A particularly simple chaotic nonequilibrium open system with two Cartesian degrees of freedom, characterized by two distinct temperatures T(x) and T(y), is introduced. The two temperatures are maintained by Nose-Hoover canonical-ensemble thermostats. Both the equilibrium (no net heat transfer) and nonequilibrium (dissipative) Lyapunov spectra are characterized for this simple system.

We describe and illustrate methods for treating many-body irreversible processes using time-reversible deterministic Nosé-Hoover thermostats. In phase space, Lyapunov-unstable multifractal strange attractors are the common feature representing any of these nonequilibrium flows, be they steady, periodic, or transient. This generic behavior is illustrated here for three prototypical one-body problems: steady field-driven diffusive flow in a Galton Board, time-periodic boundary-driven viscous flow of a Lorentz gas, and transient, but time-periodic, compressible flow characterizing a one-dimensional free expansion followed by compression and thermalization.