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October  2001, 7(4): 855-871. doi: 10.3934/dcds.2001.7.855

Unstable equilibria of Hamiltonian systems

Antonio Giorgilli 1,

1. 

Università di Milano Bicocca, Dipartimento di Matematica e Applicazioni, Via degli Arcimboldi 8, 20126 - Milano, Italy

Received  November 2000 Revised  January 2001 Published  July 2001

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., $\lambda_1/\lambda_2 \notin \mathbf R$. 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 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.
Keywords: equilibria of Hamiltonian systems., Normal form.
Mathematics Subject Classification: Primary: 58F36, 70K15, 34C30, 34A3.
Citation: Antonio Giorgilli. Unstable equilibria of Hamiltonian systems. Discrete and Continuous Dynamical Systems, 2001, 7 (4) : 855-871. doi: 10.3934/dcds.2001.7.855
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