🤖 AI Summary
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.
📝 Abstract
Detecting transient chaos from scalar observations without governing equations represents a fundamental challenge in nonlinear dynamics. We propose a geometry-guided machine learning framework that unifies predictive trajectory divergence with macroscopic attractor morphology to track abrupt regime shifts. The methodology extracts a local instability scale via out-of-sample k-nearest neighbor forecast errors to establish the ML-FTLE estimator, subsequently mapping this temporal divergence onto a structural closeness matrix derived from a minimal dictionary of Poincare occupancy grids. By employing partial least squares regression, we extract a latent geometric component calibrated directly to the empirical finite-time Lyapunov spectrum, yielding the Poincare-based geometric-guided FTLE. Validation against analytical QR-FTLE baselines confirms that fusing topological state spaces with predictive divergence systematically improves continuous transition tracking. The Structural Similarity Index optimally resolves gradual damping, while Hausdorff Distance exhibits extreme resilience during abrupt phase-space collapses. Furthermore, macroscopic spatial discretization acts as a robust topological regularizer against additive Gaussian noise, preserving deterministic signatures even at moderate signal thresholds. This equation-free framework provides a highly accurate, noise-resilient diagnostic for monitoring structural transitions in complex non-stationary systems.