On the periodic solutions of the Michelson continuous and discontinuous piecewise linear differential system (Q725791)

The authors consider periodic solutions of the Michelson continuous and discontinuous piecewise linear differential systems. Here, the Michelson continuous piecewise linear differential system is given by: \[ \dot{x}=y, \dot{y}=z, \dot{z}=-y+\varepsilon(2d^2-|x|), \] where \(d\geq0\) and \(\varepsilon>0\), sufficiently small. The Michelson discontinuous piecewise linear differential system has the form: \[ \dot{x}=y, \dot{y}=z, \dot{z}=-y+\varepsilon(2d^2-|x|-\text{sign} x), \] where \(d\geq0\), \(\varepsilon>0\), sufficiently small and \(\text{sign} x\) is a sign function. The main method in this article is based on new results from the averaging theory. Then they obtain the periodic solutions of the Michelson continuous and discontinuous piecewise linear differential systems. The method used in this article can also be applied to general nonsmooth dynamical systems.