Parameter identification on abelian integrals to achieve Chebyshev property
The authors consider the Hamiltonian system: \[ \dot{x} \, = \, H_y, \qquad \dot{y} \, = \, -H_x, \] where the subindexes denote partial derivatives, and where \[ H(x,y) \, = \, \frac{y^2}{2} + \frac{ \vartheta \alpha \beta}{2} x^2 + \frac{\vartheta (\alpha+\beta-\alpha\beta)}{2} x^4+ \frac{\vartheta(1-\alpha-\beta)}{4} x^6- \frac{\vartheta}{2} x^8, \] with \(\alpha\), \(\beta\) and \(\vartheta\) are real parameters. For several values of these parameters the system exhibits a period annulus surrounding the origin of coordinates and whose outer boundary is a heteroclinic loop. The aim of the paper is to study the bifurcation of limit cycles from this period annulus under perturbations of the form: \[ \dot{x} \, = \, y, \quad \dot{y} \, = \, \vartheta x(x^2-1)(x^2+\alpha)(x^2+\beta)+\varepsilon(a_0+a_1x^2+a_2x^4+a_3x^6)y, \] where \(a_0\), \(a_1\), \(a_2\) and \(a_3\) are real parameters and \(\varepsilon\) is the perturbation parameter. \par Let \(\Upsilon^\dagger\) be the heteroclinic loop of the unperturbed system satisfying \(H(\Upsilon^\dagger)=H(-1,0)\) and let \(\Upsilon_h\) be the closed orbit of the unperturbed system satisfying \(\Upsilon_h \, = \, \{ (x,y): x \in (-1,1), H(x,y)=h \}\) where \(0<h<H(\Upsilon^\dagger)\). It is well-known that the following abelian integral controls the bifurcation of limit cycles at first order of \(\varepsilon\) in the sense that its simple zeroes ensure the existence of a hyperbolic limit cycle when \(\varepsilon\) is small enough. This abelian integral is defined in the following way, \[ I(h,a_0,a_1,a_2,a_3) \, = \, a_0 I_0(h) + a_1 I_1(h) + a_2 I_2(h) +a_3 I_3(h), \] with \[ I_i(h) \, = \, \oint_{\Upsilon_h} x^{2i}ydx, \quad i=0,1,2,3. \] The study of the zeroes of this abelian integral is difficult and several criterions have been developed in the previous years. One seeks to find a Chebyshev property on the set of functions \(I_0(h), I_1(h), I_2(h), I_3(h)\). The particular problem tackled in this paper is specially difficult as there are several parameters in the non-perturbed system, namely \(\alpha\), \(\beta\) and \(\vartheta\), which make the study much more difficult. The authors try to use the known criteria but it turns out to be computationally unapproachable. Then they develop some transformations and use polynomial boundary theory which allows to provide the following results for this particular problem. \newline \textbf{Theorem 1.} For \(\vartheta=\beta=-1\) and \(h \in (0,(1+4\alpha)/24)\), with multiplicity counted, \(I(h,a_0,a_1,a_2,a_3)\) has at most three zeros if \(\alpha \in [1,3]\) and four zeros if \(\alpha \in (3,+\infty)\). \newline \textbf{Theorem 2.} For \(\vartheta=1\), \(\beta=\alpha\) and \(h \in (0,(1+4\alpha+6\alpha^2)/24)\), with multiplicity counted, \(I(h,a_0,a_1,a_2,a_3)\) has at most three zeros if \(\alpha \in (\alpha_4,\alpha_{10})\) and four zeros if \(\alpha \in (-\infty,\alpha_4) \cup (\alpha_{10},-1) \cup (\alpha^{*}, +\infty)\), where \(\alpha_4\), \(\alpha_{10}\) and \(\alpha^*\) are given in the proof. \newline \textbf{Theorem 3.} For \(\vartheta=\beta=1\) and \(h \in (0,(3+8\alpha)/24)\), \(I(h,a_0,a_1,a_2,a_3)\) has at most four zeros counting multiplicity if \(\alpha \in [3/8, +\infty)\). \newline The statement of these theorems have been quoted from the paper.
scientific article; zbMATH DE number 7360894
scientific article