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We analyze the time-reversible mechanics of two irreversible simulationtypes. The first is a dissipative one-dimensional heat-conducting oscillatorexposed to a temperature gradient in a three-dimensional phase space withcoordinate $q$, momentum $p$, and thermostat control variable $\zeta$. Thesecond type simulates a conservative two-dimensional $N$-body fluid with $4N$phase variables $\{q,p\}$ undergoing shock compression. Despite thetime-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. Theirreversible oscillator seeks out an attractive dissipative limit cycle. Thelikewise irreversible, but thoroughly conservative, Newtonian shockwaveeventually generates a reversible near-equilibrium pair of rarefaction fans.Both problem types illustrate interesting features of Lyapunov instability.

Time-Reversible Thermodynamic Irreversibility : One-Dimensional Heat-Conducting Oscillators and Two-Dimensional Newtonian Shockwaves