Local Lyapunov analysis via micro-ensembles: finite-time Lyapunov exponent estimation and KNN-based predictive comparison in complex-valued BAM neural networks

Abstract

Complex-valued bidirectional associative memory (BAM) neural networks with fractional-order dynamics and delays can exhibit transient instabilities that degrade synchronization and short-horizon predictability. This paper develops a unified analytical and data-driven framework to assess stability, synchronization, and predictability in such networks. First, using Caputo fractional calculus and Lyapunov-Mittag-Leffler techniques, we derive sufficient conditions for global Mittag-Leffler synchronization of a drive-response BAM pair under a linear error-feedback controller and obtain an explicit time-to-tolerance bound. Second, to quantify local transient instability from finite trajectory data, we propose a micro-ensemble finite-time Lyapunov exponent (FTLE) estimator based on the geometric-mean growth of small perturbations over short windows, avoiding variational equations. We further introduce k-nearest-neighbor prediction-error Lyapunov proxies, including full-state and modulus-based variants, to connect local instability to forecasting performance. Numerical experiments on fractional-order complex-valued BAM benchmarks confirm effective synchronization under the proposed control and demonstrate a clear correspondence between FTLE levels and prediction errors. The resulting framework provides practical and reproducible diagnostics for complex-valued neural systems in data-limited settings.