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Dynamic Behavior of a Large Ring of Coupled Self-Excited Oscillators

[+] Author and Article Information
Miguel A. Barron

Departamento de Materiales,
Universidad Autonoma Metropolitana-Azcapotzalco,
Av. San Pablo 180,
Mexico City 02200, Mexico
e-mail: bmma@correo.azc.uam.mx

Mihir Sen

Department of Aerospace and Mechanical Engineering,
University of Notre Dame,
Notre Dame, IN 46556
e-mail: Mihir.Sen.1@nd.edu

Contributed by the Design Engineering Division of ASME for publication in the Journal of Computational and Nonlinear Dynamics. Manuscript received February 3, 2012; final manuscript received October 10, 2012; published online December 19, 2012. Assoc. Editor: D. Dane Quinn.

J. Comput. Nonlinear Dynam 8(3), 034501 (Dec 19, 2012) (5 pages) Paper No: CND-12-1021; doi: 10.1115/1.4023008 History: Received February 03, 2012; Revised October 10, 2012

Interconnected, self-excited oscillators are often found in nature and in engineered devices. In this work, a ring of van der Pol oscillators, each of which is connected to its immediate neighbors, is considered. The focus is on the emergent behavior of a large number of oscillators. Conditions are determined under which time-independent solutions are obtained, and the linear stability of these solutions is investigated. The effect of the singularity of the coupling matrix on the ring dynamics is explored. When this becomes singular, an infinite number of steady states is present, and the phenomenon of oscillation death arises. It is also possible to have, depending on initial conditions, all oscillators with in-phase synchrony, metachronal traveling waves with different wavelengths going around the ring, or standing waves. Interconnected oscillators can propagate information at a group velocity, and the information signal is present as an amplitude modulation.

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Figures

Grahic Jump Location
Fig. 1

N self-excited oscillators in a ring

Grahic Jump Location
Fig. 2

Time series for k=1. (a) Thick i=1, thin i=21, dashed i=41, dotted i=61, dash-dot i=81. (b) Neighboring oscillators. Lines: solid i=40, dashed i=41, dotted i=42.

Grahic Jump Location
Fig. 3

(a) Time series for a=1, k=2, i=1. (b) Corresponding power spectrum. (c) Detail of envelope to show modulated signal traveling through ring; lines: solid i=1, dashed i=2, dotted i=3.

Grahic Jump Location
Fig. 4

(a) Time series for a=10, k=2, i=1. (b) Corresponding power spectrum.

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