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An Introduction to Nonlinear Chemical DynamicsOscillations, Waves, Patterns, and Chaos$
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Irving R. Epstein and John A. Pojman

Print publication date: 1998

Print ISBN-13: 9780195096705

Published to Oxford Scholarship Online: November 2020

DOI: 10.1093/oso/9780195096705.001.0001

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PRINTED FROM OXFORD SCHOLARSHIP ONLINE (oxford.universitypressscholarship.com). (c) Copyright Oxford University Press, 2021. All Rights Reserved. An individual user may print out a PDF of a single chapter of a monograph in OSO for personal use. date: 17 January 2022

Coupled Oscillators

Coupled Oscillators

Chapter:
(p.255) 12 Coupled Oscillators
Source:
An Introduction to Nonlinear Chemical Dynamics
Author(s):

Irving R. Epstein

John A. Pojman

Publisher:
Oxford University Press
DOI:10.1093/oso/9780195096705.003.0018

We have thus far learned a great deal about chemical oscillators, but, except in Chapter 9, where we looked at the effects of external fields, our oscillatory systems have been treated as isolated. In fact, mathematicians, physicists, and biologists are much more likely than are chemists to have encountered and thought about oscillators that interact with one another and with their environment. Forced and coupled oscillators, both linear and nonlinear, are classic problems in mathematics and physics. The key notions of resonance and damping that arise from studies of these systems have found their way into several areas of chemistry as well. Although biologists rarely consider oscillators in a formal sense, the vast variety of interdependent oscillatory processes in living systems makes the representation of an organism by a system of coupled oscillators a less absurd caricature than one might at first think. In this chapter, we will examine some of the rich variety of behaviors that coupled chemical oscillators can display. We will consider two approaches to coupling oscillatory chemical reactions, and then we will look at the phenomenology of coupled systems. We begin with some general considerations about forced oscillators, which constitute a limiting case of asymmetric coupling, in which the linkage between two oscillators is infinitely stronger in one direction than in the other. As an aid to intuition, picture a child on a swing or a pendulum moving periodically. The forcing consists of an impulse that is applied, either once or periodically, generally with a frequency different from that of the unforced oscillator. In a chemical oscillator, the forcing might occur through pulsed addition of a reactive species or variation of the flow rate in a CSTR. Mathematically, we can write the equations describing such a system as . . . dX/dt = f(x) + ε g(X, t) (12.1) . . . where the vector x contains the concentrations, the vector function f(x) contains all the rate and flow terms in the absence of forcing, g(x) represents the appropriately scaled temporal dependence of the forcing, and the scalar parameter e specifies the strength of the forcing.

Keywords:   acetylcholine, birhythmicity, chlorite-iodide reaction, gluing bifurcation, limit cycle, neurotransmitter, oscillator death, phase of oscillator, quasiperiodicity, rhythmogenesis

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