By Perrine C. D.
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However, we prefer to state the main result in a bit more general framework. Namely, the very well known Lyapunov's Theorem 12 ensures that if we have an n-degrees of freedom analytic Hamiltonian system, with the origin being an equilibrium and eigenvalues ±Aj, ±A2,. • •, ±A n satisfying that • Aj is purely imaginary, • none of the quocients j 2 , . . ^ is an integer, then there exists a one-parameter family of periodic orbits accumulating to the origin. In other words, one can prove the existence of a transformation leading such system into (Birkhoff) normal form with respect to the variables associated to the imaginary eigenvalue Ai In our context, we have the following result, which represents the simplest situation where Lyapunov's Theorem would apply, that is, when the 48 spectrum of the differential of the field at the equilibrium consists on two pairs of eigenvalues.
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