A Deep Learning Framework for Identifying Weakly Chaotic, Strongly Chaotic, Resonant and Non-resonant Orbits in the Generalized Kicked Rotator

TL;DR

This paper addresses the challenge of distinguishing weakly chaotic, strongly chaotic, resonant, and non-resonant orbits in the generalized kicked rotator. It combines a hierarchical data-labeling pipeline based on the weighted Birkhoff average, Lyapunov exponents, and correlation dimension to reliably identify weak chaos, with a 2D-CNN that excels at classifying orbits from image-like trajectory representations, achieving over 99% accuracy. The approach demonstrates robust generalization across parameter regimes and offers a scalable framework for applying ML to other nonlinear dynamical systems. The work provides both a practical classification tool and insight into the dynamical structure of the GKR, potentially informing analyses of stability and resonance in complex systems.

Abstract

Identifying the types of orbits is an important topic in the study of chaotic dynamical systems. Beyond the well-known distinctly chaotic and regular motions, we focus on dynamics occurring in regions where regular and chaotic motions coexist and intertwine, which potentially indicating weakly chaotic orbits. This intermediate regime lies between strongly chaotic dynamics, characterized by exponential sensitivity and completely non-chaotic, purely regular behavior. In this paper, we introduce a deep learning framework to identify the types of orbits in the generalized kicked rotator system, which is challenging to study due to its complex and mixed chaotic behaviors. Our deep learning framework can be divided into two steps. First, we propose a novel algorithm that integrates the weighted Birkhoff average, the Lyapunov exponent, and the correlation dimension to identify weakly chaotic orbits. The algorithm categorizes orbits into four types: weakly chaotic, strongly chaotic, and regular orbits (which are further subdivided into resonant and non-resonant orbits), thereby creating a valuable dataset required for deep learning models. Second, we demonstrate that a well-trained 2D-CNN achieves high performance in accurately classifying orbits, largely because it effectively leverages the 2D structural information of the phase space relation. To our knowledge, this is the first paper to identify weakly chaotic orbits using deep learning methods. The method can be easily extend to other models.

Figures (20)

• Figure 1: The orbits of the GKR system for $K=0.01,0.03,0.1$ with $M=3$ (Top) and $M=5$ (Bottom) from left to right. Here, we take $400$ different initial conditions, randomly distributed throughout the phase space and iterate each of these initial conditions through the generalized kicked rotator $1000$ steps. Different colors represent orbits with different initial conditions.
• Figure 3: The histogram of Lyapunov exponents for the region $[0, 0.3] \times [0.2, 0.4]$ with the case $K=0.03, M=3$.
• Figure 4: The framework outlines the classification process of different types of orbits in the Eq.\ref{['1']}.
• Figure 5: Convergence behavior of the $WB_N$ series terms
• Figure 6: Left panel: $\Delta_N$ as a function of the number of iterations $N$ for 10 initial conditions. Right panel: $\mathrm{dig}_N$ as a function of $N$ for the corresponding orbits.