Predicting Phase Ordering in Chaotic Maps and Coupled Map Lattices

Abstract

Coupled logistic maps exhibit collective ordering of their directional phases. As the system parameter varies, the directional phases can undergo a transition from an in-phase state to an anti-phase state, while the individual map trajectories remain chaotic. In this work, we propose a data-driven machine learning (ML) framework based on parameter-aware reservoir computing (PARC) to predict order-parameter dynamics in two representative systems: a logistic map and a two-dimensional coupled map lattice (CML). For the logistic map, the reservoir is trained using only pre-crisis time series data at bifurcation parameter $\mu$ values below the attractor-merging crisis ($\mu_0 = 3.6786$). The trained reservoir reconstructs the full bifurcation diagram and correctly predicts the transition in the directional order parameter $M(\mu)$, from an ordered state ($M \approx 0$) to a disordered state ($M \neq 0$) across the crisis point. For the CML, we exploit the spatial homogeneity of the lattice: a single reservoir is trained on the dynamics of one representative lattice site and is then replicated across all $L^2$ sites during prediction, where $L$=50. The replicated reservoir correctly predicts the transition from in-phase synchronization ($\theta \approx 1$) to anti-phase clustered states ($\theta \approx 0$) at $\mu \approx 3.82$, where $\theta$ quantifies phase coherence across lattice sites.