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A Pikovsky

Publications and source records attributed to A Pikovsky.

18 recordsLinked to original sources
PubMed2005

Continuous approach for the random-field Ising chain.

We study the random-field Ising chain in the limit of strong exchange coupling. In order to calculate the free energy we apply a continuous Langevin-type approach. This continuous model can be solved exactly, whereupon we are able to locate the crossover between an exponential and a power-law decay of the free energy with increasing coupling strength. In terms of magnetization, this crossover restricts the validity of the linear scaling. The known analytical results for the free energy are recovered in the corresponding limits. The outcomes of numerical computations for the free energy are presented, which confirm the results of the continuous approach. We also discuss the validity of the replica method which we then utilize to investigate the sample-to-sample fluctuations of the finite size free energy.

Journal Article↗
PubMed2003

From multiplicative noise to directed percolation in wetting transitions.

A simple one-dimensional microscopic model of the depinning transition of an interface from an attractive hard wall is introduced and investigated. Upon varying a control parameter, the critical behavior observed along the transition line changes from a directed-percolation type to a multiplicative-noise type. Numerical simulations allow for a quantitative study of the multicritical point separating the two regions. Mean-field arguments and the mapping on yet a simpler model provide some further insight on the overall scenario.

Journal Article↗
PubMed2003

Multiscaling of noise-induced parametric instability.

We describe the statistical properties of growth rates of a linear oscillator driven by a parametric noise. We show that in general the fluctuations of local Lyapunov exponents are non-Gaussian and demonstrate multiscaling. Analytical calculations of the generalized Lyapunov exponents are complemented with approximative and numerical results; this allows us to identify the parameter range where the deviations from the Gaussian statistics become important.

Journal Article↗
PubMed2003

Controlling oscillator coherence by delayed feedback.

We demonstrate that the coherence of a noisy or chaotic self-sustained oscillator can be efficiently controlled by the delayed feedback. We develop a theory of this effect, considering noisy systems in the Gaussian approximation. We obtain a closed equation system for the phase diffusion constant and the mean frequency of oscillation. For weak feedback and strong noise, the theory is in good agreement with the numerics. We discuss possible applications of the effect for the synchronization control.

Journal Article↗
PubMed2002

System size resonance in coupled noisy systems and in the Ising model.

We consider an ensemble of coupled nonlinear noisy oscillators demonstrating in the thermodynamic limit an Ising-type transition. In the ordered phase and for finite ensembles stochastic flips of the mean field are observed with the rate depending on the ensemble size. When a small periodic force acts on the ensemble, the linear response of the system has a maximum at a certain system size, similar to the stochastic resonance phenomenon. We demonstrate this effect of system size resonance for different types of noisy oscillators and for different ensembles---lattices with nearest neighbors coupling and globally coupled populations. The Ising model is also shown to demonstrate the system size resonance.

Journal Article↗
PubMed2001

Noise-induced dynamics in bistable systems with delay.

Noise-induced dynamics of a prototypical bistable system with delayed feedback is studied theoretically and numerically. For small noise and magnitude of the feedback, the problem is reduced to the analysis of the two-state model with transition rates depending on the earlier state of the system. Analytical solutions for the autocorrelation function and the power spectrum have been found. The power spectrum has a peak at the frequency corresponding to the inverse delay time, whose amplitude has a maximum at a certain noise level, thus demonstrating coherence resonance. The linear response to the external periodic force also has maxima at the frequencies corresponding to the inverse delay time and its harmonics.

Models, Theoretical↗
PubMed2001

Comment on "Intermittency in chaotic rotations".

Lai et al. [Phys. Rev. E 62, R29 (2000)] claim that the angular velocity of the phase point moving along the chaotic trajectory in a properly chosen projection (the instantaneous frequency) is intermittent. Using the same examples, namely the Rössler and the Lorenz systems, we show the absence of intermittency in the dynamics of the instantaneous frequency. This is confirmed by demonstrating that the phase dynamics exhibits normal diffusion. We argue that the nonintermittent behavior is generic.

Letter↗
PubMed2001

Comment on "Simple approach to the creation of a strange nonchaotic attractor in any chaotic system".

We address the problem of existence of strange nonchaotic attractors (SNAs) in quasiperiodically forced dynamical systems. Recently, Shuai and Wong [Phys. Rev. E 59, 5338 (1999)] suggested a universal method for constructing a SNA in an arbitrary system possessing chaos. We demonstrate here that, in general, this method fails. For arbitrary systems, it gives a SNA only in a vicinity of transition to chaos. We discuss also a special example, where the method by Shuai and Wong indeed produces a SNA.

Letter↗
PubMed2001

Transition to coherence in populations of coupled chaotic oscillators: a linear response approach.

We consider the collective dynamics in an ensemble of globally coupled chaotic maps. The transition to the coherent state with a macroscopic mean field is analyzed in the framework of the linear response theory. The linear response function for the chaotic system is obtained using the perturbation approach to the Frobenius-Perron operator. The transition point is defined from this function by virtue of the self-excitation condition for the feedback loop. Analytical results for the coupled Bernoulli maps are confirmed by the numerics.

Journal Article↗
PubMed2001

Resolving clusters in chaotic ensembles of globally coupled identical oscillators.

Clustering in ensembles of globally coupled identical chaotic oscillators is reconsidered using a twofold approach. Stability of clusters towards "emanation" of the elements is described with the evaporation Lyapunov exponents. It appears that direct numerical simulations of ensembles often lead to spurious clusters that have positive evaporation exponents, due to a numerical trap. We propose a numerical method that surmounts the spurious clustering. We also demonstrate that clustering can be very sensitive to the number of elements in the ensemble.

Journal Article↗
PubMed2001

Lyapunov exponents in disordered chaotic systems: avoided crossing and level statistics.

The behavior of the Lyapunov exponents (LEs) of a disordered system consisting of mutually coupled chaotic maps with different parameters is studied. The LEs are demonstrated to exhibit avoided crossing and level repulsion, qualitatively similar to the behavior of energy levels in quantum chaos. Recent results for the coupling dependence of the LEs of two coupled chaotic systems are used to explain the phenomenon and to derive an approximate expression for the distribution functions of LE spacings. The depletion of the level spacing distribution is shown to be exponentially strong at small values. The results are interpreted in terms of the random matrix theory.

Journal Article↗
PubMed2001

Dynamic localization of Lyapunov vectors in Hamiltonian lattices.

The convergence of the Lyapunov vector toward its asymptotic shape is investigated in two different one-dimensional Hamiltonian lattices: the so-called Fermi-Pasta-Ulam and Phi(4) chains. In both cases, we find an anomalous behavior, i.e., a clear difference from the previously conjectured analogy with the Kardar-Parisi-Zhang equation. The origin of the discrepancy is eventually traced back to the existence of nontrivial long-range correlations both in space and time. As a consequence of this anomaly, we find that, in a Hamiltonian lattice, the largest Lyapunov exponent is affected by stronger finite-size corrections than standard space-time chaos.

Journal Article↗
PubMed2001

Transcritical riddling in a system of coupled maps.

The transition from fully synchronized behavior to two-cluster dynamics is investigated for a system of N globally coupled chaotic oscillators by means of a model of two coupled logistic maps. An uneven distribution of oscillators between the two clusters causes an asymmetry to arise in the coupling of the model system. While the transverse period-doubling bifurcation remains essentially unaffected by this asymmetry, the transverse pitchfork bifurcation is turned into a saddle-node bifurcation followed by a transcritical riddling bifurcation in which a periodic orbit embedded in the synchronized chaotic state loses its transverse stability. We show that the transcritical riddling transition is always hard. For this, we study the sequence of bifurcations that the asynchronous point cycles produced in the saddle-node bifurcation undergo, and show how the manifolds of these cycles control the magnitude of asynchronous bursts. In the case where the system involves two subpopulations of oscillators with a small mismatch of the parameters, the transcritical riddling will be replaced by two subsequent saddle-node bifurcations, or the saddle cycle involved in the transverse destabilization of the synchronized chaotic state may smoothly shift away from the synchronization manifold. In this way, the transcritical riddling bifurcation is substituted by a symmetry-breaking bifurcation, which is accompanied by the destruction of a thin invariant region around the symmetrical chaotic state.

Journal Article↗
PubMed2000

Scaling of lyapunov exponents of coupled chaotic systems

We develop a statistical theory of the coupling sensitivity of chaos. The effect was first described by Daido [Prog. Theor. Phys. 72, 853 (1984)]; it appears as a logarithmic singularity in the Lyapunov exponent in coupled chaotic systems at very small couplings. Using a continuous-time stochastic model for the coupled systems we derive a scaling relation for the largest Lyapunov exponent. The singularity is shown to depend on the coupling and the systems' mismatch. Generalizations to the cases of asymmetrical coupling and three interacting oscillators are considered, too. The analytical results are confirmed by numerical simulations.

Journal Article↗
PubMed2000

Critical point of tori collision in quasiperiodically forced systems

We report on a type of scaling behavior in quasiperiodically forced systems. On the parameter plane the critical point appears as a terminal point of the tori-collision bifurcation curve; its location is found numerically with high precision for two basic models, the forced supercritical circle map and the forced quadratic map. The hypothesis of universality, based on renormalization group arguments, is advanced to explain the observed scaling properties for the critical attractor and for the parameter plane arrangement in the neighborhood of the criticality.

Journal Article↗