TWO-TO-ONE RESONANT HOPF BIFURCATIONS IN A QUADRATICALLY NONLINEAR OSCILLATOR INVOLVING TIME DELAY

被引:12
作者
Ji, J. C. [1 ]
Li, X. Y. [2 ]
Luo, Z. [1 ]
Zhang, N. [1 ]
机构
[1] Univ Technol Sydney, Sch Elect Mech & Mechatron Syst, Sydney, NSW 2007, Australia
[2] Hebei Univ Technol, Sch Mech Engn, Tianjin 300130, Peoples R China
来源
INTERNATIONAL JOURNAL OF BIFURCATION AND CHAOS | 2012年 / 22卷 / 03期
基金
中国国家自然科学基金;
关键词
Two-to-one internal resonances; time delay; resonant Hopf bifurcations; Hopf-Hopf interactions; nonlinear oscillator; quadratic nonlinearities; feedback control; POL-DUFFING OSCILLATOR; STABILITY; VAN; FEEDBACK; DYNAMICS; SYSTEMS;
D O I
10.1142/S0218127412500605
中图分类号
O1 [数学];
学科分类号
0701 ; 070101 ;
摘要
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.
引用
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页数:14
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