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Limit cycles of a kind of nonsmooth differential systems with two switching lines. (English) Zbl 1461.34053

Summary: This paper is concerned with the number of limit cycles of nonsmooth differential systems \[ \begin{cases} \dot{x}=y,\\ \dot y=-cx^2-dx-e,\end{cases} x\geq 0;\begin{cases} \dot x = y, \\ \dot y=-ax-b,\end{cases}x<0 \] under nonsmooth perturbations of polynomials of degree at most \(n\), where \(a,b,c,d,e\in\mathbb{R}\). We first obtain the detailed expansion of the first Melnikov function by computing its generators for \(ac\neq 0\). Then by using the expansion, we give the upper bounds for the number of limit cycles bifurcating from each period annulus for two cases: \(c>0\), \(ab\neq 0\), \(d^2=4ec\) and \(c=0,abde\neq 0\).

MSC:

34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
37G15 Bifurcations of limit cycles and periodic orbits in dynamical systems
34C23 Bifurcation theory for ordinary differential equations
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