Complete Synchronization and Pattern Selection through Amplitude Dynamics and Diffusion in Heterogeneous Oscillatory Media

Abstract: We study an intricate mechanism of pattern formation in globally coupled heterogeneous oscillatory media. In anodic electrochemical etching of silicon, the electrode surface splits into two amplitude-phase regions, while all oscillators remain frequency-locked. Additionally, the relative ratio of the pattern can be tuned via a coupling term. We introduce a heterogeneous, complex Ginzburg-Landau equation with global coupling to reproduce these patterns and study their formation. Neglecting diffusion shows that frequency entrainment arises from amplitude adaptation. Diffusion, on the other hand, enforces the selection of a unique cluster ratio. In quantitative agreement with simulations, a center-manifold reduction yields a Lyapunov functional that predicts the selected ratio. Both of these results establish a theoretical framework that connects experiment and theory. Moreover, they show how heterogeneity and diffusion convert degenerate cluster dynamics into robust pattern selection.