Desynchronization Waves and Localized Instabilities in Oscillator Arrays

We consider a ring of identical or near-identical coupled periodic oscillators in which the connections have randomly heterogeneous strength. We use the master stability function method to determine the possible patterns at the desynchronization transition that occurs as the coupling strengths are increased. We demonstrate Anderson localization of the modes of instability and show that such localized instability generates waves of desynchronization that spread to the whole array. Similar results should apply to other networks with regular topology and heterogeneous connection strengths.

Figure 1

Master stability function $\Psi (\alpha )$ versus $\alpha$ for Eqs. (5).

Figure 2

(a) Eigenvector $u_i$ for the largest eigenvalue $\lambda =3.61$ for a particular realization of the matrix $G$ in (6) with $N=500$. (b) Localization length ${\gamma }^{-1}$ calculated using Eq. (8).

Figure 3

Plots of the $x$ coordinate of oscillator $i$ versus the site index $i$, at times (a) 0, (b) 1400, (c) 2800, (d) 4200, (e) 5600, and (f) 10 000. All the plots have the same scale as (e).

Figure 4

Plots (a) and (b) show the $x$ coordinate of oscillator $i$ versus the site index $i$ for times 3750 and 9660. Plots (c) and (d) show the phase of oscillator $i$ at the same times as for (a) and (b), respectively. Compare with Eq. (10).

Figure 5

Each plot shows the $x$ coordinate of oscillator $i$ as a function of the site index $i$. The time is 0, 1400, 2800, 4200, 5600, and 9970 for plots (a) to (f) and similarly for plots (g) to (l). A parameter mismatch was introduced in the oscillators. (a)–(f): All the modes are stable. (g)–(l): The pattern is organized by an unstable mode as in Fig. 3f. All the plots have the same scale as (e).