Parabolic Resonances in near Integrable Hamiltonian Systems I Introduction

When an integrable Hamiltonian system, possessing an m-resonant lower dimensional normally parabolic torus is perturbed, a parabolic m-resonance occurs. If, in addition, the iso-energetic non-degeneracy condition for the integrable system fails, the near integrable Hamiltonian exhibits a at parabolic m-resonance. It is established that most kinds of parabolic resonances are persistent in n (n 3) d.o.f. near integrable Hamiltonians, without the use of external parameters. Analytical and numerical study of a phenomenological model of a 3 degrees of freedom (d.o.f.) near integrable Hamiltonian system reveals that in 3 d.o.f. systems new types of parabolic resonances appear. Numerical study suggests that some of them cause instabilities in several directions of the phase space and a new type of complicated chaotic behavior. A model describing weather balloons motion exhibits the same dynamical behavior as the phenomenological model. Hamiltonian systems which exhibit parabolic resonances (PR) appear in many applications; in fact PR typically appears in n degrees of freedom (d.o.f.) near integrable Hamiltonian systems depending on p parameters provided n 2 and n+p 3, see section IV. For example, PR arises in a model describing the motion of weather balloons on geopotential surfaces of the earth atmosphere, see sections II and III. Here we extend the 2 d.o.f. model of 8,11,12] to 3 d.o.f. by considering the vertical oscillations of the weather balloon. Other examples where PR appear are the Duung equation with a k-torus attached to it (the tori represent the eeect of neutral modes), and a reduction of the nonlinear Srr odinger equation (4] and references therein) to a six dimensional Hamiltonian ODE. What is a parabolic resonance? Let us rst brieey review the relevant results regarding the phase space structure of near integrable Hamiltonians. In the completely integrable case the phase space is foliated by invariant tori on which quasi-1)