Localized synchronous patterns in weakly coupled bistable oscillators

TL;DR

This work analyzes localized time-periodic patterns in one-dimensional chains of weakly coupled bistable oscillators governed by a Ginzburg--Landau (Lambda--Omega) framework. It develops a matched asymptotic/bifurcation approach to classify localization geometry under weak coupling: dissipative coupling ($c=1$) yields in-phase, snaking branches of localized synchrony patterns, while conservative coupling ($c=i$) produces discrete isola branches with fixed phase offsets. A key finding is that frequency matching between coexisting oscillatory states is essential for localization; mismatches can destroy localized patterns. Application to a chain of mechanical oscillators and a harmonic-balance reduction clarifies how HBM predicts isola structures that may not persist in the full system, highlighting the role of coupling structure in shaping localized dynamics and suggesting avenues for controlled localization in engineered oscillator networks.

Abstract

Motivated by numerical continuation studies of coupled mechanical oscillators, we investigate branches of localized time-periodic solutions of one-dimensional chains of coupled oscillators. We focus on Ginzburg--Landau equations with nonlinearities of Lambda-Omega type and establish the existence of localized synchrony patterns in the case of weak coupling and weak-amplitude dependence of the oscillator periods. Depending on the coupling, localized synchrony patterns lie on a discrete stack of isola branches or on a single connected snaking branch.

Figures (7)

• Figure 1: Panel (i) illustrates our assumption that there are two oscillatory states ($r_-(\mu)$ is unstable, and $r_+(\mu)$ is stable) and one stationary stable state $r=0$ inside the bistability region $0<\mu<1$. The stationary state $r=0$ exhibits a Hopf bifurcation at $\mu=0$, while the two oscillatory states merge in a fold bifurcation at $\mu=1$. Panels (ii) and (iii) show numerical continuation results of, respectively, isola and snaking branches of \ref{['i:model']} with conservative ($c=\mathrm{i}$) and dissipative ($c=1$) coupling for $N=10$ nodes with coupling strength $\varepsilon=0.01$ and the nonlinearity given in \ref{['e:exnonl']}.
• Figure 2: Panels (i) and (ii) show how the localized synchrony patterns change along the isola and snaking branches, where the tall (blue), medium (green), and low (red) rectangles represent $r_+$, $r_-$, and $0$, respectively. Panel (iii) contains numerical continuation results that indicate how nodes are recruited along the snaking branch, starting with a single small-amplitude oscillator and ending with a fully oscillatory pattern.
• Figure 3: Panel (i) illustrates our parametrization of localized synchrony patterns along the snaking branch $\Gamma_0(N)$. Panel (ii) indicates how the localized synchrony patterns change along the snaking branch. The oscillatory states along the snaking branch share the same phase so that the phase differences $\phi_n$ vanish.
• Figure 4: We illustrate the isola branch $\Gamma_k$ for $k=2$. The colors indicate the phase difference between adjacent nodes with blue and red corresponding to $\phi_n=-\frac{\pi}{2}$ and $\phi_n=\frac{\pi}{2}$, respectively. In particular, the phase difference between consecutive nodes along the isola is always $-\frac{\pi}{2}$ except for the last node at $n=k+2$ which has phase difference $\frac{\pi}{2}$ with the node at $n=k+1$.
• Figure 5: Panels (i) and (ii) contain numerical continuation results for, respectively, a single oscillator and a chain of $N=20$ oscillators for \ref{['e:papa']} with $\varepsilon=0.01$. The frequencies of these solutions are shown in panel (iii). Panel (iv) contains two localized synchrony patterns along the branch shown in panel (ii), and we note that the nodes not shown have $x_n=0$ within computing accuracy.

Theorems & Definitions (6)

• Theorem 2.1: Snaking for dissipative coupling
• Lemma 2.3
• Definition 3.1
• Lemma 3.2: bergland
• Lemma 3.3
• proof