http://www.maths.qmul.ac.uk/%7Elatora/report_06.pdf

Complex networks: Structure and dynamics

The synchronization of chaotic systems

Mean phase coherence as a measure for phase synchronization and its application to the EEG of epilepsy patients

We agree with the Comment by Duckrow and Albano [Phys. Rev. E 67, 063901 (2003)] that mutual information, estimated with an optimized algorithm, can be a useful tool for studying synchronization in real data. However, we point out that the improvement they found is mainly due to an interesting but nonstandard embedding technique used, and not so much due to the algorithm used for the estimation of mutual information itself. We also address the issue of stationarity of electroencephalographic (EEG) data.

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Reply to ``Comment on `Performance of different synchronization measures in real data: A case study on electroencephalographic signals' ''

We study the synchronization between left and right hemisphere rat electroencephalographic (EEG) channels by using various synchronization measures, namely nonlinear interdependences, phase synchronizations, mutual information, cross correlation, and the coherence function. In passing we show a close relation between two recently proposed phase synchronization measures and we extend the definition of one of them. In three typical examples we observe that except mutual information, all these measures give a useful quantification that is hard to be guessed beforehand from the raw data. Despite their differences, results are qualitatively the same. Therefore, we claim that the applied measures are valuable for the study of synchronization in real data. Moreover, in the particular case of EEG signals their use as complementary variables could be of clinical relevance.

Performance of different synchronization measures in real data: A case study on electroencephalographic signals

In periodically driven chaotic dynamical systems with a broad distribution of intrinsic time scales, perfect phase synchronization cannot be reached. Long segments of evolution during which the phase of a chaotic variable follows the phase of the driving force are interrupted by short segments of phase drift. We demonstrate that this drift is another short-lived synchronized state; its onset is caused by the passage near the long unstable periodic orbits whose frequencies are locked by external force in ratios different from 1:1.

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Alternating Locking Ratios in Imperfect Phase Synchronization

We demonstrate that the dynamics of phase synchronization in a chaotic system under weak periodic forcing depends crucially on the distribution of intrinsic characteristic times of this system. Under the external periodic action, the frequency of every unstable periodic orbit is locked to the frequency of the force. In systems which in the autonomous case displays nearly isochronous chaotic rotations, the locking ratio is the same for all periodic orbits; since a typical chaotic orbit wanders between the periodic ones, its phase follows the phase of the force. For the Lorenz attractor with its unbounded times of return onto a Poincare surface, such state of perfect phase synchronization is inaccessible. Analysis with the help of unstable periodic orbits shows that this state is replaced by another one, which we call "imperfect phase synchronization," and in which we observe alternation of temporal segments, corresponding to different rational values of frequency lockings.

Phase synchronization in the forced Lorenz system.

A periodic action onto a chaotic system can result in phase synchronization: Although the oscillations remain chaotic, their pace is prescribed by the rhythm of the force. We show how this adjustment of characteristic timescales to the period of the forcing is violated when, due to the presence of a saddle point in the autonomous attractor, the time intervals between returns onto a Poincare plane are unbounded. In such a situation, the segments of a chaotic trajectory in which its phase follows the phase of the force, can alternate with short segments of phase-locking in ratios different from 1:1. This phenomenon is demonstrated for several nonlinear systems.

On Phase Synchronization by Periodic Force in Chaotic Oscillators with Saddle Equilibria

In real physical problems, external parameters can only approximately be viewed as constant. A gradual variation of parameters remains harmless as long as it does not produce qualitative changes in the behavior of the system. However, it may happen that a time-dependent parameter crosses a critical value beyond which the attracting state changes. The interaction of the slow parameter variation with the fast rate of motions in the phase space is the cause of phenomena known under the name of ‘‘dynamic bifurcations’’ ~see, e.g., @1#!. One of its peculiarities is the bifurcation delay: the system fails to ‘‘notice’’ the onset of instability and tracks for a long time the unstable branch of states, before exhibiting a swift transition to another state. The influence of a slow monotonic parameter variation ~‘‘sweep’’ ! on the sequence of period-doubling bifurcations was described in @2# and later investigated in detail in @3#. Although the obtained bifurcation diagrams resemble the familiar diagram of a logistic mapping @4,5#, some differences arise, due to the inevitable presence of transients. With respect to the usual perioddoubling scenario, a certain similarity between the sweep and the action of noise has been established: under a nonzero sweep rate, only a finite number of doubling bifurcations is observable. Theoretical predictions were confirmed by experiments on electronic circuits and lasers with monotonically varied characteristics @6,7#.

Eun-Hyoung Park

Papers

Alternating Locking Ratios in Imperfect Phase Synchronization

Phase synchronization in the forced Lorenz system.

On Phase Synchronization by Periodic Force in Chaotic Oscillators with Saddle Equilibria

Self-induced slow-fast dynamics and swept bifurcation diagrams in weakly desynchronized systems.