Chaos in the Lorenz Equations: A Computer Assisted Proof Part III: Classical Parameter Values

Proving that a given ordinary differential equation possesses chaotic dynamics is a difficult task for two reasons. First, such dynamics only occurs in nonlinear systems, and therefore, the necessary analysis is extremely difficult. Because of this, for most systems which are considered to be chaotic we have no proofs only numerical evidence obtained by integrating the equations. At the same time, however, it makes no sense to talk about chaotic trajectories any accepted definition of chaos is, in fact, a statement about the existence of uncountably many orbits which together exhibit sensitive dependence on initial conditions. In this paper we discuss a general computationally inexpensive technique for obtaining a computer assisted proof of the existence of chaos in the sense of symbolic dynamics. We use the Lorenz equations to give a concrete demonstration of this technique. doi:10.1006 jdeq.2000.3894, available online at http: www.idealibrary.com on