Search Results (21 total)

We present results of physical experiments where we measure the autocorrelation function (ACF) and the spectral linewidth of the basic frequency of a spiral chaotic attractor in a generator with inertial nonlinearity both without and in the presence of external noise. It is shown that the ACF of spiral attractors decays according to an exponential law with a decrement which is defined by the phase diffusion coefficient. It is also established that the evolution of the instantaneous phase can be approximated by a Wiener random process.

Models of active Brownian motion in two-dimensional (2D) systems developed earlier are investigated with respect to the influence of linear attracting forces and external noise. Our consideration is restricted to the case that the driving is rather weak and that the forces show only weak deviations from radial symmetry. In this case an analytical study of the bifurcations of the system is possible. We show that in the presence of external linear forces with only small deviations from radial symmetry, the system develops rotational excitations with left-right symmetry, corresponding to limit cycles in the 4D phase space, the corresponding distribution has the form of a hoop or a tire in the 4D space. In the last part we apply the theory to swarms of Brownian particles that are held together by weak and attracting forces, which lead to cluster formation. Since near the center the potential is at least approximately parabolic and near to the radial symmetry, the swarm develops rotational modes of motion with left-right symmetry.

The recently proposed approach to detect synchronization from univariate data is applied to heart-rate-variability (HRV) data from ten healthy humans. The approach involves introducing angles for return times map and studying their behavior. For filtered human HRV data, it is demonstrated that: (i) in many of the subjects studied, interactions between different processes within the cardiovascular system can be considered as weak, and the angles can be well described by the derived model; (ii) in some of the subjects the strengths of the interactions between the processes are sufficiently large that the angles map has a distinctive structure, which is not captured by our model; (iii) synchronization between the processes involved can often be detected; (iv) the instantaneous radii are rather disordered.

A general approach is developed for the detection of phase relationships between two or more different oscillatory processes interacting within a single system, using one-dimensional time series only. It is based on the introduction of angles and radii of return times maps, and on studying the dynamics of the angles. An explicit unique relationship is derived between angles and the conventional phase difference introduced earlier for bivariate data. It is valid under conditions of weak forcing. This correspondence is confirmed numerically for a nonstationary process in a forced Van der Pol system. A model describing the angles’ behavior for a dynamical system under weak quasiperiodic forcing with an arbitrary number of independent frequencies is derived.

We study the relaxation to an invariant probability measure on quasihyperbolic and nonhyperbolic chaotic attractors in the presence of noise. We also compare different characteristics of the rate of mixing and show numerically that the rate of mixing for nonhyperbolic chaotic attractors can significantly change under the influence of noise. A mechanism of the noise influence on mixing is presented, which is associated with the dynamics of the instantaneous phase of chaotic trajectories. We also analyze how the synchronization effect can influence the rate of mixing in a system of two coupled chaotic oscillators.

The subject of our study is clustering in a population of excitable systems driven by Gaussian white noise and with randomly distributed coupling strength. The cluster state is frequency-locked state in which all functional units run at the same noise-induced frequency. Cooperative dynamics of this regime is described in terms of effective synchronization and noise-induced coherence.

We study the influence of external noise on the relaxation to an invariant probability measure for two types of nonhyperbolic chaotic attractors, a spiral (or coherent) and a noncoherent one. We find that for the coherent attractor the rate of mixing changes under the influence of noise, although the largest Lyapunov exponent remains almost unchanged. A mechanism of the noise influence on mixing is presented which is associated with the dynamics of the instantaneous phase of chaotic trajectories. This also explains why the noncoherent regime is robust against the presence of external noise.

We study numerically the effects of noise and periodic forcings on cluster synchronization in a chain of Van der Pol oscillators. We generalize the notion of effective synchronization to the case of a spatially extended system. It is shown that the structure of synchronized clusters can be effectively controlled by applying local external forcings. The effect of amplitude relations on the phase dynamics is also explored.

A novel approach is suggested for detecting the presence or absence of synchronization between two or three interacting processes with different time scales in univariate data. It is based on an angle-of-return-time map. A model is derived to describe analytically the behavior of angles for a periodic oscillator under weak periodic and quasiperiodic forcing. An explicit connection is demonstrated between the return angle and the phase of the external periodic forcing. The technique is tested on simulated nonstationary data and applied to human heart rate variability data.

Considering two different mathematical models describing chaotic spiking phenomena, namely, an integrate-and-fire and a threshold-crossing model, we discuss the problem of extracting dynamics from interspike intervals (ISIs) and show that the possibilities of computing the largest Lyapunov exponent (LE) from point processes differ between the two models. We also consider the problem of estimating the second LE and the possibility to diagnose hyperchaotic behavior by processing spike trains. Since the second exponent is quite sensitive to the structure of the ISI series, we investigate the problem of its computation.

We analyze effects of bounded white and colored noise on nonhyperbolic chaotic attractors in two-dimensional invertible maps. It is shown that first the nonhyperbolic nature is kept even in the presence of strong noise, but secondly already due to weak noise some properties of nonhyperbolic chaos can become similar to those of hyperbolic and almost hyperbolic chaos. We also estimate the stationary probability measure of noisy nonhyperbolic attractors. For this purpose two different methods for calculating the probability density are applied and the obtained results are compared in detail.

In this paper we estimate dynamical characteristics of chaotic attractors from sequences of threshold-crossing interspike intervals, and study how the choice of the threshold level (which sets the equation of a secant plane) influences the results of the numerical computations. Under quite general conditions we show that the largest Lyapunov exponent can be estimated from a series of return times to the secant plane, even in the case when some of the loops of the phase space trajectory fail to cross this plane.

In this paper we investigate the motion of a particle in the overdamped one-dimensional system with a spatially periodic potential under the influence of a sinusoidal wave and dichotomic (binary) noise. We demonstrate the effect of synchronization between the mean velocity of a particle and the phase velocity of the running wave controlled by the noise. The results of numerical simulation are in good agreement with a full-scale experiment.

We reconstruct the largest Lyapunov exponent and fractal dimension of a chaotic attractor using threshold-crossing interspike intervals alone. We show that in certain cases one may reconstruct from this data a set looking very similar to the initial attractor. We also give an explanation of this possibility based on the concept of instantaneous frequency.

We propose a method for restoration of time-dependent control parameters of dynamical system using the technique of global reconstruction. The technique presented is applied to multichannel confidential communication by means of parameter modulation.

Synchronization of two symmetrically coupled Lorenz systems, each of them considered a chaotic bistable system, is investigated numerically. A phenomenon of synchronization of the mean frequencies of switchings in coupled chaotic bistable systems is found. Bifurcations taking place in the system are analyzed. It is shown that there is the region on the “coupling-detuning” parameter plane where the mean frequencies of switchings coincide with a certain accuracy.

We investigate complex dynamics along the chain of quasiperiodically forced circle maps. We present numerical evidence for the development of a strange nonchaotic behavior within a wide range of spatial parameters. The bifurcations and properties of nontrivial attracting sets are studied.

We discuss the appearance of a strange nonchaotic attractor in a wide class of dynamical systems when the destruction of ergodic torus takes place.

We demonstrate that a strange nonchaotic attractor can be realized not only in quasiperiodically driven systems but also in autonomous and periodically forced systems. We show that the destruction of an ergodic torus via a band-merging crisis and the appearance of a strange nonchaotic attractor are applicable to a wide class of dynamical systems. © 1996 The American Physical Society.

The electrical activity of the heart on the short time range is studied numerically on the base of high-resolution electrocardiograms. We find that the low-amplitude part of the signal is well approximated by a superposition of two time exponents, one of them being complex. This serves as a justification to embed the whole process into a low-dimensional space. A combination of a noise reduction with time delay technique recovers a phase portrait in four-dimensional space. Its fine structure is resolved by projecting into a three-dimensional subspace, where the process resembles a nearly homoclinic motion in a system with a saddle-focus fixed point. A statistical description based on the computation of respective Shannon entropies provides a sharp distinction between healthy persons and patients with high risk for sudden cardiac death. © 1996 The American Physical Society.

We present the results of the computer simulation of the dynamics of an invertible two-dimensional ring map with quasiperiodic excitation. The bifurcation diagram of the system on the parameter plane has been constructed. We analyze the mechanisms of the destruction of quasiperiodic regimes and the role of strange nonchaotic attractors (SNA) in this process. Two mechanisms of SNA appearance are discussed. We verify the existence of SNA via bifurcational analysis of the approximating sets of the attractor.