Two families of super-harmonic resonances in a time-delayed nonlinear oscillator

Two stable bifurcating periodic solutions are numerically found to coexist in a time-delayed nonlinear oscillator by using different initial conditions, after the trivial equilibrium loses its stability via two-to-one resonant Hopf bifurcations. These two coexisting solutions have different amplitudes and frequency components with one having the frequencies of Hopf bifurcations while the other containing different frequencies from those of Hopf bifurcations. The dynamic interaction of the periodic excitation and the two coexisting solutions can induce two families of super-harmonic resonances, when the forcing frequency is approximately at half the lower frequency component of the stable bifurcating solutions. It is found that the forced response under two families of super-harmonic resonances exhibits qualitatively different dynamic behaviour. In addition, one family of super-harmonic resonances may suddenly disappear when the excitation magnitude reaches a certain value and then the forced response becomes non-resonant response. The other family of super-harmonic resonances can be established by adjusting the forcing frequency accordingly. Time trajectories, phase portraits, frequency spectra, basin of attraction and bifurcation diagrams are given to characterise the different dynamic behaviours of the time-delayed nonlinear oscillator.