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...