Chaos in the Lorenz equations: A computer assisted proof. III: Classical parameter values (Q5929069)

A technique of obtaining computer assisted proofs of existence of chaotic dynamics is developed in the paper. It is based on rigorous estimates of error bounds arising in computer calculation of Poincaré maps, representing the errors as multivalued maps, and application of the theory of isolated invariant sets of multivalued discrete-time dynamical systems. The technique is an improvement of the one from the previous papers in the series [part I in Bull. Am. Math. Soc., New Ser. 32, No. 1, 66-72 (1995; Zbl 0820.58042), and part II in Math. Comput. 67, No. 223, 1023-1046 (1998; Zbl 0913.58038)]. In the present paper it is applied to the Lorenz system \(\dot x=\sigma(y-x)\), \(\dot y=Rx-y-xz\), \(\dot z=xy-bz\) for parameter values \((\sigma,R,b)\in \{(10,28,8/3),(10,60,8/3),(10,54,45)\}\). For those parameters the existence of semiconjugacy of restrictions of the Poincaré maps over the section \(\{x,y,z)\colon z=R-1\}\) to some subshifts of finite type is proved.