On the Maximum Number of Limit Cycles in a Class of 3D-Nonlinear Piecewise Differential Systems

This paper investigates the maximum number of limit cycles in a class of three-dimensional discontinuous nonlinear piecewise differential system formed by two Euler systems separated by a family of quadric surfaces. Each subsystem possesses two independent first integrals, which constrain the dynamics and enable an analytical study. We consider switching manifolds such as elliptic and hyperbolic paraboloids, one- and two-sheeted hyperboloids, and right circular cylinders, and provide for each case examples exhibiting one, two, three, and four limit cycles. We also provide examples exhibiting a continuum of periodic solutions with no limit cycles. In addition, we consider a parabolic cylinder as a switching manifold and prove that the discontinuous nonlinear piecewise differential system has at most two limit cycles. Moreover, we prove that these upper bounds are sharp and provide an example for each corresponding case with one and two limit cycles, respectively. Furthermore, we present examples exhibiting a continuum of periodic solutions with no limit cycles. These results establish a precise upper bound on the oscillatory behavior of three-dimensional discontinuous systems and extend the theory of piecewise smooth dynamical systems.