Local Lyapunov Analysis via Micro-Ensembles: finite-time Lyapunov exponent Estimation and KNN-Based Predictive Comparison in Complex-Valued BAM Neural Networks

Abstract: Finite-time Lyapunov exponents (FTLEs) quantify short-horizon trajectory divergence and provide a local, spatially resolved view of transient instabilities and synchronization behavior in nonlinear dynamics. This work studies a class of fractional-order complex-valued bidirectional associative memory (BAM) neural networks and proposes a unified analytical and data-driven framework for synchronization and local stability assessment. Using fractional Lyapunov stability theory together with Mittag-Leffler functions, sufficient conditions are derived to guarantee global Mittag-Leffler synchronization of the drive-response systems under a linear error-feedback controller. In addition, an explicit conservative time-to-tolerance estimate is obtained via a standard upper bound on the Mittag-Leffler function. Numerical simulations corroborate the theory and demonstrate rapid decay of synchronization errors in both real and imaginary state components. To complement the model-based guarantees, two trajectory-driven Lyapunov proxies are introduced: (i) micro-ensemble FTLE estimation based on the geometric-mean growth of small perturbations, and (ii) a k-nearest neighbors (kNN) prediction-error index that quantifies local instability through short-term forecast errors. Both proxies reveal oscillatory transient divergence patterns and consistently reflect the stabilizing effect of the designed controller. The proposed integration of fractional calculus, synchronization control, and data-driven Lyapunov diagnostics provides a robust methodology for complex-valued fractional-order neural networks, with potential applications in secure communications and nonlinear signal processing.