Composition conditions and center problem (Q2765895)

The family of analytic plane vector fields \(X_{(\varepsilon)}= \varepsilon{}X_1\) is considered, which is a perturbation of the analytic Hamiltonian system \(X_H\) NEWLINE\[NEWLINE\dot{x}=-\partial_{y}H+\varepsilon A(x,y),\qquad \dot{y}=-\partial_{x}H+\varepsilon B(x,y),NEWLINE\]NEWLINE at the origin. Here, \(A\), \(B\) are analytic perturbations of the field \(X_H\) at the origin. It is supposed that the Hamiltonian \(H\) is analytic, the level lines of \(H\), \(H^{-1}(c)\), contain connected components \(\gamma_{(c)}\) which are ovals around the origin and that, for all analytic 1-forms \(\omega\), \(\int_{\gamma_{(c)}}\omega=0\) iff there are two analytic functions \(g\) and \(R\) such that \(\omega=g dH+dR\). The main result is a composition condition ensuring the vanishing of all successive derivatives \(P_k\) in the first return map \(P:H|\Sigma\to{}H|\Sigma\), NEWLINE\[NEWLINE P(c):=c+\varepsilon{}P_{1}(c) +\varepsilon^2P_{2}(c) +\varepsilon^3P_{3}(c)+\cdots, NEWLINE\]NEWLINE hence ensuring that the perturbed vector field \(X_{(\varepsilon)}\) has a center at the origin for sufficiently small \(\varepsilon\). A relation to the composition condition of Alwash and Lloyd is discussed as well and a new composition condition is announced which may be generalized to any dimension.