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

TL;DR

This work addresses local stability and synchronization in fractional-order complex-valued BAM neural networks by integrating analytical Mittag-Leffler-based synchronization guarantees with data-driven trajectory diagnostics. It establishes global Mittag-Leffler synchronization under a linear error-feedback controller and provides a conservative time-to-tolerance bound, grounded in fractional calculus. Complementing the theory, it introduces two data-driven proxies—micro-ensemble FTLE and kNN-prediction-error Lyapunov measures (full-state and modulus variants)—to quantify local instability along actual trajectories, with the modulus-based kNN offering robust, low-dimensional diagnostics. The combined approach advances robust, predictive stability analysis for complex-valued fractional-order neural networks, with implications for secure communications and nonlinear signal processing.

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.

Figures (7)

• Figure 1: State trajectories of systems (1), (4), and (6) with and without the designed controller: (a) real part of $x(t)$ for the drive and response systems (no control vs. with control); (b) imaginary part of $x(t)$ (no control vs. with control); (c) real part of $y(t)$ (no control vs. with control); (d) imaginary part of $y(t)$; (e) synchronization error without the designed controller; (f) synchronization error under the designed controller; (g) Lyapunov function evolution and the Mittag--Leffler stability bound; (h) real part of the complex-valued BAM neural network time series.
• Figure 2: Examples of ensemble-averaged growth curves $L(h)=\langle \log \|\delta x(h)\| \rangle$ and their MAE--optimal linear fits. Blue markers show the empirical profile as a function of $\tau = h\Delta t$, while the solid lines indicate the selected short horizon fit (one or two line model). The red marker at $\tau = 0$ denotes the initial ensemble deviation, plotted for completeness but excluded from the slope estimation.
• Figure 3: Local FTLE $\hat{\lambda}_{1}(t_{ref})$ estimated from micro-ensembles along the main BAM trajectory. Each marker corresponds to a reference state; the curve reveals a pronounced, nearly sinusoidal modulation of the local growth rate with no appreciable long–term drift.
• Figure 4: Examples of kNN-based prediction error curves $y(h) = \log G(h)$ and their MAE optimal linear fits for two reference states. Blue markers show the empirical profile as a function of $\tau = h\Delta t$, while the solid line indicates the selected short--horizon fit (one or two line model collapsed to its left segment). In contrast to the FTLE-based growth curves, the kNN profiles exhibit pronounced fluctuations with horizon, reflecting the difficulty of predicting all eight state components accurately from a small micro-ensemble.
• Figure 5: Local Lyapunov proxy $\hat{\lambda}_{1}(t_{\text{ref}})$ obtained from kNN prediction errors with full-state targets in $\mathbb{R}^{8}$. The time series exhibits sizeable point-to-point fluctuations and only a weakly visible oscillatory pattern, suggesting that the high-dimensional regression problem introduces substantial noise into the slope estimates.

Theorems & Definitions (9)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Lemma 2.4
• Lemma 2.5
• Definition 3.1
• Theorem 3.2
• proof
• Remark 3.3