On the Hamiltonian structure of normal forms at elliptic equilibria of reversible vector fields in R4
2020
Abstract This paper addresses the question whether normal forms of smooth reversible vector fields in R 4 at an elliptic equilibrium possess a formal Hamiltonian structure. In the non-resonant case we establish a formal conjugacy between reversible and Hamiltonian normal forms. In the case of non-semi-simple 1 : 1 resonance and p : q resonance with p + q > 2 we establish a weaker form of equivalence, namely that of a formal orbital equivalence to a Hamiltonian normal form that involves an additional time-reparametrization of orbits. Moreover, in case p + q > 3 we show that no formal conjugacy to a Hamiltonian normal form exists.
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