Vladimir N. Belykh

Persistent clusters in lattices of coupled nonidentical chaotic systems

Two-dimensional (2D) lattices of diffusively coupled chaotic oscillators are studied. In previous work, it was shown that various cluster synchronization regimes exist when the oscillators are identical. Here, analytical and numerical studies allow us to conclude that these cluster synchronization regimes persist when the chaotic oscillators have slightly different parameters. In the analytical approach, the stability of almost-perfect synchronization regimes is proved via the Lyapunov function method for a wide class of systems, and the synchronization error is estimated. Examples include a 2D lattice of nonidentical Lorenz systems with scalar diffusive coupling. In the numerical study, it is shown that in lattices of Lorenz and Rossler systems the cluster synchronization regimes are stable and robust against up to 10%-15% parameter mismatch and against small noise.

Synchronization in asymmetrically coupled networks with node balance

We study global stability of synchronization in asymmetrically connected networks of limit-cycle or chaotic oscillators. We extend the connection graph stability method to directed graphs with node balance, the property that all nodes in the network have equal input and output weight sums. We obtain the same upper bound for synchronization in asymmetrically connected networks as in the network with a symmetrized matrix, provided that the condition of node balance is satisfied. In terms of graphs, the symmetrization operation amounts to replacing each directed edge by an undirected edge of half the coupling strength. It should be stressed that without node balance this property in general does not hold.

Cluster synchronization in oscillatory networks

Synchronous behavior in networks of coupled oscillators is a commonly observed phenomenon attracting a growing interest in physics, biology, communication, and other fields of science and technology. Besides global synchronization, one can also observe splitting of the full network into several clusters of mutually synchronized oscillators. In this paper, we study the conditions for such cluster partitioning into ensembles for the case of identical chaotic systems. We focus mainly on the existence and the stability of unique unconditional clusters whose rise does not depend on the origin of the other clusters. Also, conditional clusters in arrays of globally nonsymmetrically coupled identical chaotic oscillators are investigated. The design problem of organizing clusters into a given configuration is discussed.

Dynamics of Stochastically Blinking Systems. Part I: Finite Time Properties

We consider dynamical systems whose parameters are switched within a discrete set of values at equal time intervals. Similar to the blinking of the eye, switching is fast and occurs stochastically and independently for different time intervals. There are two time scales present in such systems, namely the time scale of the dynamical system and the time scale of the stochastic process. If the stochastic process is much faster, we expect the blinking system to follow the averaged system where the dynamical law is given by the expectation of the stochastic variables. We prove that, with high probability, the trajectories of the two systems stick together for a certain period of time. We give explicit bounds that relate the probability, the switching frequency, the precision, and the length of the time interval to each other. We discover the apparent presence of a soft upper bound for the time interval, beyond which it is almost impossible to keep the two trajectories together. This comes as a surprise in view of the known perturbation analysis results. From a probability theory perspective, our results are obtained by directly deriving large deviation bounds. They are more conservative than those derived by using the action functional approach, but they are explicit in the parameters of the blinking system.

Dynamics of Stochastically Blinking Systems. Part II: Asymptotic Properties

We study stochastically blinking dynamical systems as in the companion paper (Part I). We analyze the asymptotic properties of the blinking system as time goes to infinity. The trajectories of the averaged and blinking system cannot stick together forever, but the trajectories of the blinking system may converge to an attractor of the averaged system. There are four distinct classes of blinking dynamical systems. Two properties differentiate them: single or multiple attractors of the averaged system and their invariance or noninvariance under the dynamics of the blinking system. In the case of invariance, we prove that the trajectories of the blinking system converge to the attractor(s) of the averaged system with high probability if switching is fast. In the noninvariant single attractor case, the trajectories reach a neighborhood of the attractor rapidly and remain close most of the time with high probability when switching is fast. In the noninvariant multiple attractor case, the trajectory may escape to another attractor with small probability. Using the Lyapunov function method, we derive explicit bounds for these probabilities. Each of the four cases is illustrated by a specific example of a blinking dynamical system. From a probability theory perspective, our results are obtained by directly deriving large deviation bounds. They are more conservative than those derived by using the action functional approach, but they are explicit in the parameters of the blinking system.

HYPERBOLIC PLYKIN ATTRACTOR CAN EXIST IN NEURON MODELS

Strange hyperbolic attractors are hard to find in real physical systems. This paper provides the first example of a realistic system, a canonical three-dimensional (3D) model of bursting neurons, that is likely to have a strange hyperbolic attractor. Using a geometrical approach to the study of the neuron model, we derive a flow-defined Poincare map giving an accurate account of the systems dynamics. In a parameter region where the neuron system undergoes bifurcations causing transitions between tonic spiking and bursting, this two-dimensional map becomes a map of a disk with several periodic holes. A particular case is the map of a disk with three holes, matching the Plykin example of a planar hyperbolic attractor. The corresponding attractor of the 3D neuron model appears to be hyperbolic (this property is not verified in the present paper) and arises as a result of a two-loop (secondary) homoclinic bifurcation of a saddle. This type of bifurcation, and the complex behavior it can produce, have not been previously examined.

WHEN SYMMETRIZATION GUARANTEES SYNCHRONIZATION IN DIRECTED NETWORKS

We review and illustrate our recent results on globally stable synchronization in directed oscillator networks. We consider asymmetrically connected networks with node balance, the property that the sum of the coupling coefficients of all edges directed to a node equals the sum of the coupling coefficients of all the edges directed outward from the node. We show that for such directed but node balanced networks, it is sufficient to symmetrize all connections by replacing a unidirectional coupling with a bidirectional coupling of half the coupling strength. The synchronization condition for the symmetrized network then guarantees synchronization in the original directed network. By considering an example of coupled driven pendula, we show how to prove global stability of synchronization in a concrete unidirectional network. We also discuss the relation between local and global synchronization.

Synchronization in complex networks with blinking interactions

We propose a new model of small-world networks of cells with a time-varying coupling and study its synchronization properties. In each time interval of length /spl tau/ such a coupling is switched on with probability p and the corresponding switching random variables are independent for different links and for different times. At each moment the coupling corresponds to a small-world graph, but the shortcuts change from time interval to time interval, which is a good model for many real-world dynamical networks. We prove that for the blinking model, a few random shortcut additions significantly lower the synchronization threshold together with the effective characteristic path length. Short interactions between cells, as in the blinking model, are important in practice. To cite prominent examples, computers networked over the Internet interact by sending packets of information, and neurons in our brain interact by sending short pulses, called spikes. The rare interaction between arbitrary nodes in the network greatly facilitates synchronization without loading the network much. In this respect, we believe that it is more efficient than a structure of fixed random connections.

Dynamics of the finite-dimensional Kuramoto model: Global and cluster synchronization

Synchronization phenomena in networks of globally coupled non-identical oscillators have been one of the key problems in nonlinear dynamics over the years. The main model used within this framework is the Kuramoto model. This model shows three main types of behavior: global synchronization, cluster synchronization including chimera states and totally incoherent behavior. We present new sufficient conditions for phase synchronization and conditions for an asynchronous mode in the finite-size Kuramoto model. In order to find these conditions for constant and time varying frequency mismatch, we propose a simple method of comparison which allows one to obtain an explicit estimate of the phase synchronization range. Theoretical results are supported by numerical simulations.

Bistability of patterns of synchrony in Kuramoto oscillators with inertia

We study the co-existence of stable patterns of synchrony in two coupled populations of identical Kuramoto oscillators with inertia. The two populations have different sizes and can split into two clusters where the oscillators synchronize within a cluster while there is a phase shift between the dynamics of the two clusters. Due to the presence of inertia, which increases the dimensionality of the oscillator dynamics, this phase shift can oscillate, inducing a breathing cluster pattern. We derive analytical conditions for the co-existence of stable two-cluster patterns with constant and oscillating phase shifts. We demonstrate that the dynamics, that governs the bistability of the phase shifts, is described by a driven pendulum equation. We also discuss the implications of our stability results to the stability of chimeras.