Study of double Hopf bifurcation and chaos for an oscillator with time delayed feedback

This paper is concerned with the effect of time delayed feedbacks in a nonlinear oscillator with external forcing. The particular attention is focused on the case that the corresponding linear system has two pairs of purely imaginary eigenvalues at a critical point, giving rise to double Hopf bifurcations. An analytical approach is used to find the explicit expressions for the critical values of the system parameters at which nonresonant or resonant Hopf bifurcations may occur. A fourth-order Ronge-Kutta numerical integration scheme is employed to obtain the dynamical solutions in the vicinity of the critical point. Both the cases with and without the external forcing are studied. It has been found that the system exhibits very rich complex dynamics, including periodic, quasi-periodic and chaotic motions. Moreover, a sensitivity analysis is carried out to show that chaotic motions are very sensitive to the time delay. This suggests that the time delay can be used: (1) to control bifurcations and chaos; and (2) to anticontrol (or generate) bifurcations and chaos.