Homoclinic Mechanism of Synchronization of Periodic Oscillations (original) (raw)
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Recently, complete chaotic synchronization in coupled systems has been well studied. In this paper, we study complete synchronization in coupled periodic oscillators with diffusive and gradient couplings. Eight typical types of critical curve for the transverse Lyapunov exponent of standard mode, which give rise to different synchronization-desynchronization patterns, are classified. All possible desynchronous behaviors including steady state, periodic state, quasiperiodic state, lowdimensional chaotic state, and two types of high-dimensional chaotic state are identified, and two classical synchronization-desynchronizaiton bifurcations-the shortest wavelength bifurcation and Hopf bifurcation from synchronous periodic state-are classified.
On Phase Synchronization by Periodic Force in Chaotic Oscillators with Saddle Equilibria
International Journal of Bifurcation and Chaos, 2000
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 Poincaré 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.
Physica Scripta, 2008
We study the bifurcation structure and the synchronization of a double-well Duffing oscillator coupled to a single-well one and subjected to periodic forces. Using the amplitudes and the frequencies of these driving forces as control parameters, we show that our model presents phenomena which were not observed in a similar system but with identical potentials. In the regime of relatively weak coupling, bubbles of bifurcations and chains of symmetry-breaking are identified. For much stronger couplings, Hopf bifurcations born from orbits of higher periodicity, as well as subcritical and supercritical Neimark bifurcations emerge. Varying the coupling strength, we also find a threshold for which the system remains quasiperiodic. Moreover, tori-breakdown route to a strange non-chaotic attractor is another highlight of features found in this model. In two parameter diagrams, regions of chaos and quasiperiodicity are clearly identified. Finally, threshold parameters for which synchronization occurs have been found.
Homoclinic Bifurcation as a Mechanism of Chaotic Phase Synchronization
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This paper demonstrates a mechanism of chaotic phase synchronization in which the transition from asynchronous to synchronous chaos is associated with the collision of the asynchronous chaotic attractor with an unstable periodic orbit. This gives rise to a hysteretic transition with the two chaotic regimes coexisting over a certain parameter interval.
The Dynamics of Synchronization and Phase Regulation
This is a joint project begun in 1983 and first reported in (Abraham, 1989). We were inspired by Arthur Winfree and have in mind a number of applications to medical physiology and mathematical biology. The main theme is the role of the geometry of periodic attractors -- the shape of an attractive limit cycle and its isochrons -- in determining the phase synchrony of a network of heterogeneous oscillators (periodic or chaotic). We present a profusely illustrated review of the geometric theory of the synchronization of periodic attractors, such as biological oscillators, by periodic pulsatile forcing. We think of this tutorial as "Winfree 101" but a number of new insights are included.
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Physical Review Letters, 1997
In the work we investigate the bifurcational mechanism of the loss of stability of the synchronous chaotic regime in coupled identical systems. We show that loss of synchronization is a result of the sequence of soft bifurcations of saddle periodic orbits which induce the bubbling and riddling transitions in the system. A bifurcation of a saddle periodic orbit embedded in the chaotic attractor determines the bubbling transition. The phenomenon of riddled basins occurs through a bifurcation of a periodic orbit located outside the symmetric subspace.
Synchronization and desynchronization under the influence of quasiperiodic forcing
Physical Review E, 2003
We study the influence of quasiperiodic forcing on synchronization and desynchronization using two coupled quasiperiodically forced logistic maps as a paradigm. We show that due to the forcing the synchronization region in parameter space shrinks. The loss of transverse stability of the synchronized attractors leads to desynchronization. Two types of such blowout bifurcations are described, namely, the blowout bifurcations of synchronized quasiperiodic motion on invariant curves and synchronized strange nonchaotic attractors, both yielding desynchronized chaotic attractors.
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By varying the forcing frequency and amplitude of a periodically forced planar oscillator, we can obtain a rich variety of responses. Whenever the resonance regions that are known to exist for small amplitudes of forcing terminate, we show that a fixed-point Hopf bifurcation must be involved. The main tool, whose properties we discuss in detail, is a self-rotation number for orbits in the plane. We illustrate our theorems with a numerical model.
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Lithuanian Journal of Physics
The delayed feedback control method is applied to control a quasi-periodic motion. We consider a weakly nonlinear van der Pol oscillator subjected to a periodic force. Making use of the fact that the system is close to a supercritical Hopf bifurcation we are able to treat it analytically. Our analysis shows that the domain of synchronization of a forced self-sustained oscillator can be essentially extended by delayed feedback. The main results and the approach are of general importance since they are relevant to any forced self-sustained oscillator close to the supercritical Hopf bifurcation.
Synchronization of Homoclinic Chaos
Physical Review Letters, 2001
Homoclinic chaos is characterized by regular geometric orbits occurring at erratic times. Phase synchronization at the average repetition frequency is achieved by a tiny periodic modulation of a control parameter. An experiment has been carried on a CO 2 laser with feedback, set in a parameter range where homoclinic chaos occurs. Any offset of the modulation frequency from the average induces phase slips over long times. Perfect phase synchronization is recovered by slow changes of the modulation frequency based upon the sign and amplitude of the slip rate. Satellite synchronization regimes are also realized, with variable numbers of homoclinic spikes per period of the modulation.