  Positive definiteness of a Hamiltonian expanded about an equilibrium point
provides only a necessary condition for stability, a criterion known as
Dirichlet's theorem. The reason that this criterion is not necessary for
stability is because of the possible existence of negative energy modes, which
are linearly stable modes of oscillation that have negative energy. When such
modes are present, the Hamiltonian is, in general, indefinite. Although such
systems with negative energy modes are linearly stable (spectral stable), they
are unstable to infinitesimal perturbations under the nonlinear dynamics. In
the present work we study this kind of nonlinear instability with the simplest
nontrivial four dimensional area-preserving map, which has a cubic degree of
freedom, that was designed to mimic the behavior of a Hamiltonian system with
one positive and one negative energy mode, and a quadratic degree of freedom,
that allows eventual escapes in phase space, usually called as Arnold
diffusion.
