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The number of zeros of abelian integrals for near-Hamilton system of degree three with a butterfly. (Chinese. English summary) Zbl 1413.34137

Summary: We consider the following near-Hamilton system \[ \begin{aligned} \dot{x} &= 2y\left ({a{x^2} + 2c{y^2}} \right) + \varepsilon f\left ({x, y} \right), \\ \dot{y} &= 2x\left ({1 - 2b{x^2} - a{y^2}} \right) + \varepsilon g\left ({x, y} \right), \end{aligned} \] where \(a < 0\), \(c > 0\), \(4bc > {a^2}\), \(0 < \left| \varepsilon \right| \ll 1\), \(f\left ({x, y} \right)\) and \(g\left ({x, y} \right)\) are cubic polynomials of \(x\) and \(y\). We obtain the upper bound of the number of isolated zeros of the abelian integral.

MSC:

34C08 Ordinary differential equations and connections with real algebraic geometry (fewnomials, desingularization, zeros of abelian integrals, etc.)
34E10 Perturbations, asymptotics of solutions to ordinary differential equations
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