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On the finite cyclicity of open period annuli. (English) Zbl 1205.34030

Consider a real analytic vector field \(X_0\) defined on a real analytic surface \(S\) without boundary. An open period annulus \(\Pi\) of \(X_0\) is a union of periodic orbits which is bianalytic with the standard annuli \(\mathcal{S}^1 \times (0,1)\). In this paper, relatively compact period annuli are considered, that is, the closure \(\bar{\Pi} \subset S\) is assumed to be compact. Given an analytic family of analytic vector fields \(X_\lambda\), with \(\lambda \in (\mathbb{R}^n,0)\), defined on \(S\), the cyclicity of \(\Pi\) with respect to the deformation \(X_\lambda\) is the maximal number of limit cycles of \(X_\lambda\) which tend to \(\Pi\) as \(\lambda\) tends to zero.
This paper deals with Hamiltonian vector fields, that is \(X_0\) has a first integral with isolated critical points in a complex neighborhood of \(\bar{\Pi}\), and generic Darbouxian vector fields, that is all the singular points of \(X_0\) in a neighborhood of \(\bar{\Pi}\) are orbitally analytically equivalent to linear saddles.
The main result of the paper establishes that the cyclicity of the open period annulus \(\Pi\) of a Hamiltonian or a generic Darbouxian vector field with respect to any of the described families \(X_\lambda\) is finite.
To prove the finite cyclicity of \(\Pi\), it suffices to show its finite cyclicity with respect to a given one-parameter deformation \(X_\varepsilon\) (with \(\varepsilon \in (\mathbb{R},0)\)) [L. Gavrilov, Ergodic Theory Dynam. Systems, 28, No. 5, 1497–1507 (2008; Zbl 1172.37020)]. Then, consider the first return map associated to \(\Pi\) for \(X_\varepsilon\):
\[ t \to t + \varepsilon^k M_k(t) + \cdots, \]
where \(t \in (0,1)\) and \(k \geq 1\). The function \(M_k(t)\) is called the Poincaré-Pontryagin-Melnikov function and it is analytic for \(t \in (0,1)\). The finite cyclicity of \(\Pi\) is proved if the function \(M_k(t)\) has a finite number of zeros in \((0,1)\), that is, the finite cyclicity follows from the nonaccumulation of zeros of \(M_k(t)\) at \(0\) or \(1\).
In the Hamiltonian case, \(M_k(t)\) satisfies a Fuchsian equation whose properties ensure that \(M_k(t)\) has a finite number of zeros in \((0,1)\).
In the generic Darbouxian case, the authors follow the work [D. Novikov, Geom. Funct. Anal., 18, No. 5, 1750–1773 (2008; Zbl 1167.32005)] and make use of the properties of the Mellin transformation on iterated integrals.

MSC:

34C07 Theory of limit cycles of polynomial and analytic vector fields (existence, uniqueness, bounds, Hilbert’s 16th problem and ramifications) for ordinary differential equations
34C23 Bifurcation theory for ordinary differential equations
34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
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References:

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