On Local Lyapunov Exponents of Chaotic Hamiltonian Systems

Chaos in conservative systems, particularly in Hamiltonian systems, is different from chaos in dissipative systems. For example, not only the eigenvalues of the symmetric Jacobian, but also the global Lyapunov exponents of Hamiltonian systems occur in pairs (λ,−λ). In this article, we even show that appropriately defined local Lyapunov exponents occur in pairs, and in turn this allows to give a new and easily accessible proof of the pairing property for global Lyapunov exponents. As examples of low dimensional chaotic Hamiltonian systems, we discuss the classical Hénon-Heiles system and a sixth order generalisation. For the latter, there is numerical evidence of two disjoint chaotic seas.