Formal equivalence between normal forms of reversible and Hamiltonian dynamical systems (Q380162)

Let \(\varphi :\mathbb{R}^{2n}\rightarrow \mathbb{R}^{2n}\) be an involutive diffeomorphism. A vector field \(X:\mathbb{R}^{2n}\rightarrow \mathbb{R}^{2n}\) is said to be \(\varphi \)-reversible if \(\varphi _{*}X=-X\). This implies that if \(x(t)\) is a flow curve of \(X\) then \(\varphi (x(-t))\) is also a flow curve. A famous open problem of Arnold is: ``Develop a supertheory whose even component corresponds to reversible systems and whose odd one to Hamiltonian systems''.NEWLINENEWLINEFix a smooth reversible vector field \(X\) with a simple and symmetric equilibrium at \(0\). The present paper is an important step towards a solution of Arnold's problem by studying whether there exists a sequence of changes of coordinates and time-reparametrizations around the origin that turn the truncations of the Taylor expansion of \(X\) at any given order into a Hamiltonian vector field.