Geometric transition of latent dynamics in intermittent chaos: SVD-based effective dimension for detection and proactive control

This paper investigates the geometric detection of latent dynamic transitions (where the latent space is defined as a finite-dimensional state space reconstructed via a delay-embedding Hankel matrix) in mutually coupled phase-locked loops exhibiting on-off intermittent chaos. We propose a data-driven approach to identify the transition of the effective dimension in latent dynamics as a precursor to physical bursting events. By applying Singular Value Decomposition to a Hankel matrix constructed from high-resolution oversampled data, we extract the effective dimension that reflects the dynamical shift of intrinsic variables-from a synchronized two-dimensional manifold to a higher-dimensional chaotic state-prior to the actual burst. This geometric indicator enables a proactive, event-driven intervention using Pyragas-type delayed feedback control, which is activated only when the detected dimension exceeds a critical threshold. Simulation results demonstrate that this geometric precursor effectively compensates for control latencies and substantially reduces accumulated control energy compared to conventional continuous feedback. The findings establish a methodological strategy for stabilizing infinite-dimensional delayed systems by exploiting early warning signals embedded in dimensional transitions of latent dynamics.