Some of the engineering systems and physical systems described by a lumped element system have certain symmetries in the system configuration and nonlinear characteristics. In such systems, these symmetries are reflected to the state equations which become invariant with respect to the symmetric transformation of the state variables. The authors have reported symmetries in autonomous equations [1]. Similar discussions are possible to the nonautonomous system with a periodic external force. However, the discussion has to be extended due to the periodic external force.
In this paper, the symmetry of the nonautonomous system with periodic external force and the symmetry of the periodic solution are discussed. It is shown that two types of symmetry should be defined for the analysis of a nonautonomous system. It is demonstrated also that, aside from certain exceptions, the symmetry of the autonomous systems can be discussed systematically. Further, a mapping representing the symmetry of the solution is defined and the quasi-periodic solutions and chaos are classified according to the symmetry.