Attractor-Repeller Collision and Eyelet Intermittency at the Transition to Phase Synchronization

Abstract

The chaotically driven circle map is considered as the simplest model of phase synchronization of a chaotic continuous-time oscillator by external periodic force. The phase dynamics is analyzed via phase-locking regions of the periodic cycles embedded in the strange attractor. It is shown that full synchronization, where all the periodic cycles are phase locked, disappears via the attractor-repeller collision. Beyond the transition an intermittent regime with exponentially rare phase slips, resulting from the trajectory's hits on an eyelet, is observed.

Multistable behavior above synchronization in a locally coupled Kuramoto model

“…Coupled systems with local interactions are of special importance. In particular the Kuramoto model [9] in its local version, i.e., the locally coupled Kuramoto model (LCKM), where individually linear oscillators behave chaotically under the effect of nonlinear local interactions, has raised attention since most features of systems with phase coupling appear in this particularly simple model [10][11][12][13][14].…”

Length distribution of laminar phases for type-I intermittency in the presence of noise

“…Its arousal and main statistical properties have been studied and characterized already since long time ago, and different types of intermittency have been classified as types I-III [1,2], on-off intermittency [3,4,5,6], eyelet intermittency [7,8,9] and ring intermittency [10].…”

Ring Intermittency in Coupled Chaotic Oscillators at the Boundary of Phase Synchronization

“…Indeed, close to the threshold parameter values for which the coupled systems show synchronized dynamics, it is observed that the de-synchronization mechanism involves persistent intermittent time intervals during which the synchronized oscillations are interrupted by the non-synchronous behavior. These pre-transitional intermittencies have been described in details for the case of lag synchronization [5,6,7] and for generalized synchronization [8], and their main statistical properties (following those of the on-off intermittency) have been shown to be common to other relevant physical processes.As far as intermittency phenomena near the phase synchronization onset are concerned, two types of intermittent behavior have been observed so far [9,10,11,12], namely the type-I intermittency and the super-long laminar behavior (so called "eyelet intermittency" [13]). …”

“…By analyzing the statistics of the laminar phases, it is found that the intermittent type behavior described in Refs. [9,10,11,12,13] takes place only for small differences in the natural frequencies of the drive and response systems. In particular, the eyelet intermittent phenomenon occurs in the range ω d = 0.90 ÷ 0.98.…”