Chaos in the Lorenz equations:

A computer assisted proof.

Part II: Details

Authors:
Konstantin Mischaikow and Marian Mrozek

Journal:
Math. Comp. **67** (1998), 1023-1046

MSC (1991):
Primary 58F13, 54H20, 65L99, 34-04, 68T15

DOI:
https://doi.org/10.1090/S0025-5718-98-00945-4

MathSciNet review:
1459392

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Abstract | References | Similar Articles | Additional Information

Abstract: Details of a new technique for obtaining rigorous results concerning the global dynamics of nonlinear systems is described. The technique combines abstract existence results based on the Conley index theory with rigorous computer assisted computations. As an application of these methods it is proven that for some explicit parameter values the Lorenz equations exhibit chaotic dynamics.

**1.**U. Ascher, R.M. Mattheij and D.R. Russell,*Numerical Solution of Boundary Value Problems for ODEs*, Prentice-Hall, Englewood Cliffs, N.J., 1988. MR**90h:65120****2.**Xinfu Chen, Lorenz Equations, Part III: Existence of Hyperbolic Sets, preprint 1995.**3.**C. Conley, Isolated Invariant Sets and the Morse Index, CBMS Regional Conf. Ser. Math., no 38, AMS, Providence, R.I., 1978. MR**80c:58009****4.**E. Hairer, S.P. Nørsett and G. Wanner,*Solving Ordinary Differential Equations I, Nonstiff Problems*, Springer-Verlag, Berlin Heidelberg 1987. MR**87m:65005****5.**B. Hassard, J. Zhang, S. Hastings, and W. Troy, A computer proof that the Lorenz equations have ``chaotic'' solutions,*Appl. Math. Letter***7**(1994), 79-83. MR**96d:58082****6.**S.P. Hastings and W.C. Troy, A shooting approach to the Lorenz equations,*Bulletin (New Series) of the American Mathematical Society***27**(1992) 298-303. MR**93f:58150****7.**T. Kaczy\'{n}ski and M. Mrozek, Conley index for discrete multivalued dynamical systems,*Topology & its Appl.*,**65**(1995), 83-96. MR**97d:54066****8.**R.J. Lohner, Computation of Guaranteed Enclosures for the Solutions of Ordinary Initial and Boundary Value Problems, in:*Computational Ordinary Differential Equations*, J.R. Cash, I. Gladwell Eds., Clarendon Press, Oxford, 1992. CMP**96:12****9.**J. {\L}ukasiewicz, O logice trójwarto\'{s}ciowej (On three-valued logic),*Ruch Filozoficzny***5**(1920), 169-170.**10.**K. Mischaikow, The structure of isolated invariant sets,*Contemporary Mathematics*, C. McCord ed.,AMS, (1993), 269-290. MR**94k:58083****11.**K. Mischaikow, The Conley index theory: some recent developments, CIME Lectures, preprint.

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Additional Information

**Konstantin Mischaikow**

Affiliation:
Center for Dynamical Systems and Nonlinear Studies School of Mathematics Georgia Institute of Technology Atlanta, Georgia 30332-0001

Email:
mischaik@math.gatech.edu

**Marian Mrozek**

Affiliation:
Center for Dynamical Systems and Nonlinear Studies School of Mathematics Georgia Institute of Technology Atlanta, Georgia 30332-0001

Address at time of publication:
Instytut Informatyki, Uniwersytet Jagielloński, Kraków, Poland

Email:
mrozek@ii.uj.edu.pl

DOI:
https://doi.org/10.1090/S0025-5718-98-00945-4

Received by editor(s):
August 11, 1995

Received by editor(s) in revised form:
April 12, 1996, and February 18, 1997

Additional Notes:
Research of the first author was supported in part by NSF grant 9302970.

Research of the second author was supported by KBN, Grant 0449/P3/94/06.

Article copyright:
© Copyright 1998
American Mathematical Society