An Efficient Method for Studying Weak Resonant Double Hopf Bifurcation in Nonlinear Systems with Delayed Feedbacks
An Efficient Method for Studying Weak Resonant Double Hopf Bifurcation in Nonlinear Systems with Delayed Feedbacks
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DOI：
10.1137/040614207
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发表时间：
2007-02
期刊：
影响因子：
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通讯作者：
Jian Xu;K. Chung;Chuen-Lit Chan
中科院分区：
文献类型：
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作者：
Jian Xu;K. Chung;Chuen-Lit Chan
提出了一种有效的方法，称为摄动增量格式(PIS)，用于定性和定量地研究时滞诱导的弱或高阶共振双Hopf分叉以及由时滞反馈非线性系统的分叉引起的动力学问题。该方案以时滞和反馈增益为分叉参数，分摄动和增量两步进行描述。作为应用，利用该方法研究了具有时滞反馈的van der Pol-Duffing和Stuart-Landau系统中时滞诱导的弱共振双Hopf分叉和动力学。对于接近双Hopf点的分叉参数，所有由共振分叉产生的解都被定性地分类，并由PIS的摄动步长近似地表示为闭合形式。虽然解析表达式对于远离双Hopf点的分叉参数可能不够精确，但它是正确的。
An efficient method, called the perturbation‐incremental scheme (PIS), is proposed to study, both qualitatively and quantitatively, the delay‐induced weak or high‐order resonant double Hopf bifurcation and the dynamics arising from the bifurcation of nonlinear systems with delayed feedback. The scheme is described in two steps, namely, the perturbation and the incremental steps, when the time delay and the feedback gain are taken as the bifurcation parameters. As for applications, the method is employed to investigate the delay‐induced weak resonant double Hopf bifurcation and dynamics in the van der Pol–Duffing and the Stuart–Landau systems with delayed feedback. For bifurcation parameters close to a double Hopf point, all solutions arising from the resonant bifurcation are classified qualitatively and expressed approximately in a closed form by the perturbation step of the PIS. Although the analytical expression may not be accurate enough for bifurcation parameters away from the double Hopf point, it is...