractional-order systems (FOS). However, classical discrete reorthonormalization algorithms rely on the semigroup property of integer-order flows. Directly applying these local geometric operations to non-local fractional operators therefore creates a structural incompatibility. This study shows that such incompatibility can produce a positive largest LE in parameter regimes where the original FOS is non-chaotic, giving rise to algorithmic false chaos. Two distinct underlying mechanisms are identified: lower-terminal resetting and variational-history inconsistency. To address this incompatibility, an infinite state representation (ISR)-based framework is developed for fractional LE computation. The ISR transforms the fractional convolution kernel into a continuous spectrum of integer-order ordinary differential equations, restoring a semigroup structure in the augmented state space. Classical reorthonormalization can then be applied consistently to the augmented variational equations. Because numerical ISR implementation augments the system dimension, we analyze the augmented Jacobian structure and develop a degeneracy-rejection algorithm to separate physical LEs from approximation-induced spurious clusters. Numerical experiments across multiple dynamical regimes and kernel approximations show that the proposed framework recovers consistent physical LEs and avoids false chaotic signatures produced by the conventional implementations examined in this study. This framework provides a structurally consistent route for computing fractional LEs that remains aligned with the dynamics of the target FOS rather than algorithm-induced artifacts.