  We compare the Gram-Schmidt and covariant phase-space-basis-vector
descriptions for three time-reversible harmonic oscillator problems, in two,
three, and four phase-space dimensions respectively. The two-dimensional
problem can be solved analytically. The three-dimensional and four-dimensional
problems studied here are simultaneously chaotic, time-reversible, and
dissipative. Our treatment is intended to be pedagogical, for publication in
Communications in Nonlinear Science and Numerical Computation and for use in an
updated version of our book on Time Reversibility, Computer Simulation, and
Chaos. Comments are very welcome.
