Abstract
The Lorenz equation1 of turbulence generation is a simple 3-varilable quadratic differential equation producing complicated ‘nonperiodic’1 behavior. Recent interest in this equation was triggered by a paper of May2, who showed that similar complicated oscillations can occur in an ecological difference equation. Later, Winfree3 suggested, following an observation of ‘meandering’ in a non-stirred chemical reaction system4, that chemical systems too should be capable of ‘chaos’. Subsequently several abstract reaction systems realizing a simpler (non-Lorenzian) type of chaos were described5. A complicated ‘non-isothermal’ abstract reaction system realizing a chemical analogue to the Lorenz equation itself was also found6.
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References
E.N. Lorenz, Deterministic Nonperiodic Flow. J. Atmos. Sci. 20,130–141 (1963).
R. May, Biological Populations with Nonoverlapping Generations: Stable Points, Limit Cycles, and Chaos. Science 186,645–647 (1974).
A.T. Winfree, Personal Communication (1975).
A.T. Winfree, Scroll-shaped Waves of Chemical Activity in Three Dimensions. Science 181,937–939 (1973).
O.E. Rössler, Chaotic Behavior in Simple Reaction Systems. Z. Natur-forsch. 31a, 259–264 (1976).
P.J. Ortoleva, Unpublished (1975).
O.E. Rössler, Different Types of Chaos in Two Simple Differential Equations. Z. Naturforsch. 31a, 1664–1670 (1976).
O.E. Rössler, Continuous Chaos: Four Prototype Equations. In: Bifurcation Theory and Applications (O. Gurel and O.E. Rössler, eds.), Proc. N.Y. Acad. Sci. (in press).
F.G. Heineken, H.M. Tsuchiya and R. Aris, On the Mathematical Status of the Pseudo-steady State Hypothesis in Biochemical Kinetics. Math. Biosci. 1, 95–113 (1967).
O.E. Rössler, Chaos in Abstract Kinetics: Two Prototypes. Bull. Math. Biol. 39, 275–289 (1977).
J. Guckenheimer, A Strange-Strange Attractor. In: The Hopf Bifurcation (J.E. Marsden and M. McCracken, eds.), pp. 368–381. Springer-Verlag: New York 1976.
R.F. Williams, The Structure of Lorenz Attractors (preprint).
T.Y. Li and J.A. Yorke, Period Three Implies Chaos. Am. Math. Monthly 82, 985–992 (1975).
O.E. Rössler, Continuous Chaos. In: Synergetics — A Workshop (H. Haken, ed.), pp. 184–197. Springer-Verlag: New York 1977.
O.E. Rössler, Horseshoe-map Chaos in the Lorenz Equation. Phys. Letters 60A, 392–394 (1977).
K. Nakamura, Nonlinear Fluctuations Associated with Instabilities in Dissipative Systems. Prog. Theor. Phys. 57, 1874–1885 (1977).
J.L. Kaplan and J.A. Yorke, Preturbulence: A Regime Observed in a Fluid Flow Model of Lorenz (preprint).
O.E. Rössler and K. Wegmann, Different Types of Chaos in the Belousov-Zhabotinskii Reaction. Thirteenth Symposium on Theoretical Chemistry, Münster, October 1977.
O.E. Rössler and K. Wegmann, Chaos in the Zhabotinskii Reaction. Nature (in press).
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Rössler, O.E., Ortoleva, P.J. (1978). Strange Attractors in 3-Variable Reaction Systems. In: Heim, R., Palm, G. (eds) Theoretical Approaches to Complex Systems. Lecture Notes in Biomathematics, vol 21. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-93083-6_4
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DOI: https://doi.org/10.1007/978-3-642-93083-6_4
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