Chaos

Chaos

Edited by Arun V. Holden
Copyright Date: 1986
Pages: 332
https://www.jstor.org/stable/j.ctt7ztx2z
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  • Book Info
    Chaos
    Book Description:

    This volume sets out the basic applied mathematical and numerical methods of chaotic dynamics and illustrates the wide range of phenomena, inside and outside the laboratory, that can be treated as chaotic processes.

    Originally published in 1986.

    ThePrinceton Legacy Libraryuses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These paperback editions preserve the original texts of these important books while presenting them in durable paperback editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.

    eISBN: 978-1-4008-5815-6
    Subjects: Biological Sciences, Mathematics

Table of Contents

  1. Front Matter
    (pp. i-iv)
  2. Table of Contents
    (pp. v-vi)
  3. Acknowledgements
    (pp. vii-viii)
  4. Part I Prologue
    • 1 What is the use of chaos?
      (pp. 3-14)
      M. Conrad

      Rössler’s rotating taffy puller provides a beautiful image for appreciating the origin of chaos in one of its simplest forms [16]. Stretching plus folding lead to mixing by distancing neighbouring points and bringing distant points into close proximity. The addition of rotation causes the point to follow a highly irregular path, which Rössler aptly calls a ‘disciplined tangle’. The tangle will be different for each different choice of initial conditions; nevertheless the overall impression given by any two different tangles is basically the same.

      Deterministic dynamical systems of three or more dimensions can exhibit behaviours of the type generated by...

    • 2 A graphical zoo of strange and peculiar attractors
      (pp. 15-36)
      A.V. Holden and M.A. Muhamad

      A large part of the interest in nonlinear dynamics arises from its applications: there is a strong belief that an understanding of the patterns of bifurcations in dynamical systems provides a means of understanding natural phenomena. If a variable measured in the course of an experiment settles down with time, to a constant, or a maintained oscillation, it seems reasonable to assume that it is approaching some stable, maintained course that corresponds to an equilibrium or periodic solution x(x0,t) that is obtained ast→∞, of some appropriate dynamical system.

      (2.1) dx(t)/dt= F(x(t))

      x is a vector inRn, with...

  5. Part II Iterations
    • 3 One-dimensional iterative maps
      (pp. 39-57)
      H. A. Lauwerier

      In this chapter a survey is given of the main properties of one-dimensional maps. My approach is that of an applied mathematician and I have adopted a somewhat informal style. I have tried to give the reader a better understanding of the sometimes very complicated regular and irregular behaviour of discrete dynamical systems, not by stating and proving theorems in endless succession, but by illustrating the fundamental ideas using simple worked-out cases. In the next chapter two-dimensional systems are considered in a similar way. Both chapters share a common bibliography of selected books and papers. Our models are mainly drawn...

    • 4 Two-dimensional iterative maps
      (pp. 58-96)
      H. A. Lauwerier

      This chapter is a continuation of the previous one. The same notations and definitions will be used, with obvious generalisations. Also, almost all interesting features here are illustrated by quadratic maps. Most examples considered in this chapter have their origin in population dynamics such as the logistic-delay equation, a predator–prey model and models of host–parasitoid interaction. Particular attention is given to the phenomena of Hopf bifurcation and Arnold tongues. The calculus of normal forms is presented here in a way leading to quantitative results such as the size of the Hopf circle (ellipse) and the position of the...

  6. Part III Endogenous chaos
    • 5 Chaos in feedback systems
      (pp. 99-110)
      A. Mees

      Nonlinear systems theory is of great importance to anyone interested in feedback systems. It is also true that the theory of feedback systems has made important contributions to nonlinear systems theory. This chapter discusses some chaotic feedback systems, drawn from electronic circuit theory and elsewhere, and shows how they may be analysed. So far, most of the techniques required have been taken directly from the usual differential and difference equation theory, but some results with a more control system-theoretic flavour are now available.

      Any dynamical system described, for example, by difference or differential equations may be regarded as a feedback...

    • 6 The Lorenz equations
      (pp. 111-134)
      C. Sparrow

      The Lorenz equations, named after Ed Lorenz who first introduced them as a model of a two-dimensional convection [21], have been important for a number of different reasons at various times in the past 20 years or so. Initially they were remarkable just because they are a simple three-dimensional nonlinear system of autonomous ordinary differential equations showing chaotic behaviour; though many such systems are now known (as described elsewhere in this volume), in 1963 such systems were almost unheard of. So much so, in fact, that, despite the beauty of Lorenz’s original paper, and the remarkable progress he made in...

    • 7 Instabilities and chaos in lasers and optical resonators
      (pp. 135-157)
      W. J. Firth

      Turbulence in lasers and other optical systems is a newly recognised rather than new phenomenon. What is especially interesting about such systems is the possibility that experiments can be constructed which are sufficiently simple and close to theoretical models that routes to chaos can be studied in detail. That is an as yet unfulfilled hope, but the recent pace of experimental progress has been so great that there is every reason for optimism.

      This chapter aims to introduce the reader to the most important concepts and systems in the field of optical chaos. It is necessarily selective and undoubtedly subjective...

    • 8 Differential systems in ecology and epidemiology
      (pp. 158-178)
      W. M. Schaffer and M. Kot

      This chapter is concerned with the motion of populations, including human diseases, which is to say their ups and downs. Motion, whether in biology or physics, can take many forms. In recent years, the beginnings of a taxonomy have emerged. We begin by listing some of the species in this dynamical bestiary:

      (1)Point attractors. In the absence of perturbations, the system approaches a stable point. An ecological example would be competition between two species for a common set of limiting resources (e.g. [19]).

      (2)Limit cycles. The attractor is a closed curve, topologically equivalent to a circle. Limit cycles,...

    • 9 Oscillations and chaos in cellular metabolism and physiological systems
      (pp. 179-208)
      P.E. Rapp

      It was once common wisdom that biochemical reactions inevitably converged rapidly to a thermodynamic steady state and that this steady state was unique. Similarly, at the systemic level, a restrictive view of the concept of homeostasis dominated physiological thinking, and it was supposed that physiological control functioned exclusively to restore transiently disturbed systems to a steady state. It is now recognised that this is not the case. Complex dynamical behaviour is an aspect of biological regulation. Two such behaviours, sustained oscillations and chaos, are considered here.

      Two periodic biochemical systems will be described in section 9.2, the glycolytic oscillator, and...

  7. Part IV Forced chaos
    • 10 Periodically forced nonlinear oscillators
      (pp. 211-236)
      K. Tomita

      It is now widely recognised that chaos is a basic mode of motion underlying almost all natural phenomena, and it is neither exceptional nor peripheral as had been conjectured until quite recently. However, there are a variety of situations under which chaos emerges, and chaos becomes increasingly complex as the dimension of the reference space is increased. To distinguish chaos from a mere kinematical complexity, we have been interested in examining a model of the lowest possible dimensions that leads to definite chaos. A forced nonlinear oscillator clearly suits this purpose in the sense that it provides the simplest nontrivial...

    • 11 Chaotic cardiac rhythms
      (pp. 237-256)
      Leon Glass, Alvin Shrier and Jacques Bélair

      ‘Chaos’ is a term that is used to denote dynamics in deterministic mathematical equations in which the temporal evolution is aperiodic in time and sensitive to the initial conditions. But, as the above quotation illustrates, in electrocardiography ‘chaotic’ dynamics have been recognised for a much longer period of time than the recent surge of interest among mathematicians and physicists. Of course, the use of the term ‘chaotic’ by cardiologists is purely descriptive, and does not reflect a detailed theoretical analysis of the underlying mechanisms. In this chapter we describe experimental studies on the effects of periodic electrical stimulation of spontaneously...

    • 12 Chaotic oscillations and bifurcations in squid giant axons
      (pp. 257-270)
      K. Aihara and G. Matsumoto

      The fundamental functions of neurones, such as the generation and propagation of action potentials, are supported by nonlinear dynamics peculiar to the nerve membranes. The nonlinear neural dynamics produces different attractors and bifurcations in far from equilibrium conditions. For example, stable limit cycles with Hopf bifurcations and multiple equilibrium points with saddle-node bifurcations have been analysed both experimentally and theoretically [1,2,11,14,15,17,19,20,24,33].

      Self-sustained oscillations, or the spontaneous repetitive firing of action potentials in squid giant axons, can be understood in terms of a dissipative structure that has spatio-temporal order and behaves as a nonlinear neural oscillator [1,24,26]. In this chapter, we...

  8. Part V Measuring chaos
    • 13 Quantifying chaos with Lyapunov exponents
      (pp. 273-290)
      A. Wolf

      Chaos has been discovered both in the laboratory and in the mathematical models that describe a wide variety of systems [1, 3]. In common usage chaos is taken to mean a state in which chance prevails. To the nonlinear dynamicist the word chaos has a more precise and rather different meaning. A chaotic system is one in which long-term prediction of the system’s state is impossible because the omnipresent uncertainty in determining itsinitialstate grows exponentially fast in time. The rapid loss of predictive power is due to the property that orbits (trajectories) that arise from nearby initial conditions...

    • 14 Estimating the fractal dimensions and entropies of strange attractors
      (pp. 291-312)
      P. Grassberger

      Physical systems typically involve a huge number of degrees of freedom (≳ 1020, say). Since it is impossible to treat all of them explicitly, one performs some kind of ‘coarse graining’. After this is done, one deals explicitly with few variables only, but in general these variables evolve nondeterministically with time. This was widely considered to be the only source of randomness in nature until the ‘chaos revolution’ of recent years spread the concept of deterministic chaos.

      Deterministic chaos is also related to coarse graining, but of an essentially different kind. It results from the fact that we not only...

  9. Part VI Epilogue
    • 15 How chaotic is the universe?
      (pp. 315-320)
      O. E. Rössler

      The title of the chapter is better than anything can possibly follow it. Originally I planned to write something nice and grandiose about Anaxagoras’ invention of chaos as an explanation of the universe, and his ideas about transfinite iteration and the subtlety of the single ‘immiscible’ substance, the mind. Diesel automobile engines, the geyser Old Faithful, X-ray bursters in the sky, and autonomous nerve equations (including a 3-variable FitzHugh equation whose chaotic analogue computer solutions were shown to me by its inventor in late 1976) were then to follow suit—to illustrate the ubiquity of trajectorial mixing in simple differential...

  10. Index
    (pp. 321-324)