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Limit cycle bifurcations in perturbations of a reversible quadratic system with a non-rational first integral. (English) Zbl 1461.34063

In this paper, by using Melnikov function method, the authors investigate the number of limit cycles of a reversible quadratic system whose first integral is a non-rational function, when it is perturbed inside the class of all discontinuous quadratic polynomial differential systems with the straight line of discontinuity \(y = 0\). The result shows that the piecewise smooth system can have at least four more limit cycles around the origin than the smooth one. It should be emphasized that the first integral is not a rational function, which leads to a lot of calculation.

MSC:

34C23 Bifurcation theory for ordinary differential equations
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
34A36 Discontinuous ordinary differential equations
34A05 Explicit solutions, first integrals of ordinary differential equations
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