arXiv subjects

Arkady Pikovsky

Publications and source records attributed to Arkady Pikovsky.

At least 91 records · Page 5Linked to original sources

arXiv2014

Chimera-like states in an ensemble of globally coupled oscillators

We demonstrate emergence of a complex state in a homogeneous ensemble of globally coupled identical oscillators, reminiscent of chimera states in locally coupled oscillator lattices. In this regime some part of the ensemble forms a regularly evolving cluster, while all other units irregularly oscillate and remain asynchronous. We argue that chimera emerges because of effective bistability which dynamically appears in the originally monostable system due to internal delayed feedback in individual units. Additionally, we present two examples of chimeras in bistable systems with frequency-dependent phase shift in the global coupling.

nlin.CD↗

arXiv2014

Attractor of Smale-Williams type in autonomous distributed system

We consider an autonomous system of partial differential equations for one-dimensional distributed medium with periodic boundary conditions. Dynamics in time consists of alternating birth and death of patterns with spatial phases transformed from one stage of activity to another by the doubly expanding circle map. So, attractor in the Poincaré section is uniformly hyperbolic, a kind of Smale-Williams solenoid. Finite-dimensional models are derived as ordinary differential equations for amplitudes of spatial Fourier modes (the 5D and 7D models). Correspondence of the reduced models to the original system is demonstrated numerically. Computational verification of the hyperbolicity criterion is performed for the reduced models: the distribution of angles of intersection for stable and unstable manifolds on the attractor is separated from zero, i.e. the touches are excluded. The considered example gives a partial justification to the old hopes that chaotic behavior of autonomous distributed systems may be associated with uniformly hyperbolic attractors.

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arXiv2013

Network topology reconstructed from derivative-variable correlations

A method of network reconstruction from the dynamical time series is introduced, relying on the concept of derivative-variable correlation. Using a tunable observable as a parameter, the reconstruction of any network with known interaction functions is formulated via simple matrix equation. We suggest a procedure aimed at optimizing the reconstruction from the time series of length comparable to the characteristic dynamical time scale. Our method also provides a reliable precision estimate. We illustrate the method's implementation via elementary dynamical models, and demonstrate its robustness to both model and observation errors.

physics.data-an↗

arXiv2013

Robust synchronization of spin-torque oscillators with an LCR load

We study dynamics of a serial array of spin-torque-oscillators with a parallel inductor-capacitor-resistor (LCR) load. In a large range of parameters the fully synchronous regime, where all the oscillators have the same state and the output field is maximal, is shown to be stable. However, not always such a robust complete synchronization develops from the random initial state, in many cases nontrivial clustering is observed, with a partial synchronization resulting in a quasiperiodic or chaotic mean field dynamics.

nlin.AO↗

arXiv2013

Multi-pulse phase resetting curve

In this paper, we introduce and systematically study, in terms of phase response curves (PRC), the effect of a dual pulse excitation on the dynamics of an autonomous oscillator. Specifically, we test the deviations from a linear summation of phase advances from two small perturbations. We derive analytically a correction term, which generally appears for oscillators whose intrinsic dimensionality is greater than one. We demonstrate this effect in case of the Stuart-Landau model, and also in various higher dimensional neuronal model. The non-linear correction is found to be proportional to the square of the perturbation. This deviation from the superposition principle needs to be taken into account in studies of networks of pulse-coupled oscillators. Further, this deviation could be used for verification of oscillator models via a dual pulse excitation.

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arXiv2013

Synchrony suppression in ensembles of coupled oscillators via adaptive vanishing feedback

Synchronization and emergence of a collective mode is a general phenomenon, frequently observed in ensembles of coupled self-sustained oscillators of various natures. In several circumstances, in particular in cases of neurological pathologies, this state of the active medium is undesirable. Destruction of this state by a specially designed stimulation is a challenge of high clinical relevance. Typically, the precise effect of an external action on the ensemble is unknown, since the microscopic description of the oscillators and their interactions are not available. We show, that desynchronization in case of a large degree of uncertainty about important features of the system is nevertheless possible; it can by achieved by virtue of a feedback loop with an additional adaptation of parameters. The adaptation also ensures desynchronization of ensembles with non-stationary, time-varying parameters. We perform the stability analysis of the feedback-controlled system and demonstrate efficient destruction of synchrony for several models, including those of spiking and bursting neurons.

nlin.AO↗

arXiv2013

Synchronization of a Josephson junction array in terms of global variables

We consider an array of Josephson junctions with a common LCR-load. Application of the Watanabe-Strogatz approach [Physica D, v. 74, p. 197 (1994)] allows us to formulate the dynamics of the array via the global variables only. For identical junctions this is a finite set of equations, analysis of which reveals the regions of bistability of the synchronous and asynchronous states. For disordered arrays with distributed parameters of the junctions, the problem is formulated as an integro-differential equation for the global variables, here stability of the asynchronous states and the properties of the transition synchrony-asynchrony are established numerically.

nlin.CD↗

arXiv2013

Hyperbolic chaos at blinking coupling of noisy oscillators

We study an ensemble of identical noisy phase oscillators with a blinking mean-field coupling, where one-cluster and two-cluster synchronous states alternate. In the thermodynamic limit the population is described by a nonlinear Fokker-Planck equation. We show that the dynamics of the order parameters demonstrates hyperbolic chaos. The chaoticity manifests itself in phases of the complex mean field, which obey a strongly chaotic Bernoulli map. Hyperbolicity is confirmed by numerical tests based on the calculations of relevant invariant Lyapunov vectors and Lyapunov exponents. We show how the chaotic dynamics of the phases is slightly smeared by finite-size fluctuations.

nlin.CD↗

arXiv2013

Phase Description of Stochastic Oscillations

We introduce an invariant phase description of stochastic oscillations by generalizing the concept of standard isophases. The average isophases are constructed as sections in the state space, having a constant mean first return time. The approach allows to obtain a global phase variable of noisy oscillations, even in the cases where the phase is ill-defined in the deterministic limit. A simple numerical method for finding the isophases is illustrated for noise-induced switching between two coexisting limit cycles, and for noise-induced oscillation in an excitable system. We also discuss how to determine the isophases for experimentally observed irregular oscillations, providing a basis for a refined phase description of observed oscillatory dynamics.

nlin.CD↗

arXiv2012

Dynamics of multifrequency oscillator communities

We consider a generalization of the Kuramoto model of coupled oscillators to the situation where communities of oscillators having essentially different natural frequencies interact. General equations describing possible resonances between the communities' frequencies are derived. The mostly simple situation of three resonantly interacting groups is analyzed in details. We find conditions for the mutual coupling to promote or suppress synchrony in individual populations, and present examples where interaction between communities leads to their synchrony, or to a partially asynchronous state, or to a chaotic dynamics of order parameters.

nlin.AO↗

arXiv2012

Scaling properties of energy spreading in nonlinear Hamiltonian two-dimensional lattices

In nonlinear disordered Hamiltonian lattices, where there are no propagating phonons, the spreading of energy is of subdiffusive nature. Recently, the universality class of the subdiffusive spreading according to the nonlinear diffusion equation (NDE) has been suggested and checked for one-dimensional lattices. Here, we apply this approach to two-dimensional strongly nonlinear lattices and find a nice agreement of the scaling predicted from the NDE with the spreading results from extensive numerical studies. Moreover, we show that the scaling works also for regular lattices with strongly nonlinear coupling, for which the scaling exponent is estimated analytically. This shows that the process of chaotic diffusion in such lattices does not require disorder.

nlin.CD↗

arXiv2012

Hyperbolic Chaos of Turing Patterns

We consider time evolution of Turing patterns in an extended system governed by an equation of the Swift-Hohenberg type, where due to an external periodic parameter modulation long-wave and short-wave patterns with length scales related as 1:3 emerge in succession. We show theoretically and demonstrate numerically that the spatial phases of the patterns, being observed stroboscopically, are governed by an expanding circle map, so that the corresponding chaos of Turing patterns is hyperbolic, associated with a strange attractor of the Smale-Williams solenoid type. This chaos is shown to be robust with respect to variations of parameters and boundary conditions.

nlin.CD↗

arXiv2011

Optimal Phase Description of Chaotic Oscillators

We introduce an optimal phase description of chaotic oscillations by generalizing the concept of isochrones. On chaotic attractors possessing a general phase description, we define the optimal isophases as Poincaré surfaces showing return times as constant as possible. The dynamics of the resultant optimal phase is maximally decoupled of the amplitude dynamics, and provides a proper description of phase resetting of chaotic oscillations. The method is illustrated with the Rössler and Lorenz systems.

nlin.CD↗

arXiv2011

Network reconstruction from random phase-resetting

We propose a novel method of reconstructing the topology and interaction functions for a general oscillator network. An ensemble of initial phases and the corresponding instantaneous frequencies is constructed by repeating random phase-resets of the system dynamics. The desired details of network structure are then revealed by appropriately averaging over the ensemble. The method is applicable for a wide class of networks with arbitrary emergent dynamics, including full synchrony.

nlin.CD↗

arXiv2011

Strong and weak chaos in weakly nonintegrable many-body Hamiltonian systems

We study properties of chaos in generic one-dimensional nonlinear Hamiltonian lattices comprised of weakly coupled nonlinear oscillators, by numerical simulations of continuous-time systems and symplectic maps. For small coupling, the measure of chaos is found to be proportional to the coupling strength and lattice length, with the typical maximal Lyapunov exponent being proportional to the square root of coupling. This strong chaos appears as a result of triplet resonances between nearby modes. In addition to strong chaos we observe a weakly chaotic component having much smaller Lyapunov exponent, the measure of which drops approximately as a square of the coupling strength down to smallest couplings we were able to reach. We argue that this weak chaos is linked to the regime of fast Arnold diffusion discussed by Chirikov and Vecheslavov. In disordered lattices of large size we find a subdiffusive spreading of initially localized wave packets over larger and larger number of modes. The relations between the exponent of this spreading and the exponent in the dependence of the fast Arnold diffusion on coupling strength are analyzed. We also trace parallels between the slow spreading of chaos and deterministic rheology.

nlin.CD↗

arXiv2011

Reconstructing phase dynamics of oscillator networks

We generalize our recent approach to reconstruction of phase dynamics of coupled oscillators from data [B. Kralemann et al., Phys. Rev. E, 77, 066205 (2008)] to cover the case of small networks of coupled periodic units. Starting from the multivariate time series, we first reconstruct genuine phases and then obtain the coupling functions in terms of these phases. The partial norms of these coupling functions quantify directed coupling between oscillators. We illustrate the method by different network motifs for three coupled oscillators and for random networks of five and nine units. We also discuss nonlinear effects in coupling.

nlin.CD↗

arXiv2011

Effects of non-resonant interaction in ensembles of phase oscillators

We consider general properties of groups of interacting oscillators, for which the natural frequencies are not in resonance. Such groups interact via non-oscillating collective variables like the amplitudes of the order parameters defined for each group. We treat the phase dynamics of the groups using the Ott-Antonsen ansatz and reduce it to a system of coupled equations for the order parameters. We describe different regimes of co-synchrony in the groups. For a large number of groups, heteroclinic cycles, corresponding to a sequental synchronous activity of groups, and chaotic states, where the order parameters oscillate irregularly, are possible.

nlin.AO↗

arXiv2011

Desynchronization transitions in nonlinearly coupled phase oscillators

We consider the nonlinear extension of the Kuramoto model of globally coupled phase oscillators where the phase shift in the coupling function depends on the order parameter. A bifurcation analysis of the transition from fully synchronous state to partial synchrony is performed. We demonstrate that for small ensembles it is typically mediated by stable cluster states, that disappear with creation of heteroclinic cycles, while for a larger number of oscillators a direct transition from full synchrony to a periodic or quasiperiodic regime occurs.

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