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Limit cycles in two kinds of quadratic reversible systems with non-smooth perturbations. (English) Zbl 1478.34038

The author studies some non-smooth perturbations of degree \(n\) of two concrete differential systems showing a reversible center. The perturbations are in the four quadrants that surround the center. For these perturbations, the author detects the maximal bound for the maximum number of limit cycles which can be obtained from the period annulus of the mentioned center according to the degree of the perturbation. In the case of quadratic perturbations, he also detects that a certain number of limit cycles may be obtained for sure. The statement is then that \(11\leq H(2)\leq 23\), and there are other statements according the degree and the type of perturbation. Here \(H(n)\) means the maximum number of limit cycles which can be obtained from the period annulus of the center in these piecewise systems. One needs not to confound this \(H(n)\) with the more classical acception that it represents the maximum number of limit cyles that a polynomial differential system of degree \(n\) may have. The result seems very nice and the technique used of studying the Melnikov function is a standard tool. I feel just a pity that the author has not delighted us with a concrete example of perturbation showing the 11 limit cycles claimed.

MSC:

34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
34C14 Symmetries, invariants of ordinary differential equations
34E10 Perturbations, asymptotics of solutions to ordinary differential equations
34C23 Bifurcation theory for ordinary differential equations
34A36 Discontinuous 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
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