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 λ1,−λ1, λ2,−λ2, with λ1 and λ2 independent over the reals, i.e., λ1/λ2 /∈ R. That is, for a real Hamiltonian system and concerning the variables x1, y1, x2, y2 the equilibrium is of either type center–saddle or 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.

Unstable equilibria of Hamiltonian systems / A. Giorgilli. - In: DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS. - ISSN 1078-0947. - 7:4(2001), pp. 855-871.

### Unstable equilibria of Hamiltonian systems

#### 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 λ1,−λ1, λ2,−λ2, with λ1 and λ2 independent over the reals, i.e., λ1/λ2 /∈ R. That is, for a real Hamiltonian system and concerning the variables x1, y1, x2, y2 the equilibrium is of either type center–saddle or 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.
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Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/2434/15152`
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