Characterizing the phase synchronization transition of chaotic oscillators

Abstract

The chaotic phase synchronization transition is studied in connection with the zero Lyapunov exponent. We propose a hypothesis that it is associated with a switching of the maximal finite-time zero Lyapunov exponent, which is introduced in the framework of a large deviation analysis. A noisy sine circle map is investigated to introduce this hypothesis and it is tested in an unidirectionally coupled Rössler system by using the covariant Lyapunov vector associated with the zero Lyapunov exponent.

Covariant Lyapunov vectors

“…(17)(18)(19) or the two refinements briefly discussed above, require the solutions of certain systems of linear homogeneous equations, which in turn depend on the forward and backward GS vectors. For direct subspace intersection, the first j forward and the last (N − j + 1) backward GS vectors are required to compute the first j CLV, while only the first j forward and (j − 1) backward vectors are needed by the refined algorithms -an improvement if one is interested in either the first or last CLVs.…”

“…In principle, CLVs can be obtained by: i) a "dynamical" approach, which consists in first determining the Lyapunov basis via the GS dynamics (23) forward in time and and then following the time-reversed evolution in the corresponding subspaces [9]) ; ii) a "static" approach which consists in computing the corresponding Oseledets's subspaces via (18), (19) and subsequently intersecting them according to Eq. (17) [7].…”

“…The situation has drastically changed a few years ago, when efficient algorithms have been developed [9,10,11], so that many scientists are now aware that CLVs can offer information on the local geometric structure of chaotic attractors, as opposed to LEs, which are powerful but global quantities. Several papers appeared, where CLVs have been successfully employed to better understand many aspects of chaotic dynamics [12,13,14,15,16,17,18,19,20,21,22]. Some of the relevant questions are touched in this Special Issue.…”

Information flow in heterogeneously interacting systems

“…Much attention has been paid to the phase dynamics of coupled chaotic oscillators (for e.g., Rosenblum et al 1996Rosenblum et al , 1997Belykh et al 2001;Osipov et al 2003;Li and Zheng 2007;Wilmer et al 2010;Ouchi et al 2011). Numbers of methods has been proposed for estimation of driver-response relationships from observed time series (Quiroga et al 2000;Schreiber 2000;Rosenblum and Pikovsky 2001;Paluš and Vejmelka 2007;Wilmer et al 2010).…”

Interatomic Coulombic decay widths of helium trimer: A diatomics-in-molecules approach

“…It has been observed mainly after photoionization or photoexcitation but has also been experimentally demonstrated after electron [12] and ion impact [13][14][15]. ICD was first predicted and observed after ionization in the inner-valence shell, but it was also demonstrated (i) after two-electron processes like simultaneous ionization-excitation [16,17] and double ionization or excitation [18,19], (ii) in cascades after Auger [20][21][22][23] and resonant Auger [24,25].…”