Galina I. Strelkova

Correlation analysis of dynamical chaos

We study correlation and spectral properties of chaotic self-sustained oscillations of different types. It is shown that some classical models of stochastic processes can be used to describe behavior of autocorrelation functions of chaos. The influence of noise on chaotic systems is also considered.

Studying hyperbolicity in chaotic systems

Abstract On the basis of method [1] proposed for diagnosing 2-dimensional chaotic saddles we present a numerical procedure to distinguish hyperbolic and nonhyperbolic chaotic attractors in three-dimensional flow systems. This technique is based on calculating the angles between stable and unstable manifolds along a chaotic trajectory in R 3 . We show for three-dimensional flow systems that this serves as an efficient characteristic for exploring chaotic differential systems. We also analyze the effect of noise on the structure of angle distribution for both 2-dimensional invertible maps and a 3-dimensional continuous system.

Temporal intermittency and the lifetime of chimera states in ensembles of nonlocally coupled chaotic oscillators

We describe numerical results for the dynamics of networks of nonlocally coupled chaotic maps. Switchings in time between amplitude and phase chimera states have been first established and studied. It has been shown that in autonomous ensembles, a nonstationary regime of switchings has a finite lifetime and represents a transient process towards a stationary regime of phase chimera. The lifetime of the nonstationary switching regime can be increased to infinity by applying short-term noise perturbations.

Correlation analysis of the coherence-incoherence transition in a ring of nonlocally coupled logistic maps

We present numerical results for a set of bifurcations occurring at the transition from complete chaotic synchronization to spatio-temporal chaos in a ring of nonlocally coupled chaotic logistic maps. The regularities are established for the evolution of cross-correlations of oscillations in the network elements at the bifurcations related to the coupling strength variation. We reveal the distinctive features of cross-correlations for phase and amplitude chimera states. It is also shown that the effect of time intermittency between the amplitude and phase chimeras can be realized in the considered ensemble.

Amplitude and phase chimeras in an ensemble of chaotic oscillators

The transition from coherence to incoherence in an ensemble of nonlocally coupled logistic maps is considered. Chimera states of two types (amplitude and phase) are found. The mechanism and conditions of their appearance are determined.

Chaotic attractors of two-dimensional invertible maps

In this paper, we investigate the characteristics of quasihyperbolic attractors and quasiattractors in Invertible dissipative maps of the plane. The criteria which allow one to diagnose the indicated types of attractors in numerical experiments are formulated.

New type of chimera structures in a ring of bistable FitzHugh–Nagumo oscillators with nonlocal interaction

Abstract We study the spatiotemporal dynamics of a ring of nonlocally coupled FitzHugh–Nagumo oscillators in the bistable regime. A new type of chimera patterns has been found in the noise-free network and when isolated elements do not oscillate. The region of existence of these structures has been explored when the coupling range and the coupling strength between the network elements are varied.

“Coherence–incoherence” transition in ensembles of nonlocally coupled chaotic oscillators with nonhyperbolic and hyperbolic attractors

We consider in detail similarities and differences of the “coherence–incoherence” transition in ensembles of nonlocally coupled chaotic discrete-time systems with nonhyperbolic and hyperbolic attractors. As basic models we employ the Hénon map and the Lozi map. We show that phase and amplitude chimera states appear in a ring of coupled Hénon maps, while no chimeras are observed in an ensemble of coupled Lozi maps. In the latter, the transition to spatio-temporal chaos occurs via solitary states. We present numerical results for the coupling function which describes the impact of neighboring oscillators on each partial element of an ensemble with nonlocal coupling. Varying the coupling strength we analyze the evolution of the coupling function and discuss in detail its role in the “coherence–incoherence” transition in the ensembles of Hénon and Lozi maps.

New type of chimera and mutual synchronization of spatiotemporal structures in two coupled ensembles of nonlocally interacting chaotic maps

We study numerically the dynamics of a network made of two coupled one-dimensional ensembles of discrete-time systems. The first ensemble is represented by a ring of nonlocally coupled Henon maps and the second one by a ring of nonlocally coupled Lozi maps. We find that the network of coupled ensembles can realize all the spatio-temporal structures which are observed both in the Henon map ensemble and in the Lozi map ensemble while uncoupled. Moreover, we reveal a new type of spatiotemporal structure, a solitary state chimera, in the considered network. We also establish and describe the effect of mutual synchronization of various complex spatiotemporal patterns in the system of two coupled ensembles of Henon and Lozi maps.

INSTANTANEOUS PHASE METHOD IN STUDYING CHAOTIC AND STOCHASTIC OSCILLATIONS AND ITS LIMITATIONS

We study the behavior of an instantaneous phase and mean frequency of chaotic self-sustained oscillations and noise-induced stochastic oscillations. The results obtained by using various methods of the phase definition are compared to each other. We also compare two methods for describing synchronization of chaotic self-sustained oscillations, namely, instantaneous phase locking and locking of characteristic frequencies in power spectra. It is shown that the technique for diagnostics of the chaos synchronization based on the instantaneous phase locking is not universal.