Deterministic Chaos Theory and Its Applications to Materials Science
- PDF / 1,432,783 Bytes
- 9 Pages / 576 x 777.6 pts Page_size
- 8 Downloads / 234 Views
tense study. The answer to this question is not simple. One clue lies in the fact that theoretical studies of chaos require extensive numerical computations which, in turn, require the use of powerful computers, the likes of which have become commonly available only within the past few decades. Moreover, as far as experiments are concerned, it is most likely that chaos has indeed been observed on numerous occasions but has gone unrecognized as such and may simply have been discarded as unwanted "noise." For example, it has been noted, with specific reference to nonlinear electrochemical systems (although the stated conclusion probably applies quite generally), that "Probably chaotic oscillations were many times observed in electrochemical systems but they were treated rather as a noise and therefore all authors passed over them in silence."2 However, one does wonder why it took so long for chaos to be recognized for what it is, particularly since it can occur in very simple systems. This question was forcefully addressed with particular regard to chaos in the double pendulum: "This simple system exhibits outrageously complicated behavior. How could anyone watch the double pendulum and continue to assume that all solutions of Newton's equations could be developed in quasiperiodic perturbation expansions? Did hundreds of years really go by without anyone looking at a dynamical system in action?"3 Interest in this subject has been spurred among the general public by the very popular book on chaos by Gleick,4 as well as by a presentation on chaos on
the public television program NOVA and by numerous articles in the popular press. In many cases, the question has been "How can this concept help me in my area of research?" Although this has proved difficult to answer, practical applications are beginning to emerge, which we illustrate later in this article. In the discussion that follows, some comments are made regarding what chaos actually is and what some of the characteristics of a chaotically oscillating system are. Two simple examples are considered in order to illustrate these concepts. We then describe different ways of studying chaotic systems, some of the pitfalls that can be encountered in carrying out these studies, and general areas that are currently of great interest. Finally, we turn our attention to chaotic dynamics in materials behavior, citing some examples of existing work and suggesting what the future might hold. What Is Deterministic Chaos? At this point, we need to give a very brief and simplified outline of what deterministic chaos is. Treatments that are far more comprehensive can be found elsewhere, such as in a variety of engineering-oriented books on the subject, for example, Thompson and Stewart,5 Moon,67 and Kim and Stringer.8 Consider the state of a dynamical system, which at any given time is described by the instantaneous values of all its state variables, such as position and momentum. If N of these variables are required to fully define the state of the system, we can envisage an N-dime
Data Loading...
