Spatiotemporal Oscillation Patterns in the Collective Relaxation Dynamics of Interacting Particles in Periodic Potentials

Benno Liebchen; Peter Schmelcher

doi:10.1103/physrevlett.112.134102

Home/Archive/arXiv

arXiv · 1401.1704

Spatiotemporal Oscillation Patterns in the Collective Relaxation Dynamics of Interacting Particles in Periodic Potentials

Benno Liebchen·Peter Schmelcher

View original source ↗Read source document ↗Export metadata ↓

Abstract

We demonstrate the emergence of self-organized structures in the course of the relaxation of an initially excited, dissipative and finite chain of interacting particles in a periodic potential towards its many particle equilibrium configuration. Specifically we observe a transition from an in phase correlated motion via phase randomized oscillations towards oscillations with a phase difference $π$ between adjacent particles thereby yielding the growth of long time transient spatiotemporal oscillation patterns. Parameter modifications allow for designing these patterns, including steady states and even states that combine in phase and correlated out of phase oscillations along the chain. The complex relaxation dynamics is based on finite size effects together with an evolution running from the nonlinear to the linear regime thereby providing a highly unbalanced population of the center of mass and relative motion.

Explore related subjects

nlin.AO cond-mat.other

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Benno Liebchen, Peter Schmelcher. 2014-01-08. Spatiotemporal Oscillation Patterns in the Collective Relaxation Dynamics of Interacting Particles in Periodic Potentials. https://doi.org/10.1103/physrevlett.112.134102

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Frequency bursts in adaptive delay-coupled oscillators

We report on frequency bursting oscillations in a system of phase oscillators with adaptive and delayed coupling. Adaptation of the coupling strengths is considered slow and depends on the phase shift between the oscillators. We find due to the combined chain of adaptation, collective dynamics, and time delays, the system robustly achieves a state in which the oscillator's frequencies are nearly synchronized but detuned by an integer number of small adaptation frequencies. We demonstrate that this quantization of the detuning is caused by alternating slow and fast transitions. Moreover, the observed motions take the form of bursts of instantaneous frequency, and the number of spikes in each burst corresponds to the quantization level of the detuning. We provide a fast-slow analysis of this phenomenon and explain the mechanisms behind the emergence of bursts. Our findings indicate that these frequency bursting oscillations are robust and exist stably within finite parameter regions.

nlin.AO↗

arXiv2026

Discrete-time Kuramoto model with phase lag: Linear stability analysis and onset of synchronization

We investigate the discrete-time version of the Kuramoto model with phase lag, which comprises globally-coupled phase oscillators of distributed frequencies that are evolving under a nonlinear map. In the continuum limit of an infinite number of oscillators ($N\to \infty$), we derive the exact Frobenius-Perron equation for the time evolution of the single-oscillator probability density, and study linear stability of the incoherent state. Instability signals onset of synchronization. The corresponding synchronization threshold is obtained analytically for the case of a Lorentzian distribution of the oscillator frequencies. The threshold differs from that of the continuous-time Kuramoto model, reflecting the fundamentally different stability conditions for discrete-time maps and continuous-time flows. Beyond synchronization threshold, we observe several interesting nonlinear phenomena: Unlike the classical Kuramoto model, the discrete-time version exhibits periodic and chaotic states. Numerical simulations of the finite-$N$ system confirm the analytical prediction for the synchronization threshold, while highlighting breakdown of the celebrated Ott-Antonsen ansatz invoked to conveniently study the continuous-time Kuramoto model in terms of a low-dimensional description.

nlin.AO↗

arXiv2026

Deviations from global coupling in adaptive oscillator networks: a mean-field theory for the variance of coupling weights

A wide range of physical and biological systems are adaptive networks, in which the dynamics of the nodes and of the edges connecting them co-evolve. Mean-field reductions of such systems typically track only the average coupling strength, and therefore cannot determine when the coupling stays effectively homogeneous and when structured connectivity emerges. Here, we present a second-order moment closure that allows us to derive mean-field equations for the coupling-weight variance in networks of heterogeneous phase oscillators with adaptive coupling, starting from uniform coupling weights. In agreement with network simulations, we find a nonlinear, non-monotonic dependence of the relative weight variance on the oscillator heterogeneity that is mediated by the phase coherence. Moreover, we find that deviations from global coupling strongly depend on an interaction between the oscillator heterogeneity and the adaptation rule. Whereas symmetric adaptation causes a strongly coupled core of coherent oscillators to emerge and creates a bistable regime that is absent without adaptation, antisymmetric adaptation leads to antisymmetric coupling within the same core, thereby destabilizing it. Our equations therefore delineate the regimes in which adaptive networks behave like globally coupled systems from those in which more complex coupling patterns form.

nlin.AO↗