Ragged synchronizability of coupled oscillators

We discuss synchronization thresholds in an array of nondiagonally coupled oscillators. We argue that nondiagonal coupling can cause the appearance or disappearance of desynchronous windows in the coupling parameter space. Such a phenomenon is independent of the motion character (periodic or chaotic) of the isolated node system. A mechanism governing this phenomenon is explained and its influence on the global network dynamics is analyzed.

Figure 1

Typical examples of the MSF ${\lambda }_T\left(\alpha \right)$ for real coupling: (a), (b) bottom-limited synchronous range $({\alpha }_1,\propto )$; (c) double-limited synchronous interval $({\alpha }_1,{\alpha }_2)$.

Figure 2

Chain of Duffing oscillators coupled by springs.

Figure 3

Bifurcation diagram of the MSF ${\lambda }_T\left(\alpha \right)$ versus the amplitude of forcing $q$ (a) and its cross sections $q=7.0$ (b) and 5.6 (c).

Figure 4

Desynchronizing mechanism: a projection from the MSF ${\lambda }_T\left(\alpha \right)$ diagram, via eigenvalues ${\gamma }_k$ of the connectivity matrix $\mathbf{G}$, to the bifurcation diagram of the synchronization error $e$ versus the coupling coefficient $\sigma$ for an array of $N=4$ oscillators, $q=5.6$. Desynchronous intervals are in gray. Complete synchronization takes place in the $\sigma$ ranges where $e$ approaches zero value.

Figure 5

Desynchronous $\sigma$ intervals (bubbles) associated with different transverse modes for a varying number $N$ of oscillators in the array, $q=5.6$. All the $\sigma$ ranges correspond to the interval $0<\alpha <4.0$.

Figure 6

Desynchronous $\alpha$ region (hatched) and the corresponding desynchronous $\sigma$ regions (in gray) associated with five different transverse modes, for an array of $N=10$ oscillators working (when uncoupled) in the periodic range of the parameter $q$.

Figure 7

(Color online) Projection from the MSF ${\lambda }_T\left(\alpha \right)$ diagram, via eigenvalues ${\gamma }_k$, to the bifurcation diagram of the synchronization error $e$ versus the coupling coefficient $\sigma$ for an array of $N=10$ oscillators, $q=0.8$. Desynchronous intervals are in gray.

Figure 8

Chain of Duffing oscillators with a shortcut.

Figure 9

Desynchronous $\sigma$ regions as a function of the parameter $q$ in an array of $N=10$ oscillators without a shortcut (in black) and the additional regions of desynchronization that appeared after the shortcut creation (in gray).