Hostname: page-component-7b7567c8c-wxvwp Total loading time: 0 Render date: 2026-11-14T17:05:47.192Z Has data issue: false hasContentIssue false
  • English
  • Français

Defining Chaos

Published online by Cambridge University Press:  01 April 2022

Robert W. Batterman
Robert W. Batterman*
Affiliation:
Department of Philosophy, Ohio State University
Get accessRights & Permissions [Opens in a new window]

Abstract

This paper considers definitions of classical dynamical chaos that focus primarily on notions of predictability and computability, sometimes called algorithmic complexity definitions of chaos. I argue that accounts of this type are seriously flawed. They focus on a likely consequence of chaos, namely, randomness in behavior which gets characterized in terms of the unpredictability or uncomputability of final given initial states. In doing so, however, they can overlook the definitive feature of dynamical chaos—the fact that the underlying motion generating the behavior exhibits extreme trajectory instability. I formulate a simple criterion of adequacy for any definition of chaos and show how such accounts fail to satisfy it.

Information

Type
Research Article
Information
Philosophy of Science , Volume 60 , Issue 1 , March 1993, pp. 43 - 66
Copyright
Copyright © Philosophy of Science Association 1993

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

Footnotes

†

I would like to thank Mark Wilson for numerous helpful discussions, without which I would still be confused on many points. Naturally, any mistakes and confusions which still remain are entirely my own. I would also like to thank Jim Joyce for comments on a very early draft of this paper. This work was supported by the National Science Foundation under grant No. DIR-9012010.

‡

Send reprint requests to the author, Department of Philosophy, Ohio State University, 350 University Hall, 230 N. Oval Mall, Columbus, OH 43210, USA.

References

Alekseev, V. M. and Yakobson, M. V. (1981), “Symbolic Dynamics and Hyperbolic Dynamical Systems”, Physics Reports 75: 287–325.10.1016/0370-1573(81)90186-1CrossRefGoogle Scholar
Arnold, V. I. (1973), Ordinary Differential Equations. Cambridge, MA: MIT Press.Google Scholar
Arnold, V. I. and Avez, A. (1968), Ergodic Problems of Classical Mechanics. New York: Addison Wesley.Google Scholar
Arrowsmith, D. K. and Place, C. M. (1990), An Introduction to Dynamical Systems. Cambridge, England: Cambridge University Press.Google Scholar
Berry, M. V. (1983), “Semi-Classical Mechanics of Regular and Irregular Motion”, in Ioos, G., Helleman, R. G. H., and Stora, R. (eds.), Chaotic Behavior of Deterministic Systems (Les Houches, Session 36) Amsterdam: North Holland, pp. 171–271.Google Scholar
Brudno, A. A. (1978), “The Complexity of the Trajectories of a Dynamical System”, Russian Mathematical Surveys 33: 197–198.10.1070/RM1978v033n01ABEH002243CrossRefGoogle Scholar
Earman, J. (1986), A Primer on Determinism. Dordrecht: Reidel.10.1007/978-94-010-9072-8CrossRefGoogle Scholar
Fine, T. (1973), Theories of Probability: An Examination of Foundations. New York: Academic Press.Google Scholar
Ford, J. (1981), “How Random is a Coin Toss?” in C. W. Horton, L. E. Reichl, and V. G. Szebehely (eds.), Proceedings of the Workshop on Long-Time Predictions in Nonlinear Conservative Dynamical Systems. New York: Wiley, pp. 79–92.Google Scholar
Ford, J. (1983), “How Random is a Coin Toss?” Physics Today 36: 40–47.10.1063/1.2915570CrossRefGoogle Scholar
Ford, J. (1988), “Quantum Chaos, Is There Any?” in H. Bai-Lin (ed.), Directions in Chaos 2. Singapore: World Scientific, pp. 128–147.10.1142/9789814415729_0006CrossRefGoogle Scholar
Ford, J. (1989), “What is Chaos, That We Should be Mindful of It?” in P. Davies (ed.), The New Physics. Cambridge, England: Cambridge University Press, pp. 348–371.Google Scholar
Hadamard, J. (1923), Lectures on Cauchy's Problem in Linear Partial Differential Equations. New Haven: Yale University Press.Google Scholar
Hunt, G. M. K. (1987), “Determinism, Predictability and Chaos”, Analysis 47: 129–132.10.1093/analys/47.3.129CrossRefGoogle Scholar
Lichtenberg, A. J. and Lieberman, M. A. (1983), Regular and Stochastic Motion. New York: Springer-Verlag.10.1007/978-1-4757-4257-2CrossRefGoogle Scholar
Martin-Löf, P. (1966), “The Definition of a Random Sequence”, Information and Control 9: 602–619.10.1016/S0019-9958(66)80018-9CrossRefGoogle Scholar
Stone, M. A. (1989), “Chaos, Prediction, and Laplacean Determinism”, American Philosophical Quarterly 26: 123–131.Google Scholar
Wilson, M. (1989), “Critical Notice: John Earman's A Primer on Determinism”, A Primer on Determinism 56: 502–532.Google Scholar