arXiv subjects

A. Pikovsky

Publications and source records attributed to A. Pikovsky.

At least 19 recordsLinked to original sources

arXiv2026

Coherence properties of collective modes in ensembles of oscillators

Synchronization transition in oscillatory networks manifests itself as the appearance of a periodic global mode. While perfect in the thermodynamic limit, this mode fluctuates for finite ensembles. We characterize the coherence of this mode in terms of the phase diffusion constant. In several examples, we always observed normal diffusion, but the dependence of the diffusion constant on the system size $D\sim N^{-\mu}$ depends on the nature of coupled units: for coupled chaotic systems $\mu=1$, while for coupled periodic oscillators we observe, depending on the particular model, $\mu=2$ and $\mu=2.5$. These large values of the power index are attributed to the size-dependence of collective chaos in the finite ensemble, which disappears in the thermodynamic limit. We also show that in the standard Kuramoto model for a symmetric set of frequencies, there is an additional transition to a symmetric chaotic state with vanishing diffusion of the global phase.

nlin.CD↗

arXiv2023

Attractor-repeller collision and the heterodimensional dynamics

We study the heterodimensional dynamics in a simple map on a three-dimensional torus. This map consists of a two-dimensional driving Anosov map and a one-dimensional driven M\"obius map, and demonstrates the collision of a chaotic attractor with a chaotic repeller if parameters are varied. We explore this collision by following tangent bifurcations of the periodic orbits, and establish a regime where periodic orbits with different numbers of unstable directions coexist in a chaotic set. For this situation, we construct a heterodimensional cycle connecting these periodic orbits. Furthermore, we discuss properties of the rotation number and of the nontrivial Lyapunov exponent at the collision and in the heterodimensional regime.

nlin.CD↗

arXiv2022

Traveling chimeras in oscillator lattices with advective-diffusive coupling

We consider a one-dimensional array of phase oscillators coupled via an auxiliary complex field. While in the seminal chimera studies by Kumamoto and Battogtokh only diffusion of the field was considered, we include advection which makes the coupling left-right asymmetric. Chimera starts to move and we demonstrate, that a weakly turbulent moving pattern appears. It possesses a relatively large synchronous domain where the phases are nearly equal, and a more disordered domain where the local driving field is small. For a dense system with a large number of oscillators, there are strong local correlations in the disordered domain, which at most places looks like a smooth phase profile. We find also exact regular traveling wave chimera-like solutions of different complexity, but only some of them are stable.

nlin.PS↗

arXiv2021

Finite-density-induced motility and turbulence of chimera solitons

We consider a one-dimensional oscillatory medium with a coupling through a diffusive linear field. In the limit of fast diffusion this setup reduces to the classical Kuramoto-Battogtokh model. We demonstrate that for a finite diffusion stable chimera solitons, namely localized synchronous domain in an infinite asynchronous environment, are possible. The solitons are stable also for finite density of oscillators, but in this case they sway with a nearly constant speed. This finite-density-induced motility disappears in the continuum limit, as the velocity of the solitons is inverse proportional to the density. A long-wave instability of the homogeneous asynchronous state causes soliton turbulence, which appears as a sequence of soliton mergings and creations. As the instability of the asynchronous state becomes stronger, this turbulence develops into a spatio-temporal intermittency.

nlin.PS↗

arXiv2021

Disorder fosters chimera in an array of motile particles

We consider an array of non-locally coupled oscillators on a ring, which for equally spaced units possesses a Kuramoto-Battogtokh chimera regime and a synchronous state. We demonstrate that disorder in oscillators positions leads to a transition from the synchronous to the chimera state. For a static (quenched) disorder we find that the probability of synchrony survival changes, in dependence on the number of particles, from nearly zero at small populations to one in the thermodynamic limit. Furthermore, we demonstrate how the synchrony gets destroyed for randomly (ballistically or diffusively) moving oscillators. We show that, depending on the number of oscillators, there are different scalings of the transition time with this number and the velocity of the units.

nlin.PS↗

arXiv2019

Scaling of energy spreading in a disordered Ding-Dong lattice

We study numerically propagation of energy in a one dimensional Ding-Ding lattice, composed of linear oscillators with ellastic collisions. Wave propagation is suppressed by breaking translational symmetry, we consider three way to do this: a position disorder, a mass disorder, and a dimer lattice with alternating distances between the units. In all cases the spreading of an initially localized wavepacket is irregular, due to appearance of chaos, and subdiffusive. Guided by a nonlinear diffusion equation, we establish that the mean waiting times of spreading obey a scaling law in dependence on energy. Moreover, we show that the spreading exponents very weakly depend on the level of disorder.

cond-mat.dis-nn↗

arXiv2019

Microscopic Cross-Correlations in the Finite-Size Kuramoto Model of Coupled Oscillators

Super-critical Kuramoto oscillators with distributed frequencies separate into two disjoint groups: an ordered one locked to the mean field, and a disordered one consisting of effectively decoupled oscillators -- at least so in the thermodynamic limit. In finite ensembles, in contrast, such clear separation fails: The mean field fluctuates due to finite-size effects and thereby induces order in the disordered group. To our best knowledge, this publication is the first to reveal such an effect, similar to noise-induced synchronization, in a purely deterministic system. We start by modeling the situation as a stationary mean field with additional white noise acting on a pair of unlocked Kuramoto oscillators. An analytical expression shows that the cross-correlation between the two increases with decreasing ratio of natural frequency difference and noise intensity. In a deterministic finite Kuramoto model, the strength of the mean field fluctuations is inextricably linked to the typical natural frequency difference. Therefore, we let a fluctuating mean field, generated by a finite ensemble of active oscillators, act on pairs of passive oscillators with a microscopic natural frequency difference between which we then measure the cross-correlation, at both super- and sub-critical coupling.

nlin.AO↗

arXiv2019

Nonlinear phase coupling functions: a numerical study

Phase reduction is a general tool widely used to describe forced and interacting self-sustained oscillators. Here we explore the phase coupling functions beyond the usual first-order approximation in the strength of the force. Taking the periodically forced Stuart-Landau oscillator as the paradigmatic model, we determine and numerically analyse the coupling functions up to the fourth order in the force strength. We show that the found nonlinear phase coupling functions can be used for predicting synchronization regions of the forced oscillator.

physics.comp-ph↗

arXiv2019

Dynamical disentanglement in an analysis of oscillatory systems: an application to respiratory sinus arrhythmia

We develop a technique for the multivariate data analysis of perturbed self-sustained oscillators. The approach is based on the reconstruction of the phase dynamics model from observations and on a subsequent exploration of this model. For the system, driven by several inputs, we suggest a dynamical disentanglement procedure, allowing us to reconstruct the variability of the system's output that is due to a particular observed input, or, alternatively, to reconstruct the variability which is caused by all the inputs except for the observed one. We focus on the application of the method to the vagal component of the heart rate variability caused by a respiratory influence. We develop an algorithm that extracts purely respiratory-related variability, using a respiratory trace and times of R-peaks in the electrocardiogram. The algorithm can be applied to other systems where the observed bivariate data can be represented as a point process and a slow continuous signal, e.g. for the analysis of neuronal spiking.

physics.med-ph↗

arXiv2018

Solitary synchronization waves in distributed oscillators populations

We demonstrate existence of solitary waves of synchrony in one-dimensional arrays of identical oscillators with Laplacian coupling. Coarse-grained description of the array leads to nonlinear equations for the complex order parameter, in the simplest case to lattice equations similar to those of the discrete nonlinear Schrodinger lattice. Close to full synchrony, we find solitary waves for the order parameter perturbatively, starting from the known phase compactons and kovatons; these solution are extended numerically to the full domain of possible synchrony levels. For non-identical oscillators, existence of dissipative solitons is demonstrated.

nlin.PS↗

arXiv2017

Reconstruction of a random phase dynamics network from observations

We consider networks of coupled phase oscillators of different complexity: Kuramoto-Daido-type networks, generalized Winfree networks, and hypernetworks with triple interactions. For these setups an inverse problem of reconstruction of the network connections and of the coupling function from the observations of the phase dynamics is addressed. We show how a reconstruction based on the minimization of the squared error can be implemented in all these cases. Examples include random networks with full disorder both in the connections and in the coupling functions, as well as networks where the coupling functions are taken from experimental data of electrochemical oscillators. The method can be directly applied to asynchronous dynamics of units, while in the case of synchrony, additional phase resettings are necessary for reconstruction.

nlin.AO↗

arXiv2017

Reconstruction of a scalar voltage-based neural field network from observed time series

We present a general method for reconstruction of a network of nonlinearly coupled neural fields from the observations. A prominent example of such a system is a dynamical random neural network model studied by Sompolinsky et. al [Phys. Rev. Lett., v. 61, 259 (1988)]. We develop a technique for inferring the properties of the system from the observations of the chaotic voltages. Only the structure of the model is assumed to be known, while the nonlinear gain functions of the interactions, the matrix of the coupling constants, and the time constants of the local dynamics are reconstructed from the time series.

nlin.CD↗

arXiv2017

Synchronization in Kuramoto-Sakaguchi ensembles with competing influence of common noise and global coupling

We study the effects of synchronization and desynchronization in ensembles of phase oscillators with the global Kuramoto-Sakaguchi coupling under common noise driving. Since the mechanisms of synchronization by coupling and by common noise are essentially different, their interplay is of interest. In the thermodynamic limit of large number of oscillators, employing the Ott-Antonsen approach, we derive stochastic equations for the order parameters and consider their dynamics for two cases: (i) identical oscillators and (ii) small natural frequency mismatch. For identical oscillators, the stability of the perfect synchrony state is studied; a strong enough common noise is revealed to prevail over a moderate negative (repelling) coupling and to synchronize the ensemble. An inequality between the states of maximal asynchrony (zero-value of the order parameter) and perfect synchrony; the former can be only weakly stable, while the latter can become adsorbing (the transition to the synchrony becomes unidirectional). The dependence of the temporal dynamics of the transition on the system parameters is investigated. For nonidentical oscillators the perfect synchrony state becomes impossible and an absorbing state disappears; on its place, only a weakly stable state of imperfect synchrony remains. A nontrivial effect of the divergence of individual frequencies of oscillators with different natural frequencies is revealed and studied for moderate repelling coupling; meanwhile, the order parameter remains non-small for this case. In Appendix we provide an introduction to the theories of Ott-Antonsen and Watanabe-Strogatz.

cond-mat.stat-mech↗

arXiv2016

Chimera patterns in the Kuramoto-Battogtokh model

Kuramoto and Battogtokh [Nonlinear Phenom. Complex Syst. 5, 380 (2002)] discovered chimera states represented by stable coexisting synchrony and asynchrony domains in a lattice of coupled oscillators. After reformulation in terms of local order parameter, the problem can be reduced to partial differential equations. We find uniformly rotating periodic in space chimera patterns as solutions of a reversible ordinary differential equation, and demonstrate a plethora of such states. In the limit of neutral coupling they reduce to analytical solutions in form of one- and two-point chimera patterns as well as localized chimera solitons. Patterns at weakly attracting coupling are characterized by virtue of a perturbative approach. Stability analysis reveals that only simplest chimeras with one synchronous region are stable.

nlin.PS↗

arXiv2016

Reconstruction of chaotic neural network from observed firing rates

Randomly coupled neural fields demonstrate chaotic variation of firing rates, if the coupling is strong enough, as has been shown by Sompolinsky et. al [Phys. Rev. Lett., v. 61, 259 (1988)]. We present a method for reconstruction of the coupling matrix from the observations of the chaotic firing rates. The approach is based on the particular property of the nonlinearity in the coupling, as the latter is determined by a sigmoidal gain function. We demonstrate that for a large enough data set, the method gives an accurate estimation of the coupling matrix and of other parameters of the system, including the gain function.

nlin.CD↗

arXiv2015

Marginal chimera state at cross-frequency locking of pulse-coupled neural networks

We consider two coupled populations of leaky integrate-and-fire neurons. Depending on the coupling strength, mean fields generated by these populations can have incommensurate frequencies, or become frequency locked. In the observed 2:1 locking state of the mean fields, individual neurons in one population are asynchronous with the mean fields, while in another populations they have the same frequency as the mean field. These synchronous neurons form a chimera state, where part of them build a fully synchronized cluster, while other remain scattered. We explain this chimera as a marginal one, caused by a self-organized neutral dynamics of the effective circle map.

nlin.AO↗

arXiv2014

The Kuramoto model of coupled oscillators with a bi-harmonic coupling function

We study synchronization in a Kuramoto model of globally coupled phase oscillators with a bi-harmonic coupling function, in the thermodynamic limit of large populations. We develop a method for an analytic solution of self-consistent equations describing uniformly rotating complex order parameters, both for single-branch (one possible state of locked oscillators) and multi-branch (two possible values of locked phases) entrainment. We show that synchronous states coexist with the neutrally linearly stable asynchronous regime. The latter has a finite life time for finite ensembles, this time grows with the ensemble size as a power law.

nlin.CD↗

arXiv2013

Coherence properties of cycling chaos

Cycling chaos is a heteroclinic connection between several chaotic attractors, at which switching between the chaotic sets occur at growing time intervals. Here we characterize the coherence properties of these switchings, considering nearly periodic regimes that appear close to the cycling chaos due to imperfections or to instability. Using numerical simulations of coupled Lorenz, Roessler, and logistic map models, we show that the coherence is high in the case of imperfection (so that asymptotically the cycling chaos is very regular), while it is low close to instability of the cycling chaos.

nlin.CD↗