Time-reversible dissipative attractors in three and four phase-space dimensions

We establish the dissipative nature of several three- and four-dimensional time-reversible phase-space flows and study their ergodicity. Three- and four-dimensional generalizations of the equilibrium Nos\'e-Hoover oscillator provide the simplest robust continuous models for time-reversible nonequilibrium dissipative systems. Most such systems exhibit discontinuities or periodicities. We have discovered one set of four ordinary differential equations which is simultaneously robust, time-reversible, dissipative, and ergodic, with solutions free of any discontinuities and periodicities.