Mathematics – Dynamical Systems
Scientific paper
2010-09-08
Mathematics
Dynamical Systems
Scientific paper
The aim of this paper is to introduce a class of Hamiltonian autonomous systems in dimension 4 which are completely integrable and their dynamics is described in all details. They have an equilibrium point which is stable in some rare cases and unstable in most cases. Anyhow, the eigenvalues of the linearization at the equilibrium point are imaginary and no motion is asymptotic in the past, namely no solution has the equilibrium as limit point as time goes to minus infinity. In the unstable cases, there is a sequence of initial data which converges to the origin whose corresponding solutions are unbounded and the motion is slow. So instability is quite weak and perhaps no such explicit examples of instability are known in the literature. The stable cases are also interesting since the level sets of the 2 first integrals independent and in involution keep being non-compact and stability is related to the isochronous periodicity of all orbits and the existence of a further first integral.
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