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Linear estimate for the number of zeros of Abelian integrals with cubic Hamiltonians. (English) Zbl 0921.58044

Let \(H(x,y)\), \((x,y)\in \mathbb{R}^2\), be a polynomial of degree \(m\) (called Hamiltonian) and let \(f(x,y)\), \(g(x,y)\) be polynomials of degrees not exceeding \(n\). Let \(\Sigma =\{h:h_1<h<h_2\}\subset \mathbb{R}\) be a maximal interval of existence of a continuous family of closed connected components \(\delta (h)\) of the algebraic curve \(H(x,y)=h\), \(h\in \Sigma\), free of critical points.
The infinitesimal 16th Hilbert problem [V. I. Arnol’d, Funct. Anal. Appl. 11, 85-92 (1977; Zbl 0411.58013)]is to give an estimate for the number \(Z(m,n)\) of zeros of the Abelian integral \[ I(h)=\int _{\delta (h)}[ g(x,y)dx - f(x,y)dy],\quad h\in \Sigma , \] in terms of the degrees of the polynomials \(H\), \(f\), \(g\). The infinitesimal 16th Hilbert problem is known to be closely connected with the number of limit cycles which is the main concern of Hilbert’s 16th problem.
In the paper, for cubic Hamiltonians \(H\) \((m=3)\) a linear estimate \(Z(3,n)\leq 5n+15\) for the number of zeros of \(I(h)\) is obtained. The proof is based on the properties of the Picard-Fuchs system satisfied by the four basic integrals \(\iint_{H<h}x^iy^j dx dy\), \(i,j=0,1\), generating the module of complete Abelian integrals \(I(h)\) over the ring of polynomials in \(h\).

MSC:

37G15 Bifurcations of limit cycles and periodic orbits in dynamical systems
37J20 Bifurcation problems for finite-dimensional Hamiltonian and Lagrangian systems
37K50 Bifurcation problems for infinite-dimensional Hamiltonian and Lagrangian systems
34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations

Citations:

Zbl 0411.58013
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