Local Gram–Schmidt and covariant Lyapunov vectors and exponents for three harmonic oscillator problems

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 use in an updated version of our book on Time Reversibility, Computer Simulation, and Chaos. Comments are very welcome.

► Lyapunov instability can be described by Gram–Schmidt vectors and exponents or by “covariant” vectors and exponents.
► We compare these two approaches for three different oscillator-based models.
► For nonequilibrium systems the Gram–Schmidt approach is cheaper and better than the covariant one.