# Dynamical Chaos

MICHAEL V. BERRY
IAN C. PERCIVAL
NIGEL OSCAR WEISS
Pages: 208
https://www.jstor.org/stable/j.ctt7zvb0s

1. Front Matter
(pp. [i]-[iv])
(pp. [v]-2)
3. Chairman’s introduction
(pp. 3-4)
E. C. Zeeman

The understanding of chaos and strange attractors is one of the most exciting areas of mathematics today. It is the question of how the asymptotic behaviour of deterministic systems can exhibit unpredictability and apparent chaos, due to sensitive dependence upon initial conditions, and yet at the same time preserve a coherent global structure. The field represents a remarkable confluence of several different strands of thought.

1. Firstly came the influence of differential topology, giving global geometric insight and emphasis on qualitative properties. By qualitative properties I mean invariants under differentiate changes of coordinates, as opposed to quantitative properties which are invariant...

4. Diagnosis of dynamical systems with fluctuating parameters
(pp. 5-8)
D. Ruelle

In recent years it has been ascertained that many time evolutions observed in Nature exhibit the features ofchaos. This means that they are deterministic time evolutions involving only a finite number of degrees of freedom, but that a complicated non-periodic behaviour is observed, due tosensitive dependence on initial condition. Mathematically, the deterministic time evolution corresponds to an autonomous differentiable dynamical system, sensitive dependence on initial condition means that a small perturbation of the initial condition will grow exponentially with time (as long as it does not become too large). The asymptotic evolution of the system takes place on...

5. Nonlinear dynamics, chaos and complex cardiac arrhythmias
(pp. 9-26)
L. Glass, A. L. Goldberger, M. Courtemanche and A. Schrier

The normal adult human heart at rest usually beats at a rate of between 50 and 100 times per minute. In many circumstances, some of which are life-threatening, but most of which are not, the normal rhythmicity is altered, resulting in abnormal rhythms called cardiac arrhythmias. The point of this paper is to show that a branch of mathematics called nonlinear dynamics may be useful in the analysis of physiological processes believed to underlie normal heart rate regulation and some cardiac arrhythmias.

The idea that mathematical analysis can play a role in understanding cardiac arrhythmias is not novel. Indeed, in...

6. Chaos and the dynamics of biological populations
(pp. 27-44)
R. M. May

A central task for population biologists is to disentangle, from the superimposed fluctuations caused by environmental noise and other chance events, the underlying mechanisms that regulate natural populations so that no one species of plant or animal increases without bound. Such studies lead us to consider simple equations that might describe the dynamics of natural populations if environmental noise and heterogeneity could be stripped away. A clear understanding of the dynamics of these simple and deterministic, but nonlinear, models then serves as a point of departure for evaluating the effects of various kinds of complications associated with environmental unpredictability and...

7. Fractal bifurcation sets, renormalization strange sets and their universal invariants
(pp. 45-62)
D. A. Rand

The traditional renormalization formalism as invented to study phase transitions and as used in dynamical systems to study the universal properties of the transition to chaos relies upon finding a hyperbolic saddle point for a judiciously chosen transformation of some function space. Then the geometrical and dynamical structure of the saddle point and its stable manifold is used to deduce physically and mathematically interesting consequences. In this paper I shall discuss a more general situation which has a number of interesting applications to dynamical systems and which, I believe, is of even wider interest because it will have applications in...

8. From chaos to turbulence in Bénard convection
(pp. 63-70)
A. Libchaber

Here I present some recent results on Bénard convection, limited to the understanding of the various transitions to chaos and turbulence. The experiments are restricted to fluids with low Prandtl number; mercury first and helium gas next. Also the geometry of the cell is such that the aspect ratio is small; thus above the onset of convection one or two rolls can be accommodated. As has been noted, this allows the time-dependent state to be analysed in terms of dynamical-systems theory with a small number of degrees of freedom, for low Rayleigh numbers. In the first part, results of a...

9. Dynamics of convection
(pp. 71-86)
N. O. Weiss

Most theoretical studies of temporal chaos in dissipative systems have been concerned with low-order systems whose behaviour can be analysed in detail. These simple models can be related to laboratory experiments or to numerical solutions of partial differential equations, where spatial structure is significant. Fluid dynamics provides a range of problems in which complicated spatio-temporal behaviour has been observed: thermal convection and flow between differentially rotating cylinders are classical examples (Guckenheimer 1986). In Taylor–Couette flow certain bifurcations involving changes in spatial structure and transitions to time-dependent and chaotic motion have been observed both in laboratory experiments and in precise...

10. Chaos: a mixed metaphor for turbulence
(pp. 87-96)
E. A. Spiegel

Many features of chaos are mirrored in turbulent flows and this has led some writers on chaos to use the words interchangeably. Both terms appear in ordinary language, where they do have an overlap of meaning, and both are used in scientific settings where they do not have generally accepted, precise definitions. Whether satisfactory definitions of these terms can be found before the subjects are better understood is not clear. But practitioners of both subjects can name the symptoms that should be present before a process can be considered to be chaotic or turbulent.

The property that a slight cause...

11. Arithmetical theory of Anosov diffeomorphisms
(pp. 97-108)
F. Vivaldi

Let us consider the following area-preserving map of the two-dimensional torus:${r}'=Ar;\quad A=\left[ \begin{array}{*{35}{r}} 4 & 15 \\ 1 & 4 \\ \end{array} \right];\quad \det \,A=1,\quad \text{tr}\,A=8.\caption {(1.1)}$

Ais a hyperbolic system, its eigenvalues being real. It belongs to the family of Anosov diffeomorphisms, which tipify purely chaotic motion in hamiltonian systems. They are characterized by uniform hyperbolicity and a dense set of unstable periodic orbits (for a review, see Franks 1970).

Let us now consider a rational point on the torus.

r=)n1/p,n2/p), (1.2)s

wherepis a prime number (except 2, 3 and 5), andn1andn2are integers between 0 andp−1. One finds that the orbit throughrhas the...

12. Chaotic behaviour in the Solar System
(pp. 109-130)
J. Wisdom

The Solar System is generally perceived as evolving with clockwork regularity. Indeed, it was a search for the principles that underlie the perceived regularities in the motions of the planets that culminated in Newton’s formulation of the laws of mechanics and universal gravitation 300 years ago. Recently it has been widely recognized that dynamical systems possess irregular as well as regular solutions. Irregular solutions of deterministic equations of motion are termed ‘chaotic’. The Solar System is just another dynamical system; the study of this preeminent dynamical system is not untouched by the discoveries in nonlinear dynamics. Solar System dynamics encompasses...

13. Chaos in hamiltonian systems
(pp. 131-144)
I. C. Percival

Traditionally hamiltonian systems with a finite number of degrees of freedom have been divided into those with few degrees of freedom, which were supposed to exhibit some kind of regular ordered motions, and those with large numbers of degrees of freedom for which the methods of statistical mechanics should be used.

The past few decades have seen a complete change of view, which affects almost all the practical applications. The motion of a conservative hamiltonian system is usually neither completely regular nor properly described by the methods of statistical mechanics. A typical system is mixed it exhibits regular or chaotic...

14. Particle confinement and the adiabatic invariance
(pp. 145-156)
B. V. Chirikov

The problem of controlled nuclear fusion, aimed at the peaceful use of thermonuclear power, has stimulated a broad range of fundamental research in plasma physics. Among other problems, particle dynamics in electromagnetic fields has been intensively studied. On one hand, these fields are supposed to confine particles for a fairly long time within a bounded domain of space (the so-called ‘magnetic traps’). On the other, they are expected to heat the same particles up to a very high thermonuclear temperature.

In these studies physicists have come across the first examples of a rather peculiar phenomenon, calleddynamical chaos, that is...

15. Semi-classical quantization, adiabatic invariants and classical chaos
(pp. 157-170)
W. P. Reinhardt and I. Dana

Adiabatic switching is an old idea. In 1911 Einstein is quoted (Jammer 1966) as saying, in response to Lorentz asking about a quantized pendulum retaining its quantization,

If the length of the pendulum is changed infinitely slowly, its energy remains equal tohvif it was originallyhv.

Ehrenfest (1916), whose interest in classical adiabatic invariants was stimulated by their use in the derivation of the Wien displacement law, suggested the general classical analogue dubbed ‘adiabatenhypothese’ by Einstein (1914),

If a system be affected in a reversible adiabatic way, allowed motions are transformed into allowed motions.

The elucidation of these...

16. Some geometrical models of chaotic dynamics
(pp. 171-182)
C. Series

The free motion of a particle on a surface of constant negative curvature was probably the first example of what is now described as chaotic motion. Hadamard (1898) made a detailed study of geodesies on such surfaces, and in particular noted the occurrence on certain surfaces of families of geodesies whose cross section exhibits a Cantor- or fractal-like structure. Geodesies on these surfaces diverge exponentially at a constant rate, thus among those geodesies that remain in a bounded portion of the surface (the non-wandering set) one has all the ingredients for fully chaotic motion. Moreover, one has to hand a...

17. The Bakerian Lecture, 1987 Quantum chaology
(pp. 183-200)
M. V. Berry

In Henry Baker’s day, ‘chaology’ meant ‘The history or description ofthechaos’ (O.E.D. 1893).Thechaos was the state of the world before creation (‘without form, and void’) so that chaology was a theological term. That area of theology has not been very active for the past two centuries (unless we extend its scope to include some recent speculations in cosmology) and so we are justified in reviving the term chaology, which will now refer to the study of unpredictable motion in systems with causal dynamics, as exemplified by the contributions at the meeting on ‘dynamical chaos’ of which...

18. Back Matter
(pp. 201-202)