Results about structural stability and the existence of limit cycles for piecewise smooth linear differential equations separated by the unit circle

In this article, we investigate the structural stability and the existence of limit cycles in families of piecewise smooth differential equations where the unit circle serves as the discontinuity region. Our study encompasses families featuring singularities of center or saddle type, both visible and invisible, as well as those without any singularities. For the family that admits only constant vector fields, we describe the dynamics over \(S^1\) $S^1$ and present a result regarding structural stability. For the other families, we provide an upper bound for the number of limit cycles and present examples that illustrate the maximum number of limit cycles that can be realized. In the constant-center case, we present a proof of the existence and stability of the limit cycle using elementary analytical geometry. Additionally, we discuss the presence of homoclinic cycles in saddle-center cases for such differential equations, taking into account Filippov’s convention.