Basin sizes depend on stable eigenvalues in the Kuramoto model.

We show that for the Kuramoto model (with identical phase oscillators equally coupled), its global statistics and size of the basins of attraction can be estimated through the eigenvalues of all stable (frequency) synchronized states. This result is somehow unexpected since, by doing that, one could just use a local analysis to obtain the global dynamic properties. But recent works based on Koopman and Perron-Frobenius operators demonstrate that the global features of a nonlinear dynamical system, with some specific conditions, are somehow encoded in the local eigenvalues of its equilibrium states.

Exponential distribution of lifetimes for transient bursting states in coupled noisy excitable systems.

The phenomenon of transient bursting, caused by additive noise in a set of two coupled FitzHugh-Nagumo oscillators, is studied by direct numerical integration and by measurements in the analog electronic circuit. In the parameter region where the unique global attractor of the deterministic system is the state of rest, introduction of low or moderate intensity fluctuations into the voltage dynamics results in the onset of a transient bursting state: sequences of intermittent bursts (patches of spikes), followed by ultimate relaxation to the equilibrium. Like genuine deterministic bursting, this behavior has its origin in the slow-fast character of the underlying dynamics.

Emergence and stability of periodic two-cluster states for ensembles of excitable units.

We study dynamics in ensembles of identical excitable units with global repulsive interaction. Starting from active rotators with additional higher order Fourier modes in on-site dynamics, we observe, at sufficiently strong repulsive coupling, large-scale collective oscillations in which the elements form two separate clusters. Transitions from quiescence to clustered oscillations are caused by global bifurcations involving the unstable clustered steady states.

Dynamics of spontaneous activity in random networks with multiple neuron subtypes and synaptic noise : Spontaneous activity in networks with synaptic noise.

Spontaneous cortical population activity exhibits a multitude of oscillatory patterns, which often display synchrony during slow-wave sleep or under certain anesthetics and stay asynchronous during quiet wakefulness. The mechanisms behind these cortical states and transitions among them are not completely understood. Here we study spontaneous population activity patterns in random networks of spiking neurons of mixed types modeled by Izhikevich equations.

Subdiffusive and superdiffusive transport in plane steady viscous flows.

Deterministic transport of passive tracers in steady laminar plane flows of incompressible viscous fluids through lattices of solid bodies or arrays of steady vortices can be anomalous. Motion along regular patterns of streamlines is often aperiodic: Repeated slow passages near stagnation points and/or solid surfaces serve for eventual decorrelation. Singularities of passage times near the obstacles, dictated by the boundary conditions, affect the character of transport anomalies: Flows past arrays of vortices are subdiffusive whereas tracers advected through lattices of solid obstacles can feature superdiffusion.

Transport on intermediate time scales in flows with cat's eye patterns.

We consider the advection-diffusion transport of tracers in a one-parameter family of plane periodic flows where the patterns of streamlines feature regions of confined circulation in the shape of "cat's eyes," separated by meandering jets with ballistic motion inside them. By varying the parameter, we proceed from the regular two-dimensional lattice of eddies without jets to the sinusoidally modulated shear flow without eddies. When a weak thermal noise is added, i.

Chimeras and complex cluster states in arrays of spin-torque oscillators.

We consider synchronization properties of arrays of spin-torque nano-oscillators coupled via an RC load. We show that while the fully synchronized state of identical oscillators may be locally stable in some parameter range, this synchrony is not globally attracting. Instead, regimes of different levels of compositional complexity are observed.

Anomalous transport in cellular flows: The role of initial conditions and aging.

We consider the diffusion-advection problem in two simple cellular flow models (often invoked as examples of subdiffusive tracer motion) and concentrate on the intermediate time range, in which the tracer motion indeed may show subdiffusion. We perform extensive numerical simulations of the systems under different initial conditions and show that the pure intermediate-time subdiffusion regime is only evident when the particles start at the border between different cells, i.e.

Mechanisms of Self-Sustained Oscillatory States in Hierarchical Modular Networks with Mixtures of Electrophysiological Cell Types.

In a network with a mixture of different electrophysiological types of neurons linked by excitatory and inhibitory connections, temporal evolution leads through repeated epochs of intensive global activity separated by intervals with low activity level. This behavior mimics "up" and "down" states, experimentally observed in cortical tissues in absence of external stimuli. We interpret global dynamical features in terms of individual dynamics of the neurons.

Onset of time dependence in ensembles of excitable elements with global repulsive coupling.

We consider the effect of global repulsive coupling on an ensemble of identical excitable elements. An increase of the coupling strength destabilizes the synchronous equilibrium and replaces it with many attracting oscillatory states, created in the transcritical heteroclinic bifurcation. The period of oscillations is inversely proportional to the distance from the critical parameter value.

Sustained oscillations, irregular firing, and chaotic dynamics in hierarchical modular networks with mixtures of electrophysiological cell types.

The cerebral cortex exhibits neural activity even in the absence of external stimuli. This self-sustained activity is characterized by irregular firing of individual neurons and population oscillations with a broad frequency range. Questions that arise in this context, are: What are the mechanisms responsible for the existence of neuronal spiking activity in the cortex without external input? Do these mechanisms depend on the structural organization of the cortical connections? Do they depend on intrinsic characteristics of the cortical neurons? To approach the answers to these questions, we have used computer simulations of cortical network models.

Nucleation in bistable dynamical systems with long delay.

In an asymmetric bistable dynamical system with delayed feedback, one of the stable states is usually "stronger" than the other one: The system relaxes to it not only from close initial conditions, but also from oscillatory initial configurations which contain epochs of stay near both attractors. However, if the initial nucleus of the stronger phase is shorter than a certain critical value, it shrinks, and the weaker state is established instead. We observe this effect in a paradigmatic model and in an experiment based on a bistable semiconductor laser and characterize it in terms of scaling laws governing its asymptotic properties.

Comment on "Time-averaged properties of unstable periodic orbits and chaotic orbits in ordinary differential equation systems".

A recent paper claims that mean characteristics of chaotic orbits differ from the corresponding values averaged over the set of unstable periodic orbits, embedded in the chaotic attractor. We demonstrate that the alleged discrepancy is an artifact of the improper averaging. Since the natural measure is nonuniformly distributed over the attractor, different periodic orbits make different contributions into the time averages.

Stochastic hierarchical systems: excitable dynamics.

We present a discrete model of stochastic excitability by a low-dimensional set of delayed integral equations governing the probability in the rest state, the excited state, and the refractory state. The process is a random walk with discrete states and nonexponential waiting time distributions, which lead to the incorporation of memory kernels in the integral equations. We extend the equations of a single unit to the system of equations for an ensemble of globally coupled oscillators, derive the mean field equations, and investigate bifurcations of steady states.

Ferrofluidic torsional pendulum driven by oscillating magnetic field.

A thin disk-shaped container filled with a ferrofluid and suspended in a horizontal linearly polarized ac magnetic field can perform torsional vibrations around its vertical diameter. In contrast to a recently studied spherical pendulum, the cell is sensitive to the field direction: It exposes its edge to the stationary or slowly oscillating magnetic field, and its broad side to the field of a high frequency. When the amplitude of the latter field is increased, the state of rest gets destabilized, yielding to oscillations near the equilibrium.

Nonlinear dynamics of a ferrofluid pendulum.

A ferrofluid torsion pendulum in an oscillating magnetic field exhibits a rich variety of nonlinear self-oscillatory regimes. The dynamics is governed by the system of coupled differential equations for the in- and off-axis components of the fluid magnetization and the pendulum angular deflection. In the limiting case of high driving frequency, the system reduces to the sole Rayleigh-type equation.

Phase synchronization of chaotic oscillations in terms of periodic orbits.

We consider phase synchronization of chaotic continuous-time oscillator by periodic external force. Phase-locking regions are defined for unstable periodic cycles embedded in chaos, and synchronization is described in terms of these regions. A special flow construction is used to derive a simple discrete-time model of the phenomenon.

Role of unstable periodic orbits in phase and lag synchronization between coupled chaotic oscillators.

An increase of the coupling strength in the system of two coupled Rössler oscillators leads from a nonsynchronized state through phase synchronization to the regime of lag synchronization. The role of unstable periodic orbits in these transitions is investigated. Changes in the structure of attracting sets are discussed.

Steady stokes flow with long-range correlations, fractal fourier spectrum, and anomalous transport.

We consider viscous two-dimensional steady flows of incompressible fluids past doubly periodic arrays of solid obstacles. In a class of such flows, the autocorrelations for the Lagrangian observables decay in accordance with the power law, and the Fourier spectrum is neither discrete nor absolutely continuous. We demonstrate that spreading of the droplet of tracers in such flows is anomalously fast.

Self-induced slow-fast dynamics and swept bifurcation diagrams in weakly desynchronized systems.

In systems close to the state of phase synchronization, the fast timescale of oscillations interacts with the slow timescale of the phase drift. As a result, "fast" dynamics is subjected to a slow modulation, due to which an autonomous system under fixed parameter values can imitate repeated bifurcational transitions. We demonstrate the action of this general mechanism for a set of two coupled autonomous chaotic oscillators and for a chaotic system perturbed by a periodic external force.

Multifractal Fourier spectra and power-law decay of correlations in random substitution sequences.

Binary symbolic sequences produced by randomly alternating substitution rules are considered. Exact expressions for the characteristics of autocorrelations and power spectra are derived. The decay of autocorrelation function obeys the power law.