arXiv subjects

Arkady Pikovsky

Publications and source records attributed to Arkady Pikovsky.

At least 37 records · Page 2Linked to original sources

arXiv2022

Exact finite-dimensional reduction for a population of noisy oscillators and its link to Ott-Antonsen and Watanabe-Strogatz theories

Populations of globally coupled phase oscillators are described in the thermodynamic limit by kinetic equations for the distribution densities, or equivalently, by infinite hierarchies of equations for the order parameters. Ott and Antonsen [Chaos 18, 037113 (2008)] have found an invariant finite-dimensional subspace on which the dynamics is described by one complex variable per population. For oscillators with Cauchy distributed frequencies or for those driven by Cauchy white noise, this subspace is weakly stable and thus describes the asymptotic dynamics. Here we report on an exact finite-dimensional reduction of the dynamics outside of the Ott-Antonsen subspace. We show, that the evolution from generic initial states can be reduced to that of three complex variables, plus a constant function. For identical noise-free oscillators, this reduction corresponds to the Watanabe-Strogatz system of equations [Phys. Rev. Lett. 70, 2391 (1993)]. We discuss how the reduced system can be used to explore the transient dynamics of perturbed ensembles.

cond-mat.stat-mech↗

arXiv2022

Chaos in Coupled Heteroclinic Cycles and its Piecewise-Constant Representation

We consider two stable heteroclinic cycles rotating in opposite directions, coupled via diffusive terms. A complete synchronization in this system is impossible, and numerical exploration shows that chaos is abundant at low levels of coupling. With increase of coupling strength, several symmetry-changing transitions are observed, and finally a stable periodic orbit appears via an inverse period-doubling cascade. To reveal the behavior at extremely small couplings, a piecewise-constant model for the dynamics is suggested. Within this model we construct a Poincaré map for a chaotic state numerically, it appears to be an expanding non-invertable circle map thus confirming abundance of chaos in the small coupling limit. We also show that within the piecewise-constant description, there is a set of periodic solutions with different phase shifts between subsystems, due to dead zones in the coupling.

nlin.CD↗

arXiv2022

Hierarchy of exact low-dimensional reductions for populations of coupled oscillators

We consider an ensemble of phase oscillators in the thermodynamic limit, where it is described by a kinetic equation for the phase distribution density. We propose an ansatz for the circular moments of the distribution (Kuramoto-Daido order parameters) that allows for an exact truncation at an arbitrary number of modes. In the simplest case of one mode, the ansatz coincides with that of Ott and Antonsen [Chaos 18, 037113 (2008)]. Dynamics on the extended manifolds facilitate higher dimensional behavior such as chaos, which we demonstrate with a simulation of a Josephson junction array. The findings are generalized for oscillators with a Cauchy-Lorentzian distribution of natural frequencies.

cond-mat.stat-mech↗

arXiv2021

Phase-locking dynamics of heterogeneous oscillator arrays

We consider an array of nearest-neighbor coupled nonlinear autonomous oscillators with quenched random frequencies and purely conservative coupling. We show that global phase-locked states emerge in finite lattices and study numerically their destruction. Upon change of model parameters, such states are found to become unstable with the generation of localized periodic and chaotic oscillations. For weak nonlinear frequency dispersion, metastability occur akin to the case of almost-conservative systems. We also compare the results with the phase-approximation in which the amplitude dynamics is adiabatically eliminated.

nlin.CD↗

arXiv2021

Phase reconstruction from oscillatory data with iterated Hilbert transform embeddings -- benefits and limitations

In the data analysis of oscillatory systems, methods based on phase reconstruction are widely used to characterize phase-locking properties and inferring the phase dynamics. The main component in these studies is an extraction of the phase from a time series of an oscillating scalar observable. We discuss a practical procedure of phase reconstruction by virtue of a recently proposed method termed \textit{iterated Hilbert transform embeddings}. We exemplify the potential benefits and limitations of the approach by applying it to a generic observable of a forced Stuart-Landau oscillator. Although in many cases, unavoidable amplitude modulation of the observed signal does not allow for perfect phase reconstruction, in cases of strong stability of oscillations and a high frequency of the forcing, iterated Hilbert transform embeddings significantly improve the quality of the reconstructed phase. We also demonstrate that for significant amplitude modulation, iterated embeddings do not provide any improvement.

physics.data-an↗

arXiv2021

Confinement and collective escape of active particles

Active matter broadly covers the dynamics of self-propelled particles. While the onset of collective behavior in homogenous active systems is relatively well understood, the effect of inhomogeneities such as obstacles and traps lacks overall clarity. Here, we study how interacting self-propelled particles become trapped and released from a trap. We have found that captured particles aggregate into an orbiting condensate with a crystalline structure. As more particles are added, the trapped condensate escape as a whole. Our results shed light on the effects of confinement and quenched disorder in active matter.

cond-mat.soft↗

arXiv2021

Transition to Synchrony in a Three-Dimensional Swarming Model with Helical Trajectories

We investigate the transition from incoherence to global collective motion in a three-dimensional swarming model of agents with helical trajectories, subject to noise and global coupling. Without noise this model was recently proposed as a generalization of the Kuramoto model and it was found that alignment of the velocities occurs discontinuously for arbitrarily small attractive coupling. Adding noise to the system resolves this singular limit and leads to a continuous transition, either to a directed collective motion or to center-of-mass rotations.

cond-mat.stat-mech↗

arXiv2021

Real-time estimation of phase and amplitude with application to neural data

Computation of the instantaneous phase and amplitude via the Hilbert Transform is a powerful tool of data analysis. This approach finds many applications in various science and engineering branches but is not proper for causal estimation because it requires knowledge of the signal's past and future. However, several problems require real-time estimation of phase and amplitude; an illustrative example is phase-locked or amplitude-dependent stimulation in neuroscience. In this paper, we discuss and compare three causal algorithms that do not rely on the Hilbert Transform but exploit well-known physical phenomena, the synchronization and the resonance. After testing the algorithms on a synthetic data set, we illustrate their performance computing phase and amplitude for the accelerometer tremor measurements and a Parkinsonian patient's beta-band brain activity.

q-bio.NC↗

arXiv2020

Synchronization of oscillators with hyperbolic chaotic phases

Synchronization in a population of oscillators with hyperbolic chaotic phases is studied for two models. One is based on the Kuramoto dynamics of the phase oscillators and on the Bernoulli map applied to these phases. This system possesses an Ott-Antonsen invariant manifold, allowing for a derivation of a map for the evolution of the complex order parameter. Beyond a critical coupling strength, this model demonstrates bistability synchrony-disorder. Another model is based on the coupled autonomous oscillators with hyperbolic chaotic strange attractors of Smale-Williams type. Here a disordered asynchronous state at small coupling strengths, and a completely synchronous state at large couplings are observed. Intermediate regimes are characterized by different levels of complexity of the global order parameter dynamics.

nlin.CD↗

arXiv2020

Transition to synchrony in chiral active particles

I study deterministic dynamics of chiral active particles in two dimensions. Particles are considered as discs interacting with elastic repulsive forces. An ensemble of particles, started from random initial conditions, demonstrates chaotic collisions resulting in their normal diffusion. This chaos is transient, as rather abruptly a synchronous collisionless state establishes. The life time of chaos grows exponentially with the number of particles. External forcing (periodic or chaotic) is shown to facilitate the synchronization transition.

nlin.CD↗

arXiv2020

Chimeras on a social-type network

We consider a social-type network of coupled phase oscillators. Such a network consists of an active core of mutually interacting elements, and of a flock of passive units, which follow the driving from the active elements, but otherwise are not interacting. We consider a ring geometry with a long-range coupling, where active oscillators form a fluctuating chimera pattern. We show that the passive elements are strongly correlated. This is explained by negative transversal Lyapunov exponents.

nlin.PS↗

arXiv2020

Low-dimensional description for ensembles of identical phase oscillators subject to Cauchy noise

We study an ensembles of globally coupled or forced identical phase oscillators subject to independent white Cauchy noise. We demonstrate, that if the oscillators are forced in several harmonics, stationary synchronous regimes can be exactly described with a finite number of complex order parameters. The corresponding distribution of phases is a product of wrapped Cauchy distributions. For sinusoidal forcing, the Ott-Antonsen low-dimensional reduction is recovered.

nlin.AO↗

arXiv2020

Waves in Strongly Nonlinear Gardner-like Equations on a Lattice

We introduce and study a family of lattice equations which may be viewed either as a strongly nonlinear discrete extension of the Gardner equation, or a non-convex variant of the Lotka-Volterra chain. Their deceptively simple form supports a very rich family of complex solitary patterns. Some of these patterns are also found in the quasi-continuum rendition, but the more intriguing ones, like interlaced pairs of solitary waves, or waves which may reverse their direction either spontaneously or due a collision, are an intrinsic feature of the discrete realm.

nlin.PS↗

arXiv2020

Impact of network characteristics on network reconstruction

When a network is inferred from data, two types of errors can occur: false positive and false negative conclusions about the presence of links. We focus on the influence of local network characteristics on the probability $α$ - of type I false positive conclusions, and on the probability $β$ - of type II false negative conclusions, in the case of networks of coupled oscillators. We demonstrate that false conclusion probabilities are influenced by local connectivity measures such as the shortest path length and the detour degree, which can also be estimated from the inferred network when the true underlying network is not known a priory. These measures can then be used for quantification of the confidence level of link conclusions, and for improving the network reconstruction via advanced concepts of link thresholding.

physics.soc-ph↗

arXiv2020

High-Order Phase Reduction for Coupled Oscillators

We explore the phase reduction in networks of coupled oscillators in the higher orders of the coupling parameter. For coupled Stuart-Landau oscillators, where the phase can be introduced explicitly, we develop an analytic perturbation procedure to allow for the obtaining of the higher-order approximation explicitly. We demonstrate this by deriving the second-order phase equations for a network of three Stuart-Landau oscillators. For systems where explicit expressions of the phase are not available, we present a numerical procedure that constructs the phase dynamics equations for a small network of coupled units. We apply this approach to a network of three van der Pol oscillators and reveal components in the coupling with different scaling in the interaction strength.

nlin.AO↗

arXiv2020

Coupled Moebius Maps as a Tool to Model Kuramoto Phase Synchronization

We propose Moebius maps as a tool to model synchronization phenomena in coupled phase oscillators. Not only does the map provide fast computation of phase synchronization, it also reflects the underlying group structure of the sinusoidally coupled continuous phase dynamics. We study map versions of various known continuous-time collective dynamics, such as the synchronization transition in the Kuramoto-Sakaguchi model of non-identical oscillators, chimeras in two coupled populations of identical phase oscillators, and Kuramoto-Battogtokh chimeras on a ring, and demonstrate similarities and differences between the iterated map models and their known continuous-time counterparts.

nlin.AO↗

arXiv2020

Phase reconstruction with iterated Hilbert transforms

We present a study dealing with a novel phase reconstruction method based on iterated Hilbert transform embeddings. We show results for the Stuart-Landau oscillator observed by generic observables. The benefits for reconstruction of the phase response curve a presented and the method is applied in a setting where the observed system is pertubred by noise.

math.NA↗

arXiv2020

Low-Dimensional Dynamics for Higher Order Harmonic Globally Coupled Phase Oscillator Ensemble

The Kuramoto model, despite its popularity as a mean-field theory for many synchronization phenomenon of oscillatory systems, is limited to a first-order harmonic coupling of phases. For higher-order coupling, there only exists a low-dimensional theory in the thermodynamic limit. In this paper, we extend the formulation used by Watanabe and Strogatz to obtain a low-dimensional description of a system of arbitrary size of identical oscillators coupled all-to-all via their higher-order modes. To demonstrate an application of the formulation, we use a second harmonic globally coupled model, with a mean-field equal to the square of the Kuramoto mean-field. This model is known to exhibit asymmetrical clustering in previous numerical studies. We try to explain the phenomenon of asymmetrical clustering using the analytical theory developed here, as well as discuss certain phenomena not observed at the level of first-order harmonic coupling.

nlin.AO↗