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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 H\'enon-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.

Symmetry properties of orthogonal and covariant Lyapunov vectors and their exponents