finite-time lyapunov estimation

This study addresses the challenge of detecting transient chaos and abrupt dynamical transitions in equation-free scalar time series by proposing a geometry-guided method for estimating finite-time Lyapunov exponents (FTLE). The approach uniquely integrates a structural proximity matrix derived from Poincaré section occupancy grids with predicted trajectory divergence, introduces macroscopic spatial discretization as a topological regularizer, and employs k-nearest-neighbor extrapolation errors to estimate local instability. Geometric latent variables aligned with empirical FTLE spectra are extracted via partial least squares regression. Experimental results demonstrate that, without access to governing equations, the method accurately tracks asymptotically damped dynamics and sudden phase-space collapses, significantly outperforming the QR-FTLE baseline under moderate signal-to-noise ratios while enhancing detection accuracy for continuous transitions and robustness to noise.

This work addresses the vulnerability of neural ODE models to adversarial attacks under input perturbations and their lack of interpretable input–output dynamics. It establishes, for the first time, a direct link between finite-time Lyapunov exponents (FTLE) and adversarial fragility, proposing an efficient regularization strategy that suppresses non-zero FTLE only during the early phase of input dynamics. This approach avoids the high computational cost of full-interval backward-backward propagation by characterizing the exponential divergence induced by perturbations, thereby organizing and stabilizing the dynamical evolution of neural ODEs. Experimental results demonstrate that the proposed FTLE-based regularization significantly enhances adversarial robustness while outperforming conventional full-interval regularization methods, achieving a superior trade-off between computational efficiency and model performance.

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.

Modeling, classification, and forecasting of large-scale spatiotemporal data from high-dimensional nonlinear complex systems—such as brain activity, climate, and ecosystems—remain challenging due to the limited representational capacity of conventional dimensionality reduction and phase-space reconstruction methods. Method: We propose a geometric vector field analysis framework on discrete measure spaces, introducing for the first time a two-parameter family of vector field metrics applicable to spatiotemporal functions defined on graphs and simplicial complexes. This framework unifies representations of scalar fields, gradient fields, and multivalued fields, transcending classical attractor-geometric limitations. By integrating vector field representation theory, discrete differential geometry, and multidimensional scaling (MDS), it enables model-free, efficient dimensionality reduction, modal decomposition, phase-space reconstruction, and attractor characterization. Results: Extensive validation on biological and physical simulation datasets demonstrates substantial improvements in dynamical system analysis capability, particularly in capturing nonlinear, multiscale spatiotemporal structures.

This study addresses the challenge of unified modeling of source/sink dynamics, cyclic behavior, and topology-constrained transport in complex dynamical systems. By integrating the continuous theory of Helmholtz–Hodge decomposition with discrete data-driven approaches, the authors propose a structured flow modeling paradigm based on graph vector fields (GVFs) and gradient–curl–harmonic decompositions over simplicial complexes. They develop a multi-level modeling strategy that spans from high expressivity to low computational cost through parameterized conditional models and a simplified Hodge representation. A cross-domain validation and diagnosis–simplification iterative pipeline is further designed to ensure model interpretability while achieving computational efficiency. The framework systematically elucidates the trade-offs among model complexity, interpretability, and predictive performance.

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.

This work addresses the limitations of traditional spatial network models that rely on stationary point processes and thus fail to capture the non-stationary spatiotemporal evolution of node density fields. The authors propose a fluid–spatiotemporal stochastic geometry framework, interpreting dynamic network topology as the hydrodynamic limit of discrete node configurations. By solving an inverse boundary-value problem, they identify latent dynamics and construct a scalar potential field grounded in the minimum kinetic energy principle from optimal transport theory, thereby unifying Lagrangian continuous transport with Eulerian discrete interference geometry. Their key contribution lies in establishing, for the first time, information flux vectors and material derivatives as sufficient statistics for macroscopic convection and kinematic predictors of topological deformation. This reveals intrinsic connections among coordination overhead, control signaling, and topological kinematic entropy, and yields analytical expressions for information flux and energy–density scaling laws in non-stationary networks, laying a theoretical foundation for capacity limits and control mechanism design in dynamic networks.

This work addresses the challenge of efficient and accurate ensemble forecasting in chaotic, turbulent, and stochastic systems by proposing a trajectory-aware surrogate modeling approach. The method uniquely learns the probability flow velocity directly from trajectory data, enabling modeling of trajectory-dependent dynamical quantities—such as fluxes and circulations—through first-order trajectory matching (FTM), without requiring estimation of conventional drift or diffusion coefficients, score functions, or explicit simulation. By integrating a simulation-free one-step training loss with stability analysis, the approach achieves high-fidelity ensemble predictions at low computational cost across diverse stochastic dynamical systems and partial differential equation benchmarks, significantly enhancing both trajectory resolution and predictive efficiency.

This work addresses the challenge of reconstructing full spatiotemporal dynamics from extremely sparse observations confined to short time windows. The authors propose a three-stage modular framework that first pretrains a shallow recurrent decoder (SHRED) on simulation data to map sparse sensor time series into a structured latent space, then learns temporal propagation laws governing the latent states, and finally enables joint bidirectional reconstruction and prediction using only minimal, short-duration real-world observations. The approach supports data assimilation and multiscale reconstruction, accommodates extreme constraints such as single-frame terminal inputs, and features a lightweight, deployment-efficient architecture. Validation across six complex physical systems—including turbulence, combustion transients, multiscale propulsion, and satellite environmental fields—demonstrates its superior performance over existing methods.

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.