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Positive definiteness of a Hamiltonian expanded about an equilibrium pointprovides only a necessary condition for stability, a criterion known asDirichlet's theorem. The reason that this criterion is not necessary forstability is because of the possible existence of negative energy modes, whichare linearly stable modes of oscillation that have negative energy. When suchmodes are present, the Hamiltonian is, in general, indefinite. Although suchsystems with negative energy modes are linearly stable (spectral stable), theyare unstable to infinitesimal perturbations under the nonlinear dynamics. Inthe present work we study this kind of nonlinear instability with the simplestnontrivial four dimensional area-preserving map, which has a cubic degree offreedom, that was designed to mimic the behavior of a Hamiltonian system withone positive and one negative energy mode, and a quadratic degree of freedom,that allows eventual escapes in phase space, usually called as Arnolddiffusion.

A four dimensional map for escape from resonance: negative energy modes and nonlinear instability