Synchronized Bimodal Amplitude Patterns in Heterogeneous Oscillatory Media -- Experiment and Theory

TL;DR

The paper investigates synchronized bimodal amplitude patterns in globally coupled heterogeneous oscillatory media, combining anodic silicon dissolution experiments with a heterogeneous complex Ginzburg–Landau model that includes mean-field coupling and diffusion. It shows that amplitude clustering occurs with complete frequency locking, and that the relative domain sizes can be tuned by a global coupling parameter; diffusion selects a unique cluster ratio, which is predicted by a Lyapunov functional derived from a center-manifold reduction. The work establishes a theoretical bridge between experiment and theory, revealing how heterogeneity and diffusion transform degenerate cluster dynamics into robust pattern selection and connecting cluster-singularity concepts from homogeneous systems to heterogeneous, diffusive media. These results advance understanding of amplitude-mediated synchronization and pattern formation in extended, heterogeneous oscillatory systems and point to broader implications for controlled pattern selection in reactive-diffusive media. The framework provides concrete, testable predictions about how to manipulate coupling and diffusion to steer multimodal patterns in experimental settings.

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.

Figures (8)

• Figure 1: Experimentally measured distribution of time-averaged amplitude (first row), snapshots of phase distribution (second row), and time-averaged frequency distribution (third row) on an electrode surface during anodic Si electrodissolution. From (a) to (d) the external resistance was increased: (a) $R=0.58 \rm \,k\Omega cm^2$, (b) $R=1.46 \rm \,k\Omega cm^2$, (c) $R=2.33 \rm \,k\Omega cm^2$, and (d) $R=3.65 \rm \,k\Omega cm^2$. The resistance was increased in steps, as shown in Table \ref{['tab:Ex_parameters']}. At each step, we waited 400 s after the initial transient state.
• Figure 2: Histograms of experimentally measured time-averaged amplitudes at four different resistances, (a) $R=0.58 \rm \,k\Omega cm^2$, (b) $R=1.46 \rm \,k\Omega cm^2$, (c) $R=2.33 \rm \,k\Omega cm^2$, and (d) $R=3.65 \rm \,k\Omega cm^2$.
• Figure 3: Relative size of the experimentally obtained high-amplitude region as a function of the resistance.
• Figure 4: Numerically obtained snapshots of the amplitude (first row) and phase (second row) distributions, and frequency distribution (third row) for $K=1.15, C_2=2,\epsilon=0.02,\sigma_0=0.3$. The $C_1$ parameter is adiabatically changed, and the simulation results are plotted for four different values of $C_1$. First column $C_1=-2.5$, second column $C_1=-2.1$, third column $C_1=-1.6$, and last column $C_1=-1.4$.
• Figure 5: Histograms of the amplitudes $r$ for for different values of the global coupling parameter $C_1$. The fitting has been performed using the kernel density estimator. Parameter values: $K=1.15, C_2=2,\epsilon=0.02,\sigma_0=0.3$, (a) $C_1=-2.5$, (b) $C_1=-2.1$, (c) $C_1=-1.6$, and (d) $C_1=-1.4$.