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Chaos in the Lorenz equations: A computer-assisted proof. (English) Zbl 0820.58042

A new technique to describe the global dynamics of nonlinear systems in terms of semiconjugacies is proposed. It combines, in a concrete example, abstract existence results based on topological invariants (like, e.g., the Conley index) with computer-assisted proofs of the theorems. As an application, an outline of a proof that the Lorenz equations exhibit chaotic dynamics is presented.

MSC:

37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
65L99 Numerical methods for ordinary differential equations
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References:

[1] S. P. Hastings and W. C. Troy, A shooting approach to the Lorenz equations, Bull. Amer. Math. Soc. (N.S.) 27 (1992), no. 2, 298 – 303. · Zbl 0764.58023
[2] T. Kaczynski and M. Mrozek, Conley index for discrete multi-valued dynamical systems, Topology Appl. 65 (1995), no. 1, 83 – 96. · Zbl 0843.54042 · doi:10.1016/0166-8641(94)00088-K
[3] Konstantin Mischaikow and Marian Mrozek, Isolating neighborhoods and chaos, Japan J. Indust. Appl. Math. 12 (1995), no. 2, 205 – 236. · Zbl 0840.58033 · doi:10.1007/BF03167289
[4] -, Chaos in Lorenz equations: A computer assisted proof, Part II: Details, in preparation.
[5] Marian Mrozek, Leray functor and cohomological Conley index for discrete dynamical systems, Trans. Amer. Math. Soc. 318 (1990), no. 1, 149 – 178. · Zbl 0686.58034
[6] -, Topological invariants, multivalued maps and computer assisted proofs in dynamics, in preparation. · Zbl 0861.58027
[7] Marek Ryszard Rychlik, Lorenz attractors through Šil\(^{\prime}\)nikov-type bifurcation. I, Ergodic Theory Dynam. Systems 10 (1990), no. 4, 793 – 821. · Zbl 0715.58027 · doi:10.1017/S0143385700005915
[8] B. Hassard, J. Zhang, S. P. Hastings, and W. C. Troy, A computer proof that the Lorenz equations have ”chaotic” solutions, Appl. Math. Lett. 7 (1994), no. 1, 79 – 83. · Zbl 0792.65050 · doi:10.1016/0893-9659(94)90058-2
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