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An estimate of the number of zeros of Abelian integrals for cubic vector fields with cuspidal loop. (English) Zbl 0967.34027

Summary: The upper bound \(B(n)\leq{1\over 2} \{7n+{1\over 2}((-1)^n- 1)\}\) is derived for the number of zeros of Abelian integrals \[ I(h)= \oint_{\Gamma_h} g(x,y) dy- f(x,y) dx \] on the open interval \((-{1\over 12},0)\cup (0,+\infty)\), where \(\Gamma_h\) is the compact component of the algebraic curve \[ H(x,y)= \textstyle{{1\over 2}y^2+ {1\over 3}x^3+ {1\over 4} x^4= h}, \] \(f(x,y)\) and \(g(x,y)\) are polynomials of \(x\) and \(y\), \(n= \max\{\text{deg }f(x,y),\text{ deg }g(x,y)\}\).

MSC:

34C08 Ordinary differential equations and connections with real algebraic geometry (fewnomials, desingularization, zeros of abelian integrals, etc.)
14K20 Analytic theory of abelian varieties; abelian integrals and differentials
34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
34C23 Bifurcation theory for ordinary differential equations
37G15 Bifurcations of limit cycles and periodic orbits in dynamical systems
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