Determining the boundary of dynamical chaos in the generalized Chirikov map via machine learning

Abstract

We investigate the boundary separating regular and chaotic dynamics in the generalized Chirikov map, an extension of the standard map with a control parameter $K$ and a phase-shifted (phase $τ$) secondary sequence of kicks with a control parameter $K_α$. Lyapunov maps were computed across the parameter space $(K, K_α; τ)$ and used to train a convolutional neural network (ResNet18) for binary classification of dynamical regimes. The trained model reproduces the known critical control parameter $K_c$ for the onset of global chaos in the standard map and identifies two-dimensional boundaries $(K, K_α)$ in the generalized map for varying phase shifts $τ$. The results reveal systematic deformation of the boundary as $τ$ increases, highlighting the sensitivity of the system to phase modulation and demonstrating the ability of machine learning to extract interpretable features of complex Hamiltonian dynamics. This framework allows precise characterization of stability boundaries in nontrivial nonlinear systems.