We prove that under certain conditions, the Lorenz equations support a form of chaos. The conditions have been verified for a particular set of parameter values, showing that there is a natural 1:1 correspondence between a set of solutions and the set of all sequences on countably many symbols. The computing required was modest.
1996 Academic Press, Inc.
1. Introduction
The system of equations discovered by E. N. Lorenz [11] is found in computer simulations to have chaotic behavior, by practically any definition of that term. A survey appears in [15]. However, aside from local results and various kinds of bifurcation analysis, little had been proved about these equations until very recently.
Recently, however, several authors have succeeded in giving rigorous proofs that this system of equations supports some sort of chaotic behavior [2], [9], [11], [14]. In this paper we prove and expand on the results announced in [9] and [11]. Our results here are of the form that if certain conditions are satisfied by a particular set of solutions, then there is a natural 1:1 correspondence between a set of solutions and a set of sequences of natural numbers. As corollaries we obtain information about sensitivity to initial conditions'' and about kneading theory'' [3].
Our results are global, in that there is no perturbation analysis, and in theory our conditions can be checked rigorously in any parameter range. However, the verification of our hypotheses must be done on a computer, and in practice, a rigorous proof that these hypotheses hold can only be done for very small parameter ranges. In [11], it was shown that this verification can be carried out for an open set of parameter values.
Another recent computer proof of chaos for the Lorenz equations was carried out by Mischaikow and Mrozek,in [14]. Their work used more sophisticated topology, and in particular showed that there is a ``horseshoe'' structure embedded in the flow. It may be possible to use their method to demonstrate the existence of many periodic solutions, whereas article no.