Diffusive coupling, dissipation, and synchronization

Hale, Jack K.

doi:10.1007/bf02219051

Cited by 147 publications

(111 citation statements)

References 37 publications

Supporting

Mentioning

109

Contrasting

Order By: Relevance

“…Full synchronization in twodimensional ͑2D͒ and 3D lattices of locally coupled limit-cycle 26,27 systems and chaotic oscillators [33][34][35][36]16 has recently been studied analytically.…”

Section: Introductionmentioning

confidence: 99%

“…31 Conditions of asymptotic stability of full synchronization in 2D lattices of nonidentical chaotic oscillators were obtained in a series of papers. [33][34][35][36] In particular, the stability conditions of synchronization in a square lattice of forced Duffing type nonidentical oscillators with a dissipative nearestneighbor coupling and Dirichlet boundary conditions were presented. 33 In another paper, 35 these results were extended to the case of Neumann or periodic boundary conditions.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Persistent clusters in lattices of coupled nonidentical chaotic systems

Belykh

Белых

Nevidin

et al. 2003

Chaos: An Interdisciplinary Journal of Nonlinear Science

114

View full text Add to dashboard Cite

Two-dimensional ͑2D͒ lattices of diffusively coupled chaotic oscillators are studied. In previous work, it was shown that various cluster synchronization regimes exist when the oscillators are identical. Here, analytical and numerical studies allow us to conclude that these cluster synchronization regimes persist when the chaotic oscillators have slightly different parameters. In the analytical approach, the stability of almost-perfect synchronization regimes is proved via the Lyapunov function method for a wide class of systems, and the synchronization error is estimated. Examples include a 2D lattice of nonidentical Lorenz systems with scalar diffusive coupling. In the numerical study, it is shown that in lattices of Lorenz and Rössler systems the cluster synchronization regimes are stable and robust against up to 10%-15% parameter mismatch and against small noise. © 2003 American Institute of Physics. ͓DOI: 10.1063/1.1514202͔Lattices of coupled chaotic oscillators model many systems of interest in physics, biology, and engineering. In particular, the phenomenon of cluster synchronization, i.e., the synchronization of groups of oscillators, has received much attention. This phenomenon depends heavily on the coupling configuration between the oscillators. While in a network of globally coupled identical oscillators in principle any subset of the oscillators may synchronize, in diffusively coupled oscillators only very few of the decompositions into subsets of synchronized oscillators are possible. They have been characterized in recent papers in detail "see Refs. 14-16…. In that work, the oscillators were assumed to be identical and the symmetries of the resulting system of coupled oscillators were exploited. In more realistic models of physical systems, however, the individual oscillators have slightly different parameters and therefore the perfect symmetries in the coupled systems no longer exist. Similarly, perfect cluster synchronization cannot exist anymore, but approximate synchronization is still possible. The question then arises whether the cluster synchronization regimes that are observed in systems with identical oscillators persist approximatively under small parameter mismatch and the addition of small noise. This paper gives a positive answer and explores the limit of approximate synchronization both analytically and numerically.

show abstract

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Persistent clusters in lattices of coupled nonidentical chaotic systems

Belykh

Белых

Nevidin

et al. 2003

Chaos: An Interdisciplinary Journal of Nonlinear Science

114

View full text Add to dashboard Cite

show abstract

“…It also is possible to consider equations (4.11) for a nonlinearity H(s, u t ) which depends upon s. In this case, one can show that dist The interested reader can obtain much more information about synchronization of ODE and PDE by consulting the references in [16].…”

Section: Synchronization For the Partial Nfde In Proposition 41 Wementioning

confidence: 99%

A general class of evolutionary equations

Hale

2007

Ukr Math J

View full text Add to dashboard Cite

Using observable quantities and state variable of a dynamical process, a general evolutionary equation is defined which unifies classical ordinary differential equations, partial differential equations, and hereditary systems of retarded and neutral type. Specific illustrations are given using transmission lines nearest neighbor coupled at the boundary and the theory of heat transfer in solids. Some spectral theory for linearization of the equations also is discussed.Za dopomohog spostereΩuvanyx velyçyn ta zminno] stanu dynamiçnoho procesu vyznaçeno zahal\ne evolgcijne rivnqnnq, wo uzahal\ng[ klasyçni zvyçajni dyferencial\ni rivnqnnq, dyferencial\ni rivnqnnq z çastynnymy poxidnymy ta spadkovi systemy iz zapiznennqm i systemy nejtral\noho typu. Navedeno specyfiçni ilgstraci] z vykorystannqm linij transmisi] iz zçeplennqm ,,najblyΩçyx susidiv" na meΩi ta teori] teploperenosu u tverdyx tilax. Rozhlqnuto takoΩ pevnu spektral\nu teorig dlq linearyzaci] rivnqn\. Introduction.Motivated by the fact that a dynamical system may evolve through an observable quantity rather than the state of the system, a general class of evolutionary equations is defined. This class includes standard ordinary and partial differential equations as well as functional differential equations of retarded and neutral type. In this way, the theory serves as a unification of these classical problems.Included in this general formulation is a general theory for the evolution of temperature in a solid material. In the general case, temperature is transmitted as waves with a finite speed of propagation. Special cases include a theory of delayed diffusion.We describe also in some detail a lattice on a circle where each point on the lattice is a transmission line for current and voltage whose dynamics is governed by a linear hyperbolic equation on [0, 1] with dynamic boundary conditons given by the circuitry on the line. The systems are coupled to their nearest neighbor at the end point 1 through resistors. A limiting process letting the distance between the lattice points approach zero leads to an interesting set of partial differential equations on [0, 1] × S 1 with a hyperbolic equation on [0, 1] and a parabolic equation on S 1 . These equations have not been analyzed in detail. However, we can show that the voltage at 1 satisfies a partial neutral functional differential equation. We analyze some properties of these equations including synchronization and the behavior of solutins near periodic orbits.There are other applications which involve partial differential equations on lattices for which the dynamics on each lattice point is governed by a partial differential equation on a bounded domain Ω. These systems could be coupled to neighbors through interaction on some subset of the boundary ∂Ω of Ω. If certain limiting processes are justified, one can obtain a partial differential equation on Ω together with another partial differential equation on another domain Ω 1 (determined by the nature of the lattice). We do not discuss this in the text, but ...

show abstract

“…The system dynamics can be modified through the information exchange, and, as in the consensus problem, the goal of the interconnection is to reach synchronization to a common solution of the individual dynamics [5][6][7][8].…”

mentioning

confidence: 99%

Synchronization in networks of identical linear systems

Scardovi

Sepulchre

2008

2008 47th IEEE Conference on Decision and Control

158

View full text Add to dashboard Cite

The paper investigates the synchronization of a network of identical linear state-space models under a possibly time-varying and directed interconnection structure. The main result is the construction of a dynamic output feedback coupling that achieves synchronization if the decoupled systems have no exponentially unstable mode and if the communication graph is uniformly connected. The result can be interpreted as a generalization of classical consensus algorithms. Stronger conditions are shown to be sufficient -but to some extent, also necessary -to ensure synchronization with the diffusive static output coupling often considered in the literature.

show abstract

Diffusive coupling, dissipation, and synchronization

Cited by 147 publications

References 37 publications

Persistent clusters in lattices of coupled nonidentical chaotic systems

Persistent clusters in lattices of coupled nonidentical chaotic systems

A general class of evolutionary equations

Synchronization in networks of identical linear systems

Contact Info

Product

Resources

About