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On the number of limit cycles for generic Lotka-Volterra system and Bogdanov-Takens system under perturbations of piecewise smooth polynomials. (English) Zbl 1444.34056

The authors consider the bifurcation of limit cycles for a generic Lotka-Volterra system and a Bogdanov-Takens system under perturbations of piecewise smooth polynomials with degree \(n\). By means of a linear transformation, the Picard-Fuchs equation, the Melnikov function, the authors prove that the upper bounds of the number of limit cycles (taking into account the multiplicity) for this generic L-V system and this B-T system are, respectively, \(39n-72\) \((n\geq 4)\), \(39, 59, 98\) \((n = 1, 2, 3)\) and \(12n + 3\left[\frac{n}{2}\right] + 5\).

MSC:

34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
34E10 Perturbations, asymptotics of solutions to ordinary differential equations
34C07 Theory of limit cycles of polynomial and analytic vector fields (existence, uniqueness, bounds, Hilbert’s 16th problem and ramifications) for ordinary differential equations
34C23 Bifurcation theory for ordinary differential equations
34A36 Discontinuous ordinary differential equations
34C08 Ordinary differential equations and connections with real algebraic geometry (fewnomials, desingularization, zeros of abelian integrals, etc.)
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References:

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