Symmetric Time-Reversible Flows with a Strange Attractor

A symmetric chaotic flow is time-reversible if the equations governing the flow are unchanged under the transformation t → −t except for a change in sign of one or more of the state space variables. The most obvious solution is symmetric and the same in both forward and reversed time and thus cannot be dissipative. However, it is possible for the symmetry of the solution to be broken, resulting in an attractor in forward time and a symmetric repellor in reversed time. This paper describes the simplest three-dimensional examples of such systems with polynomial nonlinearities and a strange (chaotic) attractor. Some of these systems have the unusual property of allowing the strange attractor to coexist with a set of nested symmetric invariant tori.