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

TL;DR

This paper compares Gram-Schmidt and covariant Lyapunov vectors and exponents in three harmonic oscillator problems of increasing complexity, highlighting their differences in chaotic, time-reversible, and dissipative systems.

Contribution

It provides a detailed comparison of phase-space basis vectors and exponents for harmonic oscillators in 2D, 3D, and 4D, including analytical solutions and pedagogical insights.

Findings

Analytical solution for 2D case.

Differences between Gram-Schmidt and covariant vectors.

Application to chaotic, time-reversible, dissipative systems.

Abstract

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.