Phase reduction is a dimensionality reduction scheme to describe the dynamics of nonlinear oscillators with a single phase variable. While it is crucial in synchronization analysis of coupled oscillators, analytical results are limited to few systems. In this letter, we analytically perform phase reduction for a wide class of oscillators by extending the Poincaré-Lindstedt perturbation theory. We exemplify the utility of our approach by analyzing an ensemble of Van der Pol oscillators, where the derived phase model provides analytical predictions of their collective synchronization dynamics
This paper presents a unified mathematical theory of swarms where the dynamics of social behaviors interacts with the mechanical dynamics of self-propelled particles. The term behavioral swarms is introduced to characterize the specific object of the theory which is subsequently followed by applications. As concrete examples for our unified approach, we show that several Cucker-Smale type models with internal variables fall down to our framework. Subsequently the modeling goes beyond the Cucker-Smale approach and looks ahead to research perspectives.
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.
Markus Mohr, Rainer K. Wunderlich, Kai Zweiacker
et al.
AbstractHere we present measurements of surface tension and viscosity of the bulk glass-forming alloy Pd43Cu27Ni10P20 performed during containerless processing under reduced gravity. We applied the oscillating drop method in an electromagnetic levitation facility on board of parabolic flights. The measured viscosity exhibits a pronounced temperature dependence following an Arrhenius law over a temperature range from 1100 K to 1450 K. Together with literature values of viscosity at lower temperatures, the viscosity of Pd43Cu27Ni10P20 can be well described by a free volume model. X-ray diffraction analysis on the material retrieved after the parabolic flights confirm the glassy nature after vitrification of the bulk samples and thus the absence of crystallization during processing over a wide temperature range.
Examples of one-dimensional lattice systems are considered, in which patterns of different spatial scales arise alternately, so that the spatial phase over a full cycle undergo transformation according to expanding circle map that implies occurrence of Smale-Williams attractors in the multidimensional state space. These models can serve as a basis for design electronic generators of robust chaos within a paradigm of coupled cellular networks. One of the examples is a mechanical pendulum system interesting and demonstrative for research and educational experimental studies.
We consider a glassy system of interacting spins driven by continual switching amongst a finite set of nonuniform external fields. We find that the system evolves over time towards configurations that minimize the work absorbed from this external drive. The configurations which achieve this are specific to the details of the external fields used to drive the system, and therefore act effectively as a self-organized novelty-detector that embodies accurate predictions about the typical future of its external environment.
Mathieu N. Galtier, Camille Marini, Gilles Wainrib
et al.
A method is provided for designing and training noise-driven recurrent neural networks as models of stochastic processes. The method unifies and generalizes two known separate modeling approaches, Echo State Networks (ESN) and Linear Inverse Modeling (LIM), under the common principle of relative entropy minimization. The power of the new method is demonstrated on a stochastic approximation of the El Nino phenomenon studied in climate research.
We show that the coupled complex systems can evolve into a new kind of self-organized critical state where each subsystem is not critical, however, they cooperate to be critical. This criticality is different from the classical BTW criticality where the single system itself evolves into a critical state. We also find that the outflows can be accumulated in the coupled systems. This will lead to the emergency of spatiotemporal intermittency in the critical state.
We comment on the derivation of the main equation in the bounded confidence model of opinion dynamics. In the original work, the equation is derived using an ad-hoc counting method. We point that the original derivation does contain some small mistake. The mistake does not have a large qualitative impact, but it reveals the danger of the ad-hoc counting method. We show how a more systematic approach, which we call micro to macro, can avoid such mistakes, without adding any significant complexity.
Influence systems form a large class of multiagent systems designed to model how influence, broadly defined, spreads across a dynamic network. We build a general analytical framework which we then use to prove that, while sometimes chaotic, influence dynamics of the diffusive kind is almost always asymptotically periodic. Besides resolving the dynamics of a popular family of multiagent systems, the other contribution of this work is to introduce a new type of renormalization-based bifurcation analysis for multiagent systems.
This paper introduces a new model of continuous opinion dynamics with random noise. The model belongs to the broad class of so called bounded confidence models. It differs from other popular bounded confidence models by the update rule, since it is intended to describe how the single person can influence at the same time a group of several listeners. Moreover, opinion noise is introduced to the model. Due to this noise, in some specific cases, spontaneous transitions between two states with a different number of large opinion clusters occur. Detailed analysis of these transitions is provided, with MC simulations and ME numerical integration analysis.
In this work, a new model in kinetic gas theory for deriving the Maxwellian Velocity Distribution (MVD) is proposed. We construct an operator that governs the discrete time evolution of the velocity distribution. This operator, which conserves the momentum and the energy of the ideal gas, has the MVD as a fixed point. Moreover, for any initial out-of-equilibrium velocity distribution, it is shown that the gas decays to the equilibrium distribution, that is, to the MVD.
We uncover a solvable generalization of the Kuramoto model in which shears (or nonisochronicities) and natural frequencies are distributed and statistically dependent. We show that the strength and sign of this dependence greatly alter synchronization and yield qualitatively different phase diagrams. The Ott-Antonsen ansatz allows us to obtain analytical results for a specific family of joint distributions. We also derive, using linear stability analysis, general formulae for the stability border of incoherence.
A unified approach for analyzing synchronization in coupled systems of autonomous differential equations is presented in this work. Through a careful analysis of the variational equation of the coupled system we establish a sufficient condition for synchronization in terms of the geometric properties of the local limit cycles and the coupling operator. This result applies to a large class of differential equation models in physics and biology. The stability analysis is complemented with a discussion of numerical simulations of a compartmental model of a neuron.
Gregory K. Fricke, Bruce W. Rogers, Devendra P. Garg
Representation of a swarm of independent robotic agents under graph-theoretic constructs allows for more formal analysis of convergence properties. We consider the local and global convergence behavior of an N-member swarm of agents in a modified consensus problem wherein the connectivity of agents is governed by probabilistic functions. The addition of a random walk control ensures Lyapunov stability of the swarm consensus. Simulation results are given and planned experiments are described.
We introduce an interaction mechanism between oscillators leading to exact anti-phase and in-phase synchronization. This mechanism is applied to the coupling between two nonlinear oscillators with a limit cycle in phase space, leading to a simple justification of the anti-phase synchronization observed in the Huygens's pendulum clocks experiment. If the two coupled nonlinear oscillators reach the anti-phase or the in-phase synchronized oscillatory state, the period of oscillation is different from the eigen-periods of the uncoupled oscillators.
Reproducibility of a noisy limit-cycle oscillator driven by a random piecewise constant signal is analyzed. By reducing the model to random phase maps, it is shown that the reproducibility of the limit cycle generally improves when the phase maps are monotonically increasing.
We numerically demonstrate that collective bifurcations in two-dimensional lattices of locally coupled logistic maps share most of the defining features of equilibrium second-order phase transitions. Our simulations suggest that these transitions between distinct collective dynamical regimes belong to the universality class of Miller and Huse model with synchronous update.
The sustainable biodiversity associated with a specific ecological niche as a function of land area is analysed computationally by considering the interaction of ant societies over a collection of islands. A power law relationship between sustainable species and land area is observed. We will further consider the effect a perturbative inflow of ants has upon the model.
Most avalanching systems in nature should involve diffusive processes as well which can change the behavior of such systems and should be taken into account. We examine the effects of diffusion on the model of a dissipative bi-directional burning model. It is shown that the system behavior progressively changes with the increase of diffusion. Avalanches of small sizes become suppressed and large quasiperiodic bursts develop. In contrast with sandpiles these bursts are not edge triggered and the transition is gradual.