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Analytic conjugation between planar reversible and Hamiltonian systems

F. J. S. Nascimento

TL;DR

This work establishes that analytic planar vector fields reversible about the involution $R(u,v)=(u,-v)$ and with a nondegenerate equilibrium are locally analytically conjugate to Hamiltonian systems. The conjugacy is constructed in two branches: centers map to Hamiltonians with $H=F(x^{2}+y^{2})$, while saddles map to $H=-F(x^{2}-y^{2})$, with $F$ analytic; in both cases the conjugacy can be chosen equivariant with respect to $R$. The center case leverages classical Poincaré normal form, whereas the saddle case combines formal conjugacy via homological equations with a convergence proof by Cauchy majorants to achieve an analytic, $R$-equivariant transformation. Overall, the paper strengthens prior formal equivalence results by establishing analytic conjugacy and clarifying the local reversible-Hamiltonian correspondence in the plane, while global equivalence and higher-dimensional questions remain open or substantially more delicate.

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.

Analytic conjugation between planar reversible and Hamiltonian systems

TL;DR

This work establishes that analytic planar vector fields reversible about the involution and with a nondegenerate equilibrium are locally analytically conjugate to Hamiltonian systems. The conjugacy is constructed in two branches: centers map to Hamiltonians with , while saddles map to , with analytic; in both cases the conjugacy can be chosen equivariant with respect to . The center case leverages classical Poincaré normal form, whereas the saddle case combines formal conjugacy via homological equations with a convergence proof by Cauchy majorants to achieve an analytic, -equivariant transformation. Overall, the paper strengthens prior formal equivalence results by establishing analytic conjugacy and clarifying the local reversible-Hamiltonian correspondence in the plane, while global equivalence and higher-dimensional questions remain open or substantially more delicate.

Abstract

In this work we study the local structure of analytic planar vector fields that are reversible with respect to the linear involution . 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: in the elliptic case and in the hyperbolic case, where 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 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.

Paper Structure

This paper contains 8 sections, 5 theorems, 115 equations.

Table of Contents

  1. Introduction and main result
  2. Proof of Theorem \ref{['thm:equiv_reversivel_hamiltoniano']}
  3. Center case
  4. Saddle case
  5. Formal conjugacy
  6. Convergence
  7. Conclusion of the proof of Theorem \ref{['thm:equiv_reversivel_hamiltoniano']}
  8. Conclusion

Key Result

Theorem 1.1

Let $X(u,v)$ be an analytic vector field defined in a neighbourhood of $(0,0)$, with $X(0,0)=(0,0)$ and $d=\det DX(0,0)\neq 0$, and suppose that $X$ is reversible with respect to the involution Then $X$ is analytically conjugate, in a neighbourhood of $(0,0)$, to a Hamiltonian vector field whose Hamiltonian can be written, according to the type of the singular point, as where and $G$ is real-a

Theorems & Definitions (8)

  • Theorem 1.1
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof