Hamiltonian Normal Forms

We study a new type of normal form at a critical point of an analytic Hamiltonian. Under a Bruno condition on the frequency, we prove a convergence statement. Using this result, we deduce the existence of a positive measure set of invariant tori near the critical point

Study of chaos in hamiltonian systems via convergent normal forms

We use Moser's normal forms to study chaotic motion in two-degree hamiltonian systems near a saddle point. Besides being convergent, they provide a suitable description of the cylindrical topology of the chaotic flow in that vicinity. Both aspects combined allowed a precise computation of the homoclinic interaction of stable and unstable manifolds in the full phase space, rather than just the Poincar\'e section. The formalism was applied to the H\'enon-Heiles hamiltonian, producing strong evidence that the region of convergence of these normal forms extends over that originally established by Moser.Comment: 29 pages, REVTEX, 22 postscript figures on reques

Hamiltonian Normal Forms and Galactic Potentials

The study of self-gravitating stellar systems has provided important hints to develop tools of analytical mechanics. In the present contribution we review how to exploit detuned resonant normal forms to extract information on several aspects of the dynamics in systems with self-similar elliptical equipotentials. In particular, using energy and ellipticity as parameters, we compute the instability thresholds of axial orbits, bifurcation values of low-order boxlets and phase-space fractions pertaining to the families around them. We also show how to infer something about the singular limit of the potential.Comment: To be published in "Chaos in Astronomy", G. Contopoulos, P.A. Patsis (eds.), Springe

Normal forms of vector fields on Poisson manifolds

We study formal and analytic normal forms of radial and Hamiltonian vector fields on Poisson manifolds near a singular point.Comment: Final versio

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This paper summarizes the present state of integrability of Hamiltonian normal forms and it aims at characterizing non-integrable behaviour in higher-dimensional systems. Non-generic behaviour in Hamiltonian systems can be a sign of integrability, but it is not a conclusive indication. We will discuss a few degenerations and briefly review the integrability of Hamiltonian normal forms in two and three degrees of freedom. In addition we discuss two integrable normal form Hamiltonian chains, FPU and 1:2:2:2:2:2, and three non-integrable normal form chains, with emphasis on the 1:2:3:3:3:3 resonance. To distinguish between various forms of non-integrability is a major issue; time-series and projections based on the presence of a universal quadratic integral of the normal forms can be a useful predictor