Symmetry properties of orthogonal and covariant Lyapunov vectors and their exponents

Lyapunov exponents are indicators for the chaotic properties of a classical dynamical system. They are most naturally defined in terms of the time evolution of a set of so-called covariant vectors, co-moving with the linearized flow in tangent space. Taking a simple spring pendulum and the Hénon–Heiles system as examples, we demonstrate the consequences of symplectic symmetry and of time-reversal invariance for such vectors, and study the transformation between different parameterizations of the flow. This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to 'Lyapunov analysis: from dynamical systems theory to applications'. (paper)