Loss of super-harmonic resonances in a time-delayed nonlinear oscillator

The trivial equilibrium may lose its stability via two-to-one resonant Hopf bifurcations in a time-delayed nonlinear oscillator having quadratic nonlinear terms. Two co-existing sets of stable periodic solutions are numerically found by using different histories in the vicinity of the resonant Hopf bifurcations. One set of stable periodic solutions is of small amplitude and has the Hopf bifurcation frequencies while the other is of large amplitude and has different frequencies from those of Hopf bifurcations. The presence of a periodic excitation in the time-delayed nonlinear oscillator may induce dynamic interactions between the periodic excitation and either set of the stable periodic solutions. Under hard excitations, the forced response following the resonant Hopf bifurcations can exhibit super-harmonic resonance at half the lower Hopf bifurcation frequency. With the increase of the magnitude of the external excitation, the super-harmonic resonances related to the lower Hopf bifurcation frequency may disappear and become non-resonant forced response. Frequency spectrum analysis indicates that the frequency components related to Hopf bifurcations vanish and the frequency components corresponding to the large-amplitude bifurcating periodic motion appear in the forced response. Super-harmonic resonances can be re-established by adjusting the forcing frequency according to the newly appeared frequency components of the large-amplitude bifurcating periodic solutions. The present paper uses numerical integration to study the sudden loss and re-establishment of super-harmonic resonances in the time-delayed nonlinear oscillator having periodic excitation. Time trajectories, phase portraits and frequency spectra are used to show the different forced response of the time-delayed nonlinear oscillator under super-harmonic resonances