Fractional-order (FO) chaotic systems exhibit geometric and dynamical behaviors fundamentally distinct from their integer-order counterparts due to memory-dependent dynamics. This study proposes a systematically formulated fractional-order Krause-Robert (FOKR) model and extends the competitive mode (CM) framework to the fractional domain through a curvature-based reformulation. Based on this formulation, a structural diagnostic criterion is established to characterize curvature-dominant interactions underlying memory-dependent chaos. The influence of the fractional derivative order on attractor geometry and dynamical complexity is investigated using spectral entropy (SE), the C0 complexity index, Lyapunov exponents, and the 0-1 test. The results reveal a memory-induced attenuation mechanism in which decreasing the fractional order suppresses curvature-dominant alternation and reduces spectral complexity. Furthermore, a rigorous finite-time convergence theorem is established using Lyapunov-Mittag-Leffler analysis under Caputo dynamics. Based on this result, a finite-time synchronization controller with explicit settling-time bounds is developed. Quantitative comparisons with asymptotic feedback control demonstrate significantly accelerated convergence and reduced integral error indices. Numerical simulations validate the theoretical results and confirm improved transient performance. Overall, this work provides a unified framework linking fractional calculus, curvature-based chaos diagnostics, complexity analysis, and finite-time control for fractional chaotic dynamo systems.
Lin et al. (Fri,) studied this question.
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