Generalized Controllability, Stability, and Chaos in Fractional Dynamics: A Unified Approach

This paper introduces a unified analytical and computational framework for controlling and characterizing chaos in generalized fractional-‎order dynamical systems. Despite advances in fractional calculus, existing methods face persistent challenges: high computational costs, ‎limited stability criteria, and a lack of quantitative chaos measures for generalized operators. To bridge these gaps, we propose Apc-GM, a ‎novel adaptive predictor-corrector algorithm that enhances numerical accuracy by 45% and reduces computation time by 23% compared to ‎classical approaches. Our framework integrates three core pillars: a generalized controllability theory, an extended Lyapunov stability ‎analysis valid for all orders α>0, and a quantitative chaos criterion based on generalized Lyapunov exponents. Numerical validation on a 4D ‎fractional Lorenz-Stenflo system demonstrates 92% chaos suppression efficiency under optimal control, while maintaining consistency with ‎classical results. This work provides robust, ready-to-use tools for modeling, analysis, and control of complex fractional-order systems in ‎engineering and applied sciences. Keywords: fractional calculus, chaos control, adaptive numerical methods, nonlinear dynamics, stability ‎analysis, Lyapunov exponents, predictor-corrector methods.