Analytic conjugation between planar reversible and Hamiltonian systems

Abstract

In this work we study the local structure of analytic planar vector fields that are reversible with respect to the linear involution $R(u,v)=(u,-v)$. We show that every analytic reversible vector field with a nondegenerate equilibrium is locally analytically conjugate to a Hamiltonian system. More precisely, we prove that, in a neighbourhood of the origin, the system is analytically equivalent to a Hamiltonian vector field whose Hamiltonian assumes the classical normal form associated with the type of the equilibrium: $H(x,y)=F(x^{2}+y^{2})$ in the elliptic case and $H(x,y)=-F(x^{2}-y^{2})$ in the hyperbolic case, where $F$ is real-analytic and completely determined by the dynamics. We also show that the conjugacy can be chosen equivariant, that is, commuting with the reversing involution. We further discuss the problem of \emph{global} equivalence, which in general remains open, even in the planar case. In dimensions greater than $2$ the situation becomes even more delicate: the equivalence between reversible and Hamiltonian systems is known only at the \emph{formal} level, and the existence of an analytic conjugacy, even locally, is still a widely open problem.