Super-harmonic resonance response of a quadratically nonlinear oscillator involving time delay

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Conference Proceeding
Advances in Applied Mechanics Research, Conference Proceedings - 7th Australasian Congress on Applied Mechanics, ACAM 2012, 2012, pp. 546 - 554
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Presence of time delays in nonlinear oscillators can not only cause instability of the trivial equilibriums but also generate rich dynamic behaviour including bifurcations and their interactions. For the nonlinear oscillators without external excitations, it has been shown that the trivial equilibrium of the nonlinear oscillators may lose its stability via a subcritical or a supercritical Hopf bifurcation and regain its stability via a reverse subcritical or a supercritical Hopf bifurcation as the time delay increases. The resultant periodic solutions from single Hopf bifurcations may be stable or unstable. When there are two pairs of purely imaginary solutions for the characteristic equation, double Hopf bifurcation is possible for the nonlinear oscillators involving time delay. Consequently, an interaction of two Hopf bifurcations can occur when the two critical time delays corresponding to the two Hopf bifurcations have the same value, which may result in either non-resonant or resonant Hopf-Hopf bifurcations. Two-to-one resonant Hopf bifurcations are found to exist in a quadratically nonlinear oscillator involving time delay. The two resonant Hopf bifurcations create two respective periodic solutions for the nonlinear oscillator without external excitation. The collective behaviour of the oscillator includes stable initial equilibrium, stable periodic motion, stable quasi-periodic motion and unstable motion. The presence of a periodic excitation in the nonlinear oscillator can induce dynamic interactions between the weakly periodic excitation and the stable bifurcating solution. When the frequency of the external excitation and the frequencies of Hopf bifurcations satisfy a certain relationship, the dynamic interaction can produce resonant dynamic behaviour including primary resonance, super-harmonic and sub-harmonic resonances. With the aid of normal form theory and centre manifold theorem as well as the method of multiple scales, the present paper studies the super-harmonic resonance response of a quadratically nonlinear oscillator involving time delay in the vicinity of the point of two-to-one resonances of Hopf-Hopf bifurcations. Three small perturbation parameters (which are called dummy unfolding parameters) are introduced to conveniently account for the variations of linear feedback gains and time delay near the point of double Hopf bifurcations. The effect of three small perturbation parameters on the forced response is discussed by studying the stability and bifurcations of fixed points of the averaged equations. Analytical results are validated by a comparison with those of direct numerical integration.
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