  We analyze the time-reversible mechanics of two irreversible simulation
types. The first is a dissipative one-dimensional heat-conducting oscillator
exposed to a temperature gradient in a three-dimensional phase space with
coordinate $q$, momentum $p$, and thermostat control variable $\zeta$. The
second type simulates a conservative two-dimensional $N$-body fluid with $4N$
phase variables $\{q,p\}$ undergoing shock compression. Despite the
time-reversibility of each of the three oscillator equations and all of the
$4N$ manybody motion equations both types of simulation are irreversible,
obeying the Second Law of Thermodynamics. But for different reasons. The
irreversible oscillator seeks out an attractive dissipative limit cycle. The
likewise irreversible, but thoroughly conservative, Newtonian shockwave
eventually generates a reversible near-equilibrium pair of rarefaction fans.
Both problem types illustrate interesting features of Lyapunov instability.
