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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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Complex Oscillations and Chaos

Complex Oscillations and Chaos

(p.163) 8 Complex Oscillations and Chaos
An Introduction to Nonlinear Chemical Dynamics

Irving R. Epstein

John A. Pojman

Oxford University Press

After studying the first seven chapters of this book, the reader may have come to the conclusion that a chemical reaction that exhibits periodic oscillation with a single maximum and a single minimum must be at or near the apex of the pyramid of dynamical complexity. In the words of the song that is sung at the Jewish Passover celebration, the Seder, “Dayenu” (It would have been enough). But nature always has more to offer, and simple periodic oscillation is only the beginning of the story. In this chapter, we will investigate more complex modes of temporal oscillation, including both periodic behavior (in which each cycle can have several maxima and minima in the concentrations) and aperiodic behavior, or chaos (in which no set of concentrations is ever exactly repeated, but the system nonetheless behaves deterministically). Most people who study periodic behavior deal with linear oscillators and therefore tend to think of oscillations as sinusoidal. Chemical oscillators are, as we have seen, decidedly nonlinear, and their waveforms can depart quite drastically from being sinusoidal. Even after accepting that chemical oscillations can look as nonsinusoidal as the relaxation oscillations shown in Figure 4.4, our intuition may still resist the notion that a single period of oscillation might contain two, three, or perhaps twenty-three, maxima and minima. As an example, consider the behavior shown in Figure 8.1, where the potential of a bromide-selective electrode in the BZ reaction in a CSTR shows one large and two small extrema in each cycle of oscillation. The oscillations shown in Figure 8.1 are of the mixed-mode type, in which each period contains a mixture of large-amplitude and small-amplitude peaks. Mixedmode oscillations are perhaps the most commonly occurring form of complex oscillations in chemical systems. In order to develop some intuitive feel for how such behavior might arise, we employ a picture based on slow manifolds and utilized by a variety of authors (Boissonade, 1976; Rössler, 1976; Rinzel, 1987; Barkley, 1988) to analyze mixed-mode oscillations and other forms of complex dynamical behavior.

Keywords:   Floquet exponent, Jacobian, Lyapunov exponent, Poincare section, bromide-selective electrode, firing number, hyperchaotic attractor, intcrmittency, limit point, next-amplitude map

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