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

Abstract

A new type of intermittent behavior is described to occur near the boundary of phase synchronization regime of coupled chaotic oscillators. This mechanism, called ring intermittency, arises for sufficiently high initial mismatches in the frequencies of the two coupled systems. The laws for both the distribution and the mean length of the laminar phases versus the coupling strength are analytically deduced. A very good agreement between the theoretical results and the numerically calculated data is shown. We discuss how this mechanism is expected to take place in other relevant physical circumstances. Intermittent behavior is an ubiquitous phenomenon in nonlinear science. 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] or on-off intermittency [3,4]. One of the most general and interesting manifestations of the intermittent behavior can be observed near of the boundary of chaotic synchronization regimes. 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]).In this Letter we report that a new type of intermittent behavior is observed near the phase synchronization boundary of two unidirectionally coupled chaotic oscillators, when the natural frequencies of the two oscillators are sufficiently different from one another. The system under study is represented by a pair of unidirectionally coupled Rössler systems, whose equations read aṡy r , z r )] are the cartesian coordinates of the drive (the response) oscillator, dots stand for temporal derivatives, and ε is a parameter ruling the coupling strength. The other control parameters of Eq. (1) have been set to a = 0.15, p = 0.2, c = 10.0, in analogy with previous studies [14,15]. The ω r -parameter (representing the natural frequency of the response system) has been selected to be ω r = 0.95; the analogous parameter for the drive system has been fixed to ω d = 1.0. For such a choice of parameter values, both chaotic attractors of the drive and response systems are, at zero coupling strength, phase coherent. Furthermore, the boundary of the phase synchronization regime occur around ε c ≈ 0.124.The instantaneous phase of the ch...

Horizontal visibility graphs generated by type-I intermittency

“…The study and characterization of the onset mechanisms and main statistical properties of intermittency was carried out already a long time ago; Pomeau and Manneville [5] introduced a classification as types I-III for different kinds of intermittency. Subsequently, other types have been described and typified, such as on-off intermittency [6], ring intermittency [7], etc. Our objective here is to generate networks from the time series associated with intermittency and look at how this phenomenon translates into such a different setting, and then examine the manifestation of its properties in the new context.…”

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].…”

Horizontal Visibility graphs generated by type-II intermittency

“…Intermittent behaviour can indeed be observed experimentally in many situations such as Belousov-Zhabotinski chemical reactions, Rayleigh-Benard instabilities, or turbulence [13][14][15][16] and has been deeply studied in the context of nonlinear sciences. As a result of this, a characterization of the onset mechanisms and main statistical properties of intermittency have been described and typified: from the classification of types I, II and III intermittency by Pomeau and Manneville [17] to other more recent types such as on-off intermittency [18] or ring intermittency [19]. The theoretical description of type-I intermittency from a Horizontal Visibility perspective has been advanced recently [20].…”