The cyclicity of the period annulus of the quadratic Hamiltonian systems with non-Morsean point. (English) Zbl 0960.37027

This paper deals with the number of limit cycles for small quadratic perturbations of quadratic Hamiltonian systems with non-Morsean point, that is \[ \begin{aligned} \dot x&= 2xy+ \varepsilon \Bigl( \sum_{i+j\leq 2} a_{ij} (\varepsilon) x^i y^j\Bigr),\\ \dot y&= 6x+ 6x^2- y^2+ \varepsilon \Bigl( \sum_{i+j\leq 2} b_{ij} (\varepsilon) x^i y^j \Bigr), \end{aligned} \tag \(1_\varepsilon\) \] where \(a_{ij} (\varepsilon)\) and \(b_{ij} (\varepsilon)\) depend analytically on the small parameter \(\varepsilon\). The authors prove that, for small \(\varepsilon\), the maximum number of limit cycles in \((1)_\varepsilon\) which emerge from the period annulus is equal to two.


37J45 Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods (MSC2010)
37G15 Bifurcations of limit cycles and periodic orbits in dynamical systems
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