lyapunov spectrum analysis

This study addresses the challenge of accurately estimating the positive Lyapunov exponent (LE) from one-dimensional chaotic time series. We propose a data-driven, phase-space-reconstruction-free method that uses the forward prediction error growth rate of machine learning models as a proxy for trajectory divergence, enabling direct modeling of time-series dynamics and estimation of the maximum LE. The approach integrates multiple supervised learning algorithms and operates solely on the raw scalar time series, thereby circumventing the sensitivity of traditional methods to embedding parameters and noise. Evaluated on canonical chaotic maps—including Logistic, Tent, and Chebyshev—the method achieves R²ₚₒₛ values of 0.90–0.999, requiring as few as 200 data points. It significantly improves accuracy and robustness of LE estimation under small-sample conditions. This work establishes an interpretable, deployable paradigm for chaos identification in black-box nonlinear systems.

研究通过构建Lyapunov解算子并使用Fourier神经算子来近似解决非线性系统稳定性分析中Lyapunov函数难以构建的问题。

This work addresses the challenge of convergence failure in inverse parallel solvers for nonlinear systems of equations, which often arises due to oscillatory or chaotic dynamics. To enhance stability, the authors propose an adaptive stabilization mechanism based on the local maximum Lyapunov exponent (LLE). By estimating the LLE via k-nearest neighbors and integrating it with sliding-window micro-time-series analysis, the method enables real-time detection of unstable phases along the solution trajectory. A Lyapunov-guided parameter control strategy is then developed to dynamically adjust solver parameters, thereby reinforcing numerical stability. Experimental results demonstrate strong agreement between theoretical stability diagrams and empirical Lyapunov profiles, confirming that the proposed approach significantly improves the robustness and convergence performance of solvers under perturbed initial conditions.

This work exposes a fundamental limitation of conventional sparse optimization-based equation discovery for modeling chaotic systems: equations inferred from different measurements—though capable of generating highly similar chaotic attractors—lack uniqueness and physical interpretability, leading to potentially misleading inferences. Integrating sparse regression, Koopman spectral analysis, and numerical simulations, the study systematically examines multiple chaotic systems and reveals that small-magnitude Koopman eigenvalues are highly sensitive to measurement perturbations, whereas only large-magnitude eigenvalues remain robust. This undermines the prevailing “unique correct equation” paradigm. The key contribution is the first operator-spectral proof that the deterministic assumption underlying equation discovery fails for chaotic dynamics. Consequently, the paper argues that for strongly nonlinear, highly sensitive systems, end-to-end data-driven modeling—particularly machine learning approaches—should supersede the pursuit of explicit differential equations.

This study unifies the intrinsic relationships among reactivity, contraction, and finite-time Lyapunov exponents (FTLE) in discrete-time dynamical systems. Method: We introduce a *p*-iteration system framework applicable to time-invariant, time-varying linear, and certain nonlinear maps, and rigorously establish equivalences and implication relations among the three properties. Based on this, we derive a *p*-iteration contraction criterion and extend the theory to synchronization stability analysis of coupled networks, leveraging matrix measures, operator norms, and finite-time stability theory. Contribution/Results: We prove that *p*-iteration contraction guarantees the existence of a globally asymptotically stable attractor (e.g., fixed point or limit cycle) in the original system. The proposed framework yields novel sufficient conditions for stable attractor existence and significantly improves both accuracy and applicability in synchronization analysis of coupled oscillator networks.

This work addresses the stabilization control of an inverted pendulum under vertical excitation by proposing a reinforcement learning approach that incorporates the Lyapunov Characteristic Exponent (LCE) as a physics-informed dense reward. For the first time within a reinforcement learning framework, LCE is employed to guide policy optimization, enabling the recovery of the classical Kapitza pendulum stabilization mechanism—reliant on high-frequency oscillations—and further achieving a novel stable state wherein the pendulum remains perfectly upright without any oscillation. Experimental results demonstrate that the proposed method effectively steers the agent toward discovering control policies superior to conventional mechanisms, substantially enhancing both system stability and control accuracy.

This work addresses the lack of theoretical guarantees for the reliability of Koopman eigenpairs computed from noisy data in data-driven spectral analysis. It introduces, for the first time, shadowing trajectory theory combined with backward error analysis to interpret the residual of eigenpairs obtained via Extended Dynamic Mode Decomposition (EDMD) as an operator perturbation of the original dynamical system. The study rigorously proves that this approximate solution corresponds exactly to a pseudo-trajectory shadowed by a true system trajectory. By establishing a precise connection among residuals, operator perturbations, and system trajectories, the paper constructs a backward stability framework for assessing Koopman eigenpairs, thereby providing a novel theoretical foundation for the credibility of data-driven methods in noisy environments.

This study addresses the challenge of accurately estimating negative maximal Lyapunov exponents from short scalar time series, where conventional methods are highly susceptible to noise and finite-precision effects. The authors propose a fully equation-free, periodicity-aware predictive error contraction approach that requires neither the underlying dynamical equations nor Jacobian matrices. By employing a k-nearest-neighbor predictor over phase-coherent prediction horizons, the method computes the geometric mean of absolute prediction errors and directly extracts the negative Lyapunov exponent from the slope of the logarithmic error curve. Innovatively incorporating period-synchronized prediction and a slope consensus mechanism across multiple transient lengths, the technique substantially enhances robustness. Experiments demonstrate its efficacy: on the Logistic map, it accurately recovers 92 out of 112 negative exponents with MAE = 0.0253 and R² = 0.886; in a two-dimensional system without fixed points, independent estimates from x, y, and z observables achieve MAE ≈ 0.009–0.011 and R² ≈ 0.983–0.986.

Accurately estimating the maximal Lyapunov exponent from scalar time series is highly challenging in the absence of governing equations, tangent-space dynamics, and full state information. This work proposes the FEG-Pro framework, which leverages autocorrelation-guided sparse embedding and distance-weighted k-nearest neighbor multi-step prediction to analyze the finite-horizon slope of the logarithmic growth of geometric mean prediction errors. For the first time, error growth is treated as a structured profile, incorporating multidimensional diagnostic features such as curvature, residual roughness, monotonicity, and entropy of the error distribution. The method demonstrates strong performance on scalar observations from chaotic maps, the Mackey–Glass system, and the Lorenz-63 attractor, achieving close agreement with true Lyapunov exponents in near-linear regimes and retaining interpretable characteristics even under short-data conditions, thereby offering a novel paradigm for instability rate estimation and machine learning.

This work addresses the lack of a unified computational framework for analyzing non-ergodicity, modeling heavy-tailed dynamics, and studying decision-making under uncertainty in stochastic processes. To this end, we introduce an open-source Python library that, for the first time, integrates non-ergodicity diagnostics, simulation of heavy-tailed processes—such as multiplicative Lévy growth and memory-dependent mean-reverting dynamics—and agent-based experimentation within a single platform. Built upon the scientific Python ecosystem (NumPy/SciPy), the library supports end-to-end workflows including stochastic process definition, simulation, parameter inference, and partial solution of stochastic differential equations. Through several reproducible examples—ranging from heavy-tailed ensemble diffusion to pre-asymptotic fluctuation analysis—it substantially reduces boilerplate code and enhances both reproducibility and development efficiency in the study of time-averaged behaviors of complex stochastic systems.