Periodic orbits for 2n-dimensional control piecewise smooth dynamical systems

In this paper we study the periodic orbits of a class of 2n-dimensional control dynamical systems with a perturbation of continuous piecewise smooth quadratic polynomial and a perturbation of discontinuous piecewise smooth polynomial of an arbitrarily given degree, respectively. By applying the averaging theory for continuous and discontinuous piecewise smooth systems, the number of isolated zeros of the averaging function can be determined, which provides a lower bound of the maximum number of isolated periodic orbits. At last, we give examples to show that the lower bound of the maximum number is accessible by the properties of algebraic closure and transcendental extension.