Effects of time delayed position feedback on a van der Pol–Duffing oscillator

The mechanism for the action of time delay in a non-autonomous system is investigated in this paper. The original mathematical model under consideration is a van der Pol–Duffing oscillator with excitation. A delayed system is obtained by adding both linear and nonlinear time delayed position feedbacks to the original system. Functional analysis is used to change the delayed system into a functional differential equation (FDE). The time delay is taken as a variable parameter to investigate its effect on the dynamics of the system such as the stability and bifurcation of an equilibrium point, phase locked (periodic) and phase shifting solution, period-doubling, quasi-periodic motion and chaos. A periodic solution expressed in the closed form with the aid of the center manifold and averaging theorem is found to be in good agreement with that obtained by numerical simulation. Two routes to chaos are found, namely period-doubling bifurcation and torus breaking. The results obtained in this paper show that time delay may be used as a simple but efficient “switch” to control motions of a system: either from order motion to chaos or from chaotic motion to order for different applications.