Micro-Series Lyapunov Analysis

• Micro-series Lyapunov analysis is a data-driven method that estimates the largest local Lyapunov exponent from brief trajectory segments to reveal transient stability properties.
• It utilizes a sliding-window approach combined with k-nearest neighbor regression to monitor and diagnose dynamic behavior in iterative numerical solvers.
• The technique enables real-time parameter adaptation, leading to improved convergence rates and reduced computational costs in applications like root-finding algorithms.

Micro-series Lyapunov analysis is a fine-grained, short-horizon, data-driven methodology for assessing and controlling the local stability properties of dynamical or algorithmic processes by estimating the largest local Lyapunov exponent (LLE) from short “micro-series” of trajectory data. This approach is particularly suited to optimization schemes, root-finding algorithms, and iterative numerical solvers exhibiting rich transient behavior—contractive, oscillatory, or chaotic—where traditional linearized or global Lyapunov analysis is insufficient to provide real-time diagnostics or guide parameter adaptation. Micro-series Lyapunov analysis has emerged as an effective practical tool for constructing self-stabilizing numerical schemes, with concrete applications in tuning inverse-parallel root-finding methods and other iterative algorithms (Shams et al., 20 Jan 2026).

1. Foundational Principles and Local Lyapunov Exponents

The fundamental component of micro-series Lyapunov analysis is the local (short-horizon) largest Lyapunov exponent, which quantifies the mean exponential divergence or contraction rate of neighboring trajectories over a finite, typically short, window. For a discrete-time map $x_{n+1} = f(x_n; \alpha)$ parametrized by $\alpha$ (e.g., in an inverse-parallel root-finding iteration), the global LLE is defined as

$$\lambda = \lim_{N \to \infty} \frac{1}{N} \sum_{n=1}^N \ln|f'(x_n; \alpha)|,$$

with $\lambda < 0$ indicating local contraction (stability), $\lambda > 0$ indicating local expansion (instability/chaos), and $\lambda \approx 0$ denoting the marginal or neutrally stable regime. Micro-series Lyapunov analysis restricts estimation to moving windows, yielding a local $\lambda_1(t)$ profile capturing transient contractive or expansive phases otherwise obscured in global analysis (Shams et al., 20 Jan 2026).

2. Micro-series Pipeline: Sliding Window and Estimation

The micro-series approach is operationalized through a sliding-window pipeline applied across ensembles of short trajectory segments. Given $R$ independent solver runs (e.g., from perturbed initializations), one constructs a scalar time series $x_{j,k}$ for $j=1,\ldots,R$ and $k=0,\ldots,T-1$. At each window-end index $t_\text{end}$, a batch $U^{(t_\text{end})}$ collects the most recent $L_\text{micro}$ data points. For each batch, k-nearest neighbor (kNN) regression is used to locally predict short-horizon trajectory segments and statistical errors are aggregated:

• For forecast horizon $h$, fit a kNN regressor on the last $L_b$ points to predict the $h$-step-ahead value.
• Aggregate test errors by the geometric mean $\text{GMAE}(h)$ over test runs, yielding the log error profile $y(h) = \log \text{GMAE}(h)$.
• Fit a local affine or piecewise-affine model $y(h) \approx a_1 h + b_1$ on short horizons to extract the local Lyapunov indicator $\lambda_1(t_{\text{end}}) = a_1$.

This procedure provides a real-time, fine-resolution stability profile of the solver or iterative algorithm, flagging contractive vs. unstable at each time index (Shams et al., 20 Jan 2026).

3. Lyapunov-guided Parameter Selection and Algorithmic Control

Micro-series Lyapunov analysis is directly exploited to inform parameter selection in iterative algorithms such as inverse-parallel root-finding schemes. The pipeline is used as follows:

1. For a discrete grid of candidate parameter values $\{\alpha\}$, run $R$ trajectories, collect and analyze LLE profiles $\lambda_1(t_{\text{end}}; \alpha)$.
2. Parameters are classified as "well-behaved" if the LLE drops below $0$ after a short transient and remains negative, and residual decay is monotonic.
3. Among feasible $\alpha$, select the setting with fastest residual decline or highest empirical order of convergence (COC).
4. In online adaptation, if LLE profiles indicate persistent instability (e.g., $\lambda_1(t_{\text{end}}) > \delta_{\text{pos}}$), decrease $\alpha$; if strong contraction is detected ($\lambda_1 < \delta_{\text{neg}}$), $\alpha$ can be increased for efficiency.

This Lyapunov-informed approach has demonstrated the ability to eliminate chaotic transients, accelerate convergence by an order-of-magnitude (CPU and memory efficiency), and raise convergence order, with LLE empirical profiles agreeing closely with analytical bifurcation and stability diagrams (Shams et al., 20 Jan 2026).

Step	Quantity	Role in Analysis/Adaptation
Data Window	$L_\text{micro}$	Trajectory segment for LLE estimation
Estimation	$\lambda_1(t_\text{end})$	Local stability indicator
Parameter	$\alpha$ (solver control)	Select to maintain $\lambda_1 < 0$

4. Application to Root-Finding and Solver Self-Stabilization

The methodology is instantiated for uni-parametric inverse-parallel solvers (INVM$^\alpha$ family), where the stability of the fixed point is controlled by $\alpha$. Analytical results show the transition from monotone contraction to oscillatory/chaotic regimes as $\alpha$ is varied. Micro-series Lyapunov analysis yields actionable diagnostics:

• For well-chosen $\alpha$, $\lambda_1(t_{\text{end}}; \alpha)<0$ across the post-transient region, resulting in smooth, monotonic residual decay and high COC.
• Untuned $\alpha$ yields LLE profiles near zero or positive, manifesting as residual oscillations or instability.
• Real-time LLE monitoring enables dynamic adjustment of $\alpha$ in adaptive solvers, resulting in robust, stabilized convergence even in the presence of perturbed initial conditions.

Empirically, for cubic and sextic polynomial root-finding, Lyapunov-informed $\alpha$ produced residuals below $10^{-100}$ in 20–30 iterations, with up to $90\%$ reduction in computational cost over reference methods (Shams et al., 20 Jan 2026).

5. Comparison to Coarse Lyapunov Approaches and Broader Implications

Micro-series Lyapunov analysis fundamentally differs from classical, global, or linearized Lyapunov stability analysis by:

• Capturing fine-grained, non-asymptotic, and non-stationary stability properties.
• Enabling data-driven, real-time adaptation, rather than a static or offline assessment.
• Directly supporting self-stabilizing mechanisms for iterative solvers, root-finders, and other algorithms sensitive to control parameter choices.

The methodology generalizes to higher-dimensional or noise-perturbed problems, provided scalar stability proxies can be extracted and local LLEs estimated. A plausible implication is the extensibility to pattern recognition in solver failures, online adaptation of numerical schemes, and as a diagnostic tool for emergent, data-driven control algorithms.

6. Implementation and Practical Recommendations

Efficient implementation of micro-series Lyapunov analysis requires:

• Short (5–10 step) trajectory buffers, moderate ensemble sizes ($R \sim 10^2$–$10^3$) for robust error statistics.
• kNN regression with geometric mean aggregation for LLE estimation.
• Piecewise-affine models for accurate characterization of transient local dynamics.
• Configurable thresholds for LLE-based parameter adaptation (e.g., $\delta_{\text{pos}} \sim 0.1$ for chaos flagging, $\delta_{\text{neg}} \sim -0.1$ for over-contraction).

Careful selection of micro-series length balances immediacy of detection with statistical reliability. The approach incurs negligible computational overhead relative to full solver trajectories and scales efficiently with ensemble size and window length. Online adaptation can be realized in real-time, facilitating robust stabilization protocols in practical high-speed numerical applications (Shams et al., 20 Jan 2026).