Chaos

by Arun V. Holden (Editor)

Overview

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.

The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover 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.

Product Details

ISBN-13:	9780691610535
Publisher:	Princeton University Press
Publication date:	07/14/2014
Series:	Princeton Legacy Library , #461
Pages:	334
Product dimensions:	6.10(w) x 9.10(h) x 0.80(d)

Read an Excerpt

Chaos

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By Aron V. Holden

PRINCETON UNIVERSITY PRESS

Copyright © 1986 Manchester University Press
All rights reserved.
ISBN: 978-0-691-08424-4

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CHAPTER 1

What is the use of chaos?

M. Conrad

Departments of Computer Science and Biological Sciences, Wayne State University, Detroit, Michigan 48202, USA

1.1 A functional question

Rössler's rotating taffy puller provides a beautiful image for appreciating the origin of chaos in one of its simplest forms. 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 the rotating taffy machine. Despite their determinism, the behaviours generated look extremely random. This is what it means to say that such systems are effective mixing devices. The discovery of chaos suggests that the question of whether a given random appearing behaviour is at base probabilistic or deterministic may be undecidable.

It is by now probably fair to say that many plausible dynamical models for complex biological systems become chaotic for some choice of the parameters. Chaotic solutions have been found for equations similar to chemical kinetic equations, equations governing neurone dynamics, and population dynamics equations. The question I have been asked to address is what the function, if any, of such chaotic dynamics might be. Such questions have a teleological ring. Nevertheless it is useful and justified to look at living systems from the functional point of view. This is due to the enormous asymmetry between existence and nonexistence. Some biological systems are so organised that they remain in the game of life. Others go out of existence. The asymmetry is simply that it is the existing systems which are of interest to us. It is legitimate to ask what it is about the organisation of these systems that allows them to persist. As soon as we do so, we are adopting a functional point of view.

Complex biological systems, such as neurones and the immune system, are the end result of the long process of evolution by variation and natural selection. Physiological mechanisms which control the population dynamics are also subject to variation and selection. The equations suitable for describing, say, the neurone need not be the same for all organisms, and probably are not. In biology the equations are as much the product of evolution as traits such as eye colour. If biological dynamics could be recorded in the fossil record, it would undoubtedly show evolutionary changes as dramatic as those exhibited by bones and other biological structures. The dynamics and parameter values could be selected to exhibit chaos; or they could be selected to preclude chaos.

However justified and even necessary the functional question is from the biological point of view, it is replete with dangers. What is the function of Rössler's rotating taffy puller? To make taffy, to advertise taffy, to provide employment, to earn a profit, to inspire Otto Rössler? Or, in an emergency, to serve as a lever or as a weapon? All biological entities and machines are multifunctional. How they have contributed to staying in the game of life cannot be specified completely, and how they might contribute in the future is an open question. It is dangerous to suppose that natural selection wants this or that. What is selected changes in the course of evolution and not all the phenomena of life are controlled by selection.

This caveat applies to the functional interpretation of chaos. However, in one respect, chaos is simpler to analyse functionally than most biological structures or processes. This results from the fact that it is so difficult to distinguish deterministic chaos from highly random behaviour. In so far as chaos contributes to the variability of biological matter, any analysis of the functional significance of variability a fortiori applies to the phenomenon of chaos. Fortunately, a systematic theory of biological variability is available. This is adaptability theory.

1.2 Overview of adaptability theory

A thorough review of adaptability theory can be found in my book on this subject, and more limited reviews in a number of earlier papers, It would be duplicative to re-present the theory here. It should be sufficient to state the central idea verbally and to indicate the connection to dynamical systems theory.

By adaptability I mean the ability of a system to continue to function in the face of an uncertain or unknown environment. The system of interest is usually a living system, say an organism, a population, or even a whole community. The environment is everything that influences this system. It may include other biological systems as well as the physical environment, and may in part be influenced by the activities of the system of interest. For simplicity I will call the system of interest the biological system and its surroundings the environment.

In adaptability theory the biological system is treated as a system with a set of states and a transition scheme, ω, governing the state-to-state transitions. The transition scheme is unknown, but notationally consists of a set of probabilities for the state of the biological system at time t + 1 given the state of both the biological system and environment at time t. For simplicity it is assumed that the state set is discrete and finite. The environment is also treated as a system with a set of states with a probabilistic, generally unknown, transition scheme, denoted by ω. In general, transition schemes have a deterministic aspect (for example, connected with the life cycle of the organism or the cycle of the year) and an indeterminate aspect (for example, connected with mutation or unpredictable weather patterns).

The fundamental quantities in adaptability theory are measures of behavioural uncertainty such as the following.

(1) H(ω) = behavioural uncertainty of the environment.

(2) H([??]) = potential behavioural uncertainty of the biological system ([??] is the transition scheme of the biological system in the most uncertain tolerable environment).

(3) H([??]|[??]) = potential behavioural uncertainty of the biological system given the behaviour of the environment. This increases as the ability to anticipate the environment increases or as the uncertainty which the biological system internally generates increases.

(4) H([??]|[??]) = potential behavioural uncertainty of the environment given the behaviour of the biological system. This increases as the indifference to the environment increases, for example as the organism lives in a smaller region of space.

The main question of adaptability theory is: what is the relation between the statistical properties of the biological system and the statistical properties of the environment? Ignoring for the time being the all-important question of detailed statistical structure, it is possible to summarise the situation by the following simple formula

(1.1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The left-hand side represents the adaptability of the biological system. The right-hand side represents the actual uncertainty of the environment. The arrow represents a plausible evolutionary tendency of adaptability. All forms of adaptability are costly. Adaptabilities which are never used tend to disappear in the course of evolution, so the magnitude of the left-hand side tends to drop in the direction of the actual uncertainty of the environment.

The magnitude of the terms in eqn (1.1) could be individually high, yet the adaptability low. In this case there would be a great deal of biological variability, but not much of it would appear as adaptability. These reserves of variability can be converted to adaptability in the event of a crisis. Some of the variability which contributes to the magnitudes of the entropies serves to increase reliability or to enhance the evolutionary transformability of the system.

The transition-scheme description is connected to descriptions in terms of biological variables by utilising the fact of hierarchy. Ecosystems consist of compartments such as communities, populations, organisms, cells and genes. These are associated with variables such as locations and numbers of organisms, physiological states of cells, and base sequence in DNA. Let the symbol [??]ij represent the transition scheme of compartment i at level j in terms of its subcompartments at the next lower level. The uncertainty of the whole biological system being considered can be expressed in terms of a sum of effective entropies of each compartment:

(1.2) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Each effective entropy is a sum of conditional and unconditional entropies

(1.3) He ([??]ij) = fH([??]ij) + conditional terms

where f is a normalising coefficient. The unconditional part is the behavioural uncertainty of the compartment considered in isolation, and the conditional parts express the correlation between this uncertainty and the modifiabilities of other compartments. The uncertainty taken in isolation will be called the modifiability and the conditional terms will be the independence terms.

The adaptability is not the sum of the modifiabilities. If a system is more decentralised (that is, if the parts are more independent), the adaptability is greater for given observable modifiabilities of the parts. If constraints are added to the system which decrease the conditional entropies, the adaptability must decrease or be compensated by enhanced anticipation, increased indifference, or development of new subsystems with high behavioural uncertainty (such as the central nervous system or the immune system). If no such compensation occurs, the niche must narrow or the community must absorb disturbances at the level of population fluctuations. This is an acceptable mode of adaptability for micro-organisms since these are fast growing.

Adaptability cannot always decrease. This would be incompatible with the tenure of life on Earth. Many factors control the rise and fall of adaptability in the course of evolution. When an evolutionary system loses too much adaptability, or when the uncertainty of its environment increases, it is likely to go into a crisis. The crisis instigates a series of changes which result in the renewal of the adaptability structure.

1.3 Adaptability, dynamics and the place of chaos

Many dynamical models in biology are of a continuous dynamical nature. The transition schemes of adaptability theory are discrete and probabilistic. It is possible to cross-correlate the two approaches by using the idea of a tolerance, that is of a relation on the states which is symmetric, reflexive, but not transitive. State A is similar to state B, which is similar to state C. But A is not necessarily similar to C. In this way an aspect of continuity can be conferred on discrete transition schemes. Analogues of concepts such as neutral stability, asymptotic stability and structural stability can be defined. The question can then be asked: what is the relation between the components of adaptability theory and these different dynamical concepts of stability?

Without going into details, the general situation can be pictured thus. The modifiability terms correlate to instabilities of dynamical models. This is because the alternative structures and modes of behaviour which contribute to modifiability can correspond to alternative weakly or strongly stable states. In this case disturbances are absorbed by instabilities. The modifiability terms can also correlate to asymptotic orbital stability. This occurs when the disturbance is dissipated by direct return to a strongly stable state. In this case the modifiability consists in a variation around the strongly stable state which in time is dissipated into the heat bath. The independence terms correlate either to structural stability or to weakening of the interaction of two systems. The correlation here is somewhat subtle and depends on whether one is near to the bifurcation points or far from them. If a system is structurally stable, it can undergo a variation in response to disturbance, which, however, does not qualitatively alter its structure or behaviour. In this case the behaviour of the system is less dependent on parametric compartments (see [7] for full details).

The problem of adaptability theory is to ascertain how the different forms of adaptability will be allocated to different parts of a system given the constraints which are present (such as morphological constraints). If one compartment is to be maintained in a very certain state, this must be at the expense of some other compartment being in an uncertain state. The uncertainty serves to absorb the disturbance. Corresponding to this economics of adaptability components there is an economics of stability and instability in biological systems. This is due to the fact that adaptability components correlate to forms of stability and instability in dynamical descriptions. If the dynamics of one level of organisation, say the population level, is to be very stable, it is necessary for disturbances to be absorbed by instabilities at some other level. For example, neurobehavioural instabilities or genetic variability may protect the population dynamics from disturbance and therefore allow it to appear highly stable. Alternatively, the stability of the state of internal fluids in the vertebrate may be obtained at the expense of intricate dynamical instabilities in the nervous and immune systems.

To fit chaos into this framework it is necessary to remember that the transition schemes are generally probabilistic. Not all modifiability is connected with intrinsically random processes. For example, a compartment might be able to absorb disturbances by functioning in one of several dynamical regimes. In the absence of any information about the environment, switching from one regime to another may appear virtually random. In reality the situation is completely deterministic. However, it is also possible to consider situations in which the modifiability of a compartment is due to completely intrinsic factors. This is the case in genetic variability, in variability of immunoglobin molecules in the immune system, and in exploratory processes in the nervous system. This intrinsic modifiability also makes it possible to absorb disturbances and protect other compartments. The intrinsic variability of genes is the major form of adaptability in nature. It allows organisms to protect their essential dynamical properties in the face of environmental changes by varying less essential dynamical properties.

This intrinsic modifiability could be due to Brownian motion or it could be due to chaotic dynamics. As previously stated, distinguishing these two possibilities may be effectively undecidable. For the purpose of analysing the functional significance of chaos it is not necessary to make this distinction. If the dynamics appears chaotic, and this is not due to external forcing, it will make the same contribution to adaptability whatever the mechanism. However, there may be a significant advantage in structuring the system in a manner which fulfils the stretching, folding and rotating conditions required for dynamical chaos. Brownian motion is always present, but it is damped out in some dynamical systems and magnified in others. If a biological system obeys chaotic dynamics, this will ensure that its behaviour is chaotic. If the system obeys dynamics which are not sensitive to initial conditions, it is possible that the effects of the Brownian motion will be too negligible to make a significant contribution to adaptability.

1.4 Chaotic mechanisms of adaptability

Table 1.1 classifies chaotic mechanisms of adaptability. All the mechanisms involve the diversity generation in which chaotic dynamics could conceivably play a role.

The first category includes search processes in which an ensemble of possibilities is generated and tested. The most fundamental level of diversity generation is that of mutation, crossing over, recombination, and related genetic operations. These create a combinatorial explosion of possible genotypes on which natural selection acts. Conceivably some of the chemical dynamics underlying mutation are chaotic in nature or depend on intrinsic noise processes complemented by chaotic mechanisms. At the present time, however, there is no evidence that dynamical chaos acts directly at the genetic level. Later we will argue that some indirect effects occur due to cross-level interactions with chaotic population dynamics. Another possible place of chaos in genetic diversity generation is in the origin of life. Nicolis et al have recently presented a model for the origin of prebiological polymers in which the generation of sequence diversity is driven by chaotic reaction dynamics.

(Continues...)

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Excerpted from Chaos by Aron V. Holden. Copyright © 1986 Manchester University Press. Excerpted by permission of PRINCETON UNIVERSITY PRESS.
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Table of Contents

• FrontMatter, pg. i
• Contents, pg. v
• Acknowledgements, pg. vii
• 1. What is the use of chaos?, pg. 3
• 2. A graphical zoo of strange and peculiar attractors, pg. 15
• 3. One-dimensional iterative maps, pg. 39
• 4. Two-dimensional iterative maps, pg. 58
• 5. Chaos in feedback systems, pg. 99
• 6. The Lorenz equations, pg. 111
• 7. Instabilities and chaos in lasers and optical resonators, pg. 135
• 8. Differential systems in ecology and epidemiology, pg. 158
• 9. Oscillations and chaos in cellular metabolism and physiological systems, pg. 179
• 10. Periodically forced nonlinear oscillators, pg. 211
• 11. Chaotic cardiac rhythms, pg. 237
• 12. Chaotic oscillations and bifurcations in squid giant axons, pg. 257
• 13. Quantifying chaos with Lyapunov exponents, pg. 273
• 14. Estimating the fractal dimensions and entropies of strange attractors, pg. 291
• 15. How chaotic is the universe?, pg. 315
• Index, pg. 321

Editorial Reviews

"The book is a collection of fifteen review articles on topological dynamics, mostly dealing with descriptive models arising in physiology or physics. In general, the work is remarkably well-conceived, and beautifully illustrated."

Mathematical Reviews

"The book is a collection of fifteen review articles on topological dynamics, mostly dealing with descriptive models arising in physiology or physics. In general, the work is remarkably well-conceived, and beautifully illustrated."—Mathematical Reviews

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