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A time-delayed quadratic nonlinear mechanical system can exhibit two coexisting stable bifurcating solutions ( SBSs) after two-to-one resonant Hopf bifurcations occur in the corresponding autonomous time-delayed system. One SBS is of small-amplitude and has the Hopf bifurcation frequencies ( HBFs), while the other is of large-amplitude and contains the shifted Hopf bifurcation frequencies (the shifted HBFs). When the forcing frequency is tuned to be the sum of two HBFs or the sum of two shifted HBFs, two families of additive resonances can be induced in the forced response. The forced response under the additive resonance related to the HBFs can demonstrate periodic, quasi-periodic and chaotic motion. On the contrary, the forced response under the additive resonance associated with the shifted HBFs may exhibit period-three periodic motion and quasi-periodic motion. Bifurcation diagrams, time trajectories, frequency spectra, phase portraits and Poincaré sections are presented to show periodic, quasi-periodic, and chaotic motion of the time-delayed nonlinear system under the two families of additive resonances. [ABSTRACT FROM AUTHOR]

The trivial equilibrium of a weakly nonlinear oscillator having quadratic nonlinearities under a delayed feedback control can change its stability via a single Hopf bifurcation as the time delay increases. Double Hopf bifurcation occurs when the characteristic equation has two pairs of purely imaginary solutions. An interaction of resonant Hopf-Hopf bifurcations may be possible when the two critical time delays corresponding to the two Hopf bifurcations have the same value. With the aid of normal form theory and centre manifold theorem as well as the method of multiple scales, the present paper studies the dynamics of a quadratically nonlinear oscillator involving time delay in the vicinity of the point of two-to-one resonances of Hopf-Hopf bifurcations. The ratio of the frequencies of two Hopf bifurcations is numerically found to be nearly equal to two. The two resonant Hopf bifurcations can generate two respective periodic solutions. Consequently, the centre manifold corresponding to these two solutions is determined by a set of four first-order differential equations under two-to-one internal resonances. It is shown that the amplitudes of the two bifurcating periodic solutions admit the trivial solution and two-mode solutions for the averaged equations on the centre manifolds. Correspondingly, the cumulative behavior of the original nonlinear oscillator exhibits the initial equilibrium and a quasi-periodic motion having two frequencies. Illustrative examples are given to show the unstable zero solution, stable zero solution, and stable two-mode solution of the nonlinear oscillator under the two-to-one resonant Hopf-Hopf interactions. [ABSTRACT FROM AUTHOR]

Linear-plus-nonlinear feedback control is used to stabilize Hopf bifurcation in a rotor-magnetic bearing system, for which the linearized system possesses a double zero eigenvalues. The addition of nonlinear (quadratic) terms to the original linear feedback control formulation is used to modify the coefficients of the nonlinear terms in the reduced normal forms. It is found that feedback control incorporating certain quadratic terms renders the Hopf bifurcation supercritical. Finally, illustrative examples are given to verify the analytical results. [ABSTRACT FROM AUTHOR]