Limit cycles of a class of discontinuous piecewise differential systems in \(\mathbb {R}^3\) separated by cylinders

In recent years, the study of piecewise linear differential systems has gained importance due to their applications. Many diverse natural phenomena are already modeled by this kind of system, like in physics, biology, economics, etc. It is common knowledge that limit cycles are the primary research focus in the qualitative theory of piecewise differential systems. Planar systems have been extensively taken into account in the majority of publications examining the existence and maximum number of limit cycles of piecewise differential systems. Still, a few investigations are looking into this issue in \(\mathbb {R}^3\) ${\mathbb{R}}^3$ . This study aims to answer the problem of the existence and the maximum number of limit cycles for a class of discontinuous piecewise differential systems in \(\mathbb {R}^3\) ${\mathbb{R}}^3$ formed by two Karabut systems and having one of the cylinders \(C_i =\{(x,y,z)\in \mathbb {R}^3:f_i(x,y,z)=0\}\) $C_i=\{(x,y,z)\in {\mathbb{R}}^3:f_i(x,y,z)=0\}$ with \(i=1,2\) $i=1,2$ as the switching manifold where \(f_1(x,y,z)=z-x^2\) $f_1(x,y,z)=z-x^2$ and \(f_2(x,y,z)=x^2+y^2-1\) $f_2(x,y,z)=x^2+y^2-1$ . The exact maximum number of limit cycles in this class of discontinuous piecewise differential systems is difficult to determine. Nevertheless, we find that there cannot be more than two limit cycles when the separation surface is the cylinder \(C_1\) $C_1$ and at most four limit cycles when we separate with the cylinder \(C_2\) $C_2$ . Additionally, we give instances for any separation surface that guarantees the achievement of this maximum.