Secondary resonances of a quadratic nonlinear oscillator following two-to-one resonant Hopf bifurcations

Stable bifurcating solutions may appear in an autonomous time-delayed nonlinear oscillator having quadratic nonlinearity after the trivial equilibrium loses its stability via two-to-one resonant Hopf bifurcations. For the corresponding non-autonomous time-delayed nonlinear oscillator, the dynamic interactions between the periodic excitation and the stable bifurcating solutions can induce resonant behaviour in the forced response when the forcing frequency and the frequencies of Hopf bifurcations satisfy certain relationships. Under hard excitations, the forced response of the time-delayed nonlinear oscillator can exhibit three types of secondary resonances, which are super-harmonic resonance at half the lower Hopf bifurcation frequency, sub-harmonic resonance at two times the higher Hopf bifurcation frequency and additive resonance at the sum of two Hopf bifurcation frequencies. With the help of centre manifold theorem and the method of multiple scales, the secondary resonance response of the time-delayed nonlinear oscillator following two-to-one resonant Hopf bifurcations is studied based on a set of four averaged equations for the amplitudes and phases of the free-oscillation terms, which are obtained from the reduced four-dimensional ordinary differential equations for the flow on the centre manifold. The first-order approximate solutions and the nonlinear algebraic equations for the amplitudes and phases of the free-oscillation terms in the steady state solutions are derived for three secondary resonances. Frequency-response curves, time trajectories, phase portraits and Poincare sections are numerically obtained to show the secondary resonance response. Analytical results are found to be in good agreement with those of direct numerical integrations.