Perturbed normalizers and Melnikov functions
Consider a \(C^{\infty}\) smooth family of real planar vector fields \(\left(X_\varepsilon\right)_\varepsilon\) of the form \[ X_\varepsilon (x) \, = \, X_0(x) + \varepsilon X_1(x) + \varepsilon^2 X_2(x) + \, \ldots \, + \varepsilon^n X_n(x) + \mathcal{O}\left(\varepsilon^{n+1}\right), \] for some integer \(n \geq 1\), \(x \in \mathbb{R}^2\) and \(|\varepsilon|\ll 1\). It is assumed that \(X_0(x)\) has a period annulus in a region \(\mathcal{P} \subset \mathbb{R}^2\). The number and location of the periodic orbits of \(X_0\) which ``persist'' in the family \(\left(X_\varepsilon\right)_\varepsilon\) as limit cycles is one of the questions to which more articles have been devoted. A classical method to study this bifurcation problem is the Melnikov functions. Given \(m\), an integer with \(m \geq 1\), the author provides a characterization of the property that the first \((m-1)\) Melnikov functions of \(\left(X_\varepsilon\right)_\varepsilon\) vanish identically. This characterization is given in terms of the existence of a smooth family of planar vector fields \(\left(U_\varepsilon\right)_\varepsilon\), called perturbed normalizers of order \(m\). A \textit{perturbed normalizer of order \(m\)} for the family \(\left(X_\varepsilon\right)_\varepsilon\) is a family of smooth vector fields \(\left(U_\varepsilon\right)_\varepsilon\) such that \(U_0\) is transversal to \(X_0\) in \(\mathcal{P}\) and \[ X_\varepsilon \wedge [X_\varepsilon,U_\varepsilon] \, = \, \mathcal{O}(\varepsilon^m) \, \quad \text{in \(\mathcal{P}\) for \(|\varepsilon|\ll 1\).} \] An effective formula to compute these vector fields \(\left(U_\varepsilon\right)_\varepsilon\) is provided, together with a formula for the derivative of the \(m\)-th order Melnikov function.
scientific article; zbMATH DE number 6867345
scientific article