Unstable equilibria of Hamiltonian systems

by Antonio Giorgilli

**Abstract:**
It is shown that a
Hamiltonian system in the neighbourhood of an equilibrium may be given
a special normal form in case four of the eigenvalues of the
linearized system are of the form
$\lambda_1,-\lambda_1,\lambda_2,-\lambda_2$, with $\lambda_1$ and
$\lambda_2$ independent over the reals, i.e.,
$\lambda_1/\lambda_2\notin\reali$. That is, for a real Hamiltonian
system and concerning the variables $x_1,y_1,x_2,y_2$ the equilibrium
is of either type {\it center--saddle} or {\it complex--saddle}. The
normal form exhibits the existence of a four--parameter family of
solutions which has been previously investigated by Moser. This paper
completes Moser's result in that the convergence of the transformation
of the Hamiltonian to a normal form is proven.

DCDS, to appear.

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