Formation of phase-locked states in a population of locally interacting Kuramoto oscillators

In this paper, we present an asymptotic formation of phase-locked states from the ensemble of Kuramoto oscillators with a symmetric and connected interaction topology. For a limited interaction topology that does not have an all-to-all interaction, Lyapunov type approaches based on phase and frequency diameters do not work due to the lack of completeness. Thus, we employ an energy method together with the connectedness of underlying interaction topologies to determine the complete synchronization estimates. Our synchronization estimation method consists of two parts. First we establish that the uniform boundedness of fluctuations yields the asymptotic formation of phase-locked states using ?ojasiewicz gradient inequality. Second, we show that for the initial configurations lying in the half circle, the uniform boundedness of fluctuations can be derived by a comparison with solutions to the linear Gronwall?s differential inequality for the total phase variance.     

A variational approach to nonlinear and interacting diffusions

Abstract

The article presents a novel variational calculus to analyze the stability and the propagation of chaos properties of nonlinear and interacting diffusions. This differential methodology combines gradient flow estimates with backward stochastic interpolations, Lyapunov linearization techniques as well as spectral theory. This framework applies to a large class of stochastic models including nonhomogeneous diffusions, as well as stochastic processes evolving on differentiable manifolds, such as constraint-type embedded manifolds on Euclidian spaces and manifolds equipped with some Riemannian metric. We derive uniform as well as almost sure exponential contraction inequalities at the level of the nonlinear diffusion flow, yielding what seems to be the first result of this type for this class of models. Uniform propagation of chaos properties w.r.t. the time parameter is also provided. Illustrations are provided in the context of a class of gradient flow diffusions arising in fluid mechanics and granular media literature. The extended versions of these nonlinear Langevin-type diffusions on Riemannian manifolds are also discussed.     

Long-range order in quantum lattice systems of linear oscillators

The existence of the ferromagnetic long-range order is proved for equilibrium quantum lattice systems of linear oscillators whose potential energy contains a strong ferromagnetic nearest-neighbor (nn) pair interaction term and a weak nonferromagnetic term under a special condition on a superstability bound. It is shown that the long-range order is possible if the mass of a quantum oscillator and the strength of the ferromagnetic nn interaction exceed special values. A generalized Peierls argument and a contour bound, proved with the help of a new superstability bound for correlation functions, are our main tools. Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 10, pp. 1407–1424, October, 2006.     

On lattice embeddings of a lattice into its intervals

The notion of idempotent modification of an algebra was introduced by Ježek; he proved that the idempotent modification of a group is always subdirectly irreducible. In the present note we show that the idempotent modification of a generalized MV -algebra having more than two elements is directly irreducible if and only if there exists an element in A which fails to be boolean. Some further results on idempotent modifications are also proved.     

A variational approach to chaotic dynamics in periodically forced nonlinear oscillators

We prove that a class of problems containing the classical periodically forced pendulum equation displays the main features of chaotic dynamics. The approach is based on the construction of multibump type heteroclinic solutions to periodic orbits by the use of global variational methods.     

Closed-loop suppression of chaos in nonlinear driven oscillators

Summary This paper discusses the suppression of chaos in nonlinear driven oscillators via the addition of a periodic perturbation. Given a system originally undergoing chaotic motions, it is desired that such a system be driven to some periodic orbit. This can be achieved by the addition of a weak periodic signal to the oscillator input. This is usually accomplished in open loop, but this procedure presents some difficulties which are discussed in the paper. To ensure that this is attained despite uncertainties and possible disturbances on the system, a procedure is suggested to perform control in closed loop. In addition, it is illustrated how a model, estimated from input/output data, can be used in the design. Numerical examples which use the Duffing-Ueda and modified van der Pol oscillators are included to illustrate some of the properties of the new approach.This work has been supported by CNPq (Brazil) under Grant 200597/90-6 and SERC (UK) under Grant GR/H 35286.     

On harmonic resonance in forced nonlinear oscillators exhibiting a Hopf bifurcation

The influence of a periodic forcing on a nonlinear second-orderoscillator close to a Hopf bifurcation point is investigated.The forcing frequency is close to the frequency of the Hopfbifurcation, and the forcing amplitude is assumed to be small.Second-order integral averaging is applied to reduce the givensystem to a planar autonomous system. By a bifurcation and stabilityanalysis of this system, the behaviour of the forced oscillatoris determined. It turns out that two qualitatively differenttypes of behaviour can occur. Either the system has a uniqueattractor, or the system has two competing attractors givingrise to a hysteresis phenomenon, which is known from the Duffingequation. Bifurcation diagrams are presented, and explicit formulaefor the quantities determining the behaviour are given     

An interaction of three resonant modes in a nonlinear lattice

The purpose of this paper is to discuss the Hamiltonian $\text{H}\text{= J}{}_1\text{+ 2J}{}_2\text{+ 3J}{}_3\text{+ αJ}{}_1\text{(2J}{}_2\text{)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{cos}\text{(2φ}{}_1\text{? φ}{}_2\text{) + 2β(J}{}_1\text{J}{}_2\text{J}{}_3{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{cos}\text{(φ}{}_1\text{+ φ}{}_2\text{? φ}{}_3$, where the J_k's are canonically conjugate to the φ_k's (k = 1, 2, 3). In the case β = 0 or α = 0 the corresponding Hamilton equations are integrable. A computer study of the full Hamiltonian was made by Ford and Lunsford (Phys. Rev. A. 1 (1970), 59–70). The present paper obtains analytical results that are confronted with the computer study. The results are obtained by expanding the Hamiltonian into a power series about a certain equilibrium point and constructing the corresponding Gustavson normal form up to 4th-order terms. The Gustavson normal form appears as a member of the enveloping algebra of the Lie algebra SO(2, 1). It is shown that the normal form can be used to explain certain features of Figures 5–9 of the above-mentioned computer study. Moreover the Komogloroff-Arnold-Moser theory is invoked to prove that the quasiperiodic solutions of the β = 0 case can be analytically continued to nonzero but sufficiently small $\text{β}\text{α}$.     

Regularity and decay of solutions of nonlinear harmonic oscillators

We prove sharp analytic regularity and decay at infinity of solutions of variable coefficients nonlinear harmonic oscillators. Namely, we show holomorphic extension to a sector in the complex domain, with a corresponding Gaussian decay, according to the basic properties of the Hermite functions in ${\mathbb{R}}^d$. Our results apply, in particular, to nonlinear eigenvalue problems for the harmonic oscillator associated to a real-analytic scattering, or asymptotically conic, metric in ${\mathbb{R}}^d$, as well as to certain perturbations of the classical harmonic oscillator.     

Chaos synchronization for a class of nonlinear oscillators with fractional order

In this paper, a special kind of nonlinear chaotic oscillator, the Qi oscillator, is studied in detail. Since such systems are shown to possess a relatively wide spectral bandwidth, it is considerably beneficial to practical engineering in the secure communication field. The chaos synchronization problem of the fractional-order Qi oscillators coupled in a master-slave pattern is examined by applying three different kinds of methods: the nonlinear feedback method, the one-way coupling method and the method based on the state observer. Suitable synchronization conditions are derived by using the Lyapunov stability theory, and most importantly, a sufficient and necessary synchronization condition for the case with fractional order between 1 and 2 is presented. Results of numerical simulations validate the effectiveness and applicability of the proposed schemes.     

Coupled nonlinear oscillators and the symmetries of animal gaits

Summary Animal locomotion typically employs several distinct periodic patterns of leg movements, known as gaits. It has long been observed that most gaits possess a degree of symmetry. Our aim is to draw attention to some remarkable parallels between the generalities of coupled nonlinear oscillators and the observed symmetries of gaits, and to describe how this observation might impose constraints on the general structure of the neural circuits, i.e. central pattern generators, that control locomotion. We compare the symmetries of gaits with the symmetry-breaking oscillation patterns that should be expected in various networks of symmetrically coupled nonlinear oscillators. We discuss the possibility that transitions between gaits may be modeled as symmetry-breaking bifurcations of such oscillator networks. The emphasis is on general model-independent features of such networks, rather than on specific models. Each type of network generates a characteristic set of gait symmetries, so our results may be interpreted as an analysis of the general structure required of a central pattern generator in order to produce the types of gait observed in the natural world. The approach leads to natural hierarchies of gaits, ordered by symmetry, and to natural sequences of gait bifurcations. We briefly discuss how the ideas could be extended to hexapodal gaits.     

Universal occurrence of mixed-synchronization in counter-rotating nonlinear coupled oscillators

By coupling counter-rotating coupled nonlinear oscillators, we observe a “mixed” synchronization between the different dynamical variables of the same system. The phenomenon of amplitude death is also observed. Results for coupled systems with co-rotating coupled oscillators are also presented for a detailed comparison. Results for Landau–Stuart and Rössler oscillators are presented.     

Friction dominated dynamics of interacting particles locally close to a crystallographic lattice

A system of particles, in general d‐dimensional space, that interact by means of pair potentials and adjust their positions according to the gradient flow dynamics induced by the total energy of the system is studied. The case when the range of the interaction is of the same order as the mean interparticle distance is considered. It is also assumed that particles, locally, are located close to some crystallographic lattice. An appropriate system of equations that describes the evolution of macroscopic deformation of the crystallographic lattice, as well as the system that describes the evolution of the main crystallographic directions is derived. Well‐posedness of the derived system is studied as well as the stability of the particle system. Same examples of potentials that yield stable and unstable systems are given. Copyright © 2012 John Wiley & Sons, Ltd.     

Generalized coherent states and nonlinear dynamics of a lattice quasispin model

The nonlinear lattice equation of the ϕ⁶ theory is studied by using the technique of generalized coherent states associated to a SU(2) Lie group. We analyze the discrete nonlinear equation with weak interaction between sites. The existence of saddles and centers is shown. The qualitative parametric domains which contain kinks, bubbles and plane waves were obtained. The specific implications of saddles and centers to the parametric first- and second-order phase transitions are identified and analyzed.     

On the dynamical equations of a system of linearly coupled nonlinear oscillators

We consider a system of differential equations that describes the dynamics of an infinite chain of linearly coupled nonlinear oscillators. Some results concerning the existence and uniqueness of global solutions of the Cauchy problem are obtained. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 6, pp. 723–729, June, 2006.     

System of interacting particles and nonlinear diffusion reflecting in a domain with sticky boundary

Summary We study a system of particles and the nonlinear McKean-Vlasov diffusion that is its limit for weak interactions in Statistical Mechanics, reflecting in a domain with sticky boundary. The interaction takes place in particular in the sojourn condition. We show existence and uniqueness for the nonlinear martingale problem, by a contraction argument on time-change. Then we construct the system of particles by a limiting procedure, and show propagation of chaos towards the nonlinear diffusion.

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Résumé Nous étudions un système de particules et la diffusion non-linéaire de type McKean-Vlasov qui en est la limite en Mécanique Statistique pour des interactions faibles, en réflexion dans un domaine à bord collant. L'interaction réside en particulier dans la condition de séjour. Nous montrons l'existence et l'unicité pour le problème de martingales non-linéaire, par une méthode de contraction sur le changement de temps. Nous construisons le système de particules en tant que limite en loi, et démontrons la propagation du chaos vers la diffusion non-linéaire.