The Poincaré bifurcation of a SD oscillator (Q2029749)

The authors consider the following real planar autonomous differential system \[ \begin{array}{lll} \displaystyle \frac{dy}{d\tau} & = & \displaystyle y - \epsilon \left(b\sqrt{u^2-a^2} \, + \, \sqrt{(u^2-a^2)^3}\right), \vspace{0.2cm} \\ \displaystyle \frac{du}{d\tau} & = & -(u-1), \end{array} \] where \(a\), \(b\), \(\epsilon\) are real parameters and \(\epsilon \approx 0\). This system has been studied by several authors and the bifurcation of limit cycles when \(\epsilon\) is small enough has been solved except when \(0<a<1\). The paper is devoted to solve this problem. \newline In the following, we take the notation introduced by the authors in the paper. When \(\epsilon=0\), the system has the first integral \(H(u,y) \, = \, \displaystyle \frac{y^2}{2} \, + \, \frac{(u-1)^2}{2}\) and the following period annulus can be defined \[ \Gamma_h \, = \, \left\{ (u,y) \, | \, H(u,y) = h, \ 0<h<\frac{(1-a)^2}{2} \right\}. \] The following Abelian integral controls the bifurcation of limit cycles in this period annulus \(I(h) \, = \, b I_1(h) + I_2(h)\) with \[ I_1(h) \, = \, \oint_{\Gamma_h} \frac{u}{\sqrt{u^2-a^2}} \, y \, du \ \mbox{and} \ I_2(h) \, = \, \oint_{\Gamma_h} u \sqrt{u^2-a^2} \, y\, du. \] The main result of the paper, namely Theorem 1, is the following: \begin{itemize} \item[(1)] If \(\sqrt{3}/3 \leq a < 1\), then \(I(h) \, = \, b I_1(h) + I_2(h)\) has at most one zero in \((0,(1-a)^2/2)\). \item[(2)] If \(0<a<\sqrt{3}/3\), then \(I(h) \, = \, b I_1(h) + I_2(h)\) has at most two zeros in \((0,(1-a)^2/2)\). And there exist \(b \in \mathbb{R}\) such that \(I(h)\) has exactly two zeros in \((0,(1-a)^2/2)\). \end{itemize} In order to prove this result, the authors use a criterion method. They first recall several criterions which have been developed in recent years to tackle these kind of problems. They show that these criterions do not allow the solution of the problem for the considered system. Thus, they provide an ad hoc criterion which generalizes one of the previously known criterions.