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The existence of lower-dimensional resonant bifurcating tori of parabolic, hyperbolic and elliptic normal stability types is proved to be generic and persistent in a class of n degrees of freedom (DOF) integrable Hamiltonian systems with n 3. Parabolic resonance (PR) (respectively, hyperbolic or elliptic resonance) is created when a small Hamiltonian perturbation is added to an integrable Hamiltonian system possessing a resonant torus of the corresponding normal stability. It is numerically demonstrated that PRs cause intricate behaviour and large instabilities. The role of lower-dimensional bifurcating resonant tori in creation of instability mechanisms is illustrated using phenomenological models of near integrable Hamiltonian systems with 3, 4 and 5 DOF. Critical n values for which the system first persistently possesses mechanisms for large instabilities of a certain type are found. Initial numerical studies of the rate and time of development of the most significant instabilities are presented.

Abstract.A framework for understanding the global structure of near-integrable n DOF Hamiltonian systems is proposed. To this aim two tools are developed-the energy-momentum bifurcation diagrams and the branched surfaces. Their use is demonstrated on a few near-integrable 3 DOF systems. For these systems possible sources of instabilities are identified in the diagrams, and the corresponding energy surfaces are presented in the frequency space and by the branched surfaces. The main results of this formulation are theorems which describe the connection between changes in the topology of the energy surfaces and the existence of resonant lower dimensional tori. Key words. near-integrable Hamiltonians, parabolic resonances AMS subject classifications. 70H08, 37J20DOI. 10.1137/0306001061. Introduction. The study of the structure of energy surfaces of integrable systems and the study of resonances and instabilities in near-integrable systems developed into vast disparate research fields. The relation between the two received very little attention. Indeed, near regular level sets of the integrable Hamiltonian, the standard Arnold-resonance web structure appears, and the relation between the two fields reduces to the study of Arnold conjecture regarding instabilities in phase space. Here, we demonstrate that near singular level sets of the integrable Hamiltonian much information regarding possible instabilities of the near-integrable case may be deduced from the structure of the energy surface and its relations with resonance surfaces. We suggest that by adding some information to the traditional energy-momentum diagrams [7], which we name energy-momentum bifurcation diagrams (EMBD), one achieves a global qualitative understanding of the near-integrable dynamics. We relate the geometric properties of the surfaces corresponding to lower dimensional tori in this diagram to both bifurcations in the energy surface topology and the appearance of lower dimensional resonant tori.Recall that energy surfaces of generic integrable Hamiltonian systems are foliated almost everywhere by n-tori, 1 which may be expressed locally as a product of n circles on which the dynamics reduces to simple rotations (the action-angle coordinates). A given compact regular level set (the set of phase space points with given values of the constants of motion) may be composed of several such tori. The energy surface is composed of all level sets with the same

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