Bifurcation of limit cycles for a kind of piecewise smooth differential systems with an elementary center of focus-focus type (Q6624352)

In recent years, many researchers have extended the weak Hilbert's 16th problem to piecewise smooth differential systems formed by some quadratic systems. In this paper, authors study the number of limit cycles $H(n)$ bifurcating from the piecewise smooth system with discontinuity on the $x$-axis under the perturbations of polynomials with degree $n$. Their system is formed by the quadratic reversible system (r22) and a cubic system. After analyzing the number of isolated zeros of first-order Melnikov function, the authors prove that $2n+3\leq H(n)\leq 2n+7$ for $n\geq 3$. Moreover, $H(n)=2n+3$ for $n=0,1,2$.