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Systems of rays in the presence of distribution of hyperplanes

100%

Janeczko S.

nr 1

245-260

Horizontal systems of rays arise in the study of integral curves of Hamiltonian systems $v_H$ on T*X, which are tangent to a given distribution V of hyperplanes on X. We investigate the local properties of systems of rays for general pairs (H,V) as well as for Hamiltonians H such that the corresponding Hamiltonian vector fields $v_H$ are horizontal with respect to V. As an example we explicitly calculate the space of horizontal geodesics and the corresponding systems of rays for the canonical distribution on the Heisenberg group. Local stability of systems of horizontal rays based on the standard singularity theory of Lagrangian submanifolds is also considered.

On Local Lyapunov Exponents of Chaotic Hamiltonian Systems

88%

Hofmann T. , Merker J.

tom Vol. 24, No. 2

97--111

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.