Chaotic synchronization of coupled electron-wave systems with backward waves

The chaotic synchronization of two electron-wave media with interacting backward waves and cubic phase nonlinearity is investigated in the paper. To detect the chaotic synchronization regime we use a new approach, the so-called time scale synchronization [Chaos, 14 (3) 603-610 (2004)]. This approach is based on the consideration of the infinite set of chaotic signals' phases introduced by means of continuous wavelet transform. The complex space-time dynamics of the active media and mechanisms of the time scale synchronization appearance are considered.Comment: 11 pages, 7 figures, published in CHAOS, 15 (2005) 01370

Two Scenarios of Breaking Chaotic Phase Synchronization

Two types of phase synchronization (accordingly, two scenarios of breaking phase synchronization) between coupled stochastic oscillators are shown to exist depending on the discrepancy between the control parameters of interacting oscillators, as in the case of classical synchronization of periodic oscillators. If interacting stochastic oscillators are weakly detuned, the phase coherency of the attractors persists when phase synchronization breaks. Conversely, if the control parameters differ considerably, the chaotic attractor becomes phase-incoherent under the conditions of phase synchronization break.Comment: 8 pages, 7 figure

On the attractors of two-dimensional Rayleigh oscillators including noise

We study sustained oscillations in two-dimensional oscillator systems driven by Rayleigh-type negative friction. In particular we investigate the influence of mismatch of the two frequencies. Further we study the influence of external noise and nonlinearity of the conservative forces. Our consideration is restricted to the case that the driving is rather weak and that the forces show only weak deviations from radial symmetry. For this case we provide results for the attractors and the bifurcations of the system. We show that for rational relations of the frequencies the system develops several rotational excitations with right/left symmetry, corresponding to limit cycles in the four-dimensional phase space. The corresponding noisy distributions have the form of hoops or tires in the four-dimensional space. For irrational frequency relations, as well as for increasing strength of driving or noise the periodic excitations are replaced by chaotic oscillations.Comment: 9 pages, 5 figure

Synchronization of chaotic oscillator time scales

This paper deals with the chaotic oscillator synchronization. A new approach to detect the synchronized behaviour of chaotic oscillators has been proposed. This approach is based on the analysis of different time scales in the time series generated by the coupled chaotic oscillators. It has been shown that complete synchronization, phase synchronization, lag synchronization and generalized synchronization are the particular cases of the synchronized behavior called as "time--scale synchronization". The quantitative measure of chaotic oscillator synchronous behavior has been proposed. This approach has been applied for the coupled Rossler systems.Comment: 29 pages, 11 figures, published in JETP. 100, 4 (2005) 784-79

Smooth and Non-Smooth Dependence of Lyapunov Vectors upon the Angle Variable on a Torus in the Context of Torus-Doubling Transitions in the Quasiperiodically Forced Henon Map

A transition from a smooth torus to a chaotic attractor in quasiperiodically forced dissipative systems may occur after a finite number of torus-doubling bifurcations. In this paper we investigate the underlying bifurcational mechanism which seems to be responsible for the termination of the torus-doubling cascades on the routes to chaos in invertible maps under external quasiperiodic forcing. We consider the structure of a vicinity of a smooth attracting invariant curve (torus) in the quasiperiodically forced Henon map and characterize it in terms of Lyapunov vectors, which determine directions of contraction for an element of phase space in a vicinity of the torus. When the dependence of the Lyapunov vectors upon the angle variable on the torus is smooth, regular torus-doubling bifurcation takes place. On the other hand, the onset of non-smooth dependence leads to a new phenomenon terminating the torus-doubling bifurcation line in the parameter space with the torus transforming directly into a strange nonchaotic attractor. We argue that the new phenomenon plays a key role in mechanisms of transition to chaos in quasiperiodically forced invertible dynamical systems.Comment: 24 pages, 9 figure

Generalized Synchronization in Ginzburg-Landau Equations with Local Coupling

The establishment of generalized chaotic synchronization in Ginzburg-Landau equations unidirectionally coupled at discrete points of space (local coupling) has been studied. It is shown that generalized syn-chronization regimes are also established with this type of coupling, but the necessary intensity of coupling issignificantly higher than that in the case of a spatially homogeneous couplingComment: 4 pages, 2 figure

Resonant enhancement of the jump rate in a double-well potential

We study the overdamped dynamics of a Brownian particle in the double-well potential under the influence of an external periodic (AC) force with zero mean. We obtain a dependence of the jump rate on the frequency of the external force. The dependence shows a maximum at a certain driving frequency. We explain the phenomenon as a switching between different time scales of the system: interwell relaxation time (the mean residence time) and the intrawell relaxation time. Dependence of the resonant peak on the system parameters, namely the amplitude of the driving force A and the noise strength (temperature) D has been explored. We observe that the effect is well pronounced when A/D > 1 and if A/D 1 the enhancement of the jump rate can be of the order of magnitude with respect to the Kramers rate.Comment: Published in J. Phys. A: Math. Gen. 37 (2004) 6043-6051; 6 figure

Levy stable noise induced transitions: stochastic resonance, resonant activation and dynamic hysteresis

A standard approach to analysis of noise-induced effects in stochastic dynamics assumes a Gaussian character of the noise term describing interaction of the analyzed system with its complex surroundings. An additional assumption about the existence of timescale separation between the dynamics of the measured observable and the typical timescale of the noise allows external fluctuations to be modeled as temporally uncorrelated and therefore white. However, in many natural phenomena the assumptions concerning the abovementioned properties of "Gaussianity" and "whiteness" of the noise can be violated. In this context, in contrast to the spatiotemporal coupling characterizing general forms of non-Markovian or semi-Markovian L\'evy walks, so called L\'evy flights correspond to the class of Markov processes which still can be interpreted as white, but distributed according to a more general, infinitely divisible, stable and non-Gaussian law. L\'evy noise-driven non-equilibrium systems are known to manifest interesting physical properties and have been addressed in various scenarios of physical transport exhibiting a superdiffusive behavior. Here we present a brief overview of our recent investigations aimed to understand features of stochastic dynamics under the influence of L\'evy white noise perturbations. We find that the archetypal phenomena of noise-induced ordering are robust and can be detected also in systems driven by non-Gaussian, heavy-tailed fluctuations with infinite variance.Comment: 7 pages, 8 figure

Bifurcations and chaos in semiconductor superlattices with a tilted magnetic field

We study the effects of dissipation on electron transport in a semiconductor superlattice with an applied bias voltage and a magnetic field that is tilted relative to the superlattice axis.In previous work, we showed that although the applied fields are stationary,they act like a THz plane wave, which strongly couples the Bloch and cyclotron motion of electrons within the lowest miniband. As a consequence,the electrons exhibit a unique type of Hamiltonian chaos, which creates an intricate mesh of conduction channels (a stochastic web) in phase space, leading to a large resonant increase in the current flow at critical values of the applied voltage. This phase-space patterning provides a sensitive mechanism for controlling electrical resistance. In this paper, we investigate the effects of dissipation on the electron dynamics by modifying the semiclassical equations of motion to include a linear damping term. We demonstrate that even in the presence of dissipation,deterministic chaos plays an important role in the electron transport process. We identify mechanisms for the onset of chaos and explore the associated sequence of bifurcations in the electron trajectories. When the Bloch and cyclotron frequencies are commensurate, complex multistability phenomena occur in the system. In particular, for fixed values of the control parameters several distinct stable regimes can coexist, each corresponding to different initial conditions. We show that this multistability has clear, experimentally-observable, signatures in the electron transport characteristics.Comment: 14 pages 11 figure