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Estimate of the number of zeros of abelian integrals for a kind of quartic Hamiltonians with two centers. (English) Zbl 1156.37013

Given a perturbed polynomial Hamiltonian system \[ \dot x=\frac{\partial H}{\partial y}(x,y)+\epsilon Q(x,y), \qquad \dot y=-\frac{\partial H}{\partial x}(x,y)+\epsilon P(x,y), \] suppose for \(\epsilon=0\) there exists a periodic annulus \(\Gamma_h:=H^{-1}(h)\). It is known that for small \(\epsilon\) a periodic trajectory bifurcates from \(\Gamma_{h_0}\), only if \(h=h_0\) is a zero of the abelian integral \[ I(h)=\oint_{\Gamma_h}P(x,y)dx-Q(x,y)dy. \] For the special case of a Hamiltonian \(H(x,y)=-x^2+\lambda x^4+y^4\) and real polynomials \(P,Q\) of degree \(n\), the authors give an upper bound for the number of zeros for \(I(h)\) in terms of the degree \(n\).

MSC:

37J20 Bifurcation problems for finite-dimensional Hamiltonian and Lagrangian systems
37G15 Bifurcations of limit cycles and periodic orbits in dynamical systems
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