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

Local Gram-Schmidt and Covariant Lyapunov Vectors and Exponents for Three Harmonic Oscillator Problems