Chaos in the Lorenz equations: a computer-assisted proof

Mischaikow, Konstantin; Mrozek, Marian

doi:10.1090/S0273-0979-1995-00558-6

Chaos in the Lorenz equations: a computer-assisted proof
HTML articles powered by AMS MathViewer

by Konstantin Mischaikow and Marian Mrozek PDF

Bull. Amer. Math. Soc. 32 (1995), 66-72 Request permission

Abstract:

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 computer-assisted computations. As an application of these methods it is proven that for an explicit parameter value the Lorenz equations exhibit chaotic dynamics.

References

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. MR 1161275, DOI 10.1090/S0273-0979-1992-00327-0
T. Kaczynski and M. Mrozek, Conley index for discrete multi-valued dynamical systems, Topology Appl. 65 (1995), no. 1, 83–96. MR 1354383, DOI 10.1016/0166-8641(94)00088-K
Konstantin Mischaikow and Marian Mrozek, Isolating neighborhoods and chaos, Japan J. Indust. Appl. Math. 12 (1995), no. 2, 205–236. MR 1337206, DOI 10.1007/BF03167289

Chaos in Lorenz equations

A computer assisted proof, Part

Details

Marian Mrozek, Leray functor and cohomological Conley index for discrete dynamical systems, Trans. Amer. Math. Soc. 318 (1990), no. 1, 149–178. MR 968888, DOI 10.1090/S0002-9947-1990-0968888-1

Topological invariants, multivalued maps and computer assisted proofs in dynamics

Marek Ryszard Rychlik, Lorenz attractors through Šil′nikov-type bifurcation. I, Ergodic Theory Dynam. Systems 10 (1990), no. 4, 793–821. MR 1091428, DOI 10.1017/S0143385700005915
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. MR 1349899, DOI 10.1016/0893-9659(94)90058-2