STEADY-STATE, HOPF AND STEADY-STATE-HOPF BIFURCATIONS IN DELAY DIFFERENTIAL EQUATIONS WITH APPLICATIONS TO A DAMPED HARMONIC OSCILLATOR WITH DELAY FEEDBACK

被引:11
作者
Song, Yongli [1 ]
Jiang, Jiao [2 ]
机构
[1] Tongji Univ, Dept Math, Shanghai 200092, Peoples R China
[2] Shanghai Maritime Univ, Dept Math, Shanghai 201306, Peoples R China
来源
INTERNATIONAL JOURNAL OF BIFURCATION AND CHAOS | 2012年 / 22卷 / 12期
基金
中国国家自然科学基金;
关键词
Delay differential equations; steady-state bifurcation; Hopf bifurcation; steady-state-Hopf bifurcation; NEURAL-NETWORK MODEL; PLANAR SYSTEM; NORMAL FORMS; STABILITY; NEURONS; ORBITS;
D O I
10.1142/S0218127412502860
中图分类号
O1 [数学];
学科分类号
0701 ; 070101 ;
摘要
In this paper, employing the normal form theory of delay differential equations due to Faria and Magalhaes, we present explicit formulas of the coefficients of a normal form associated with the flow on a center manifold with the unfolding for general delay differential equations under the cases of steady-state, Hopf and steady-state-Hopf singularities. The explicit conditions determining the transcritical and pitchfork bifurcations for steady-state singularity, determining the direction and stability of Hopf bifurcations, and determining the coefficients of a normal form with universal unfolding for steady-state-Hopf singularity up to third order are obtained. Using the obtained results, we give a complete description of bifurcation scenario of the damped harmonic oscillator with delay feedback near the zero equilibrium. Finally, numerical simulations are given to illustrate our theoretical results and some numerical extensions are obtained as a supplement to our theoretical analysis.
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页数:31
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