本文研究了一类具有时滞的厄尔尼诺一南方波涛动模型．得到了该模型在平衡点稳定的充分条件．通过选择时滞η为分岔参数，得到了当时滞η通过一系列的临界值时，Hopf分岔产生，然后，应用中心流形和规范型理论，得到了确定Hopf分岔特性（例如Hopf分岔方向和分岔周期解的稳定性以及Hopf分岔周期解的周期等）的计算公式．最后进行数值模拟验证了所得结果的正确性．

In this paper, a delayed sea-air oscillator coupling model for the ENSO is investigated. We obtain the sufficient condition of stability in equilibrium. By choosing delay 7/as a bifurcation parameter, we show that Hopf bifurcation can occur when delay 7/passes through a sequence of critical values. Meanwhile, based on the center manifold theory and the normal form approach, we derive the formula for determining the properties of Hopf bifurcating periodic orbit, such as the direction of Hopf bifurcation, the stability of Hopf bifurcating periodic solution and the periodic of Hopf bifurcating periodic solution. Finally, numerical simulations are carried out to illustrate the analytical results.

对一类Van der Pol-Duffing系统进行Hopf分岔分析,并基于Washout滤波器设计状态反馈控制器,讨论控制增益对Hopf分岔的存在性及其极限环幅值的影响.结果表明,选取适当的控制增益可以控制Hopf分岔的发生并改变极限环幅值的大小.

We studied the Hopf bifurcation and Hopf bifurcation control for a Van der Pol-Duffing system.First of all,we analyzed the Hopf bifurcation and designed the linear and nonlinear state feedback controller for the autonomous system.Then,we designed a state feedback controller based on Washout filter, and discussed the impact of the control gain on the existence of the Hopf bifurcation and the limit cycle amplitude.Results show that selecting appropriate control gain can not only control the occurrence of Hopf bifurcation,but also change the size of the limit cycle amplitude.

为了控制周期系数微分系统平衡点失稳后的分岔行为，基于Floquet-Lyapunov理论，将控制常系数系统分岔行为的方法（线性法、参数法、平移法）应用于一类具有周期系数的力学微分系统，设计了相应的控制器，研究了其控制平衡点分岔行为的有效性。研究结果表明：平移法不能有效控制周期系数微分系统的平衡点失稳后发生的Flip分岔和Hopf分岔行为。若平衡点失稳发生Flip分岔形成周期2点，可分别采用线性法和参数法将周期2点控制到周期1点；若平衡点失稳发生Hopf分岔形成Hopf圈，可分别采用线性法和参数法将Hopf圈控制到周期1点。

In order to control the bifurcation behavior at the equilibrium point of the differential system with periodic coefficients losing its stability,the methods for bifurcation control for the dynamical system with constant coefficients, such as using the linear controller, parameter method, and translation,were applied to a mechanical system with periodic coefficients by the Floquet-Lyapunov theory. Then,the related controllers were designed,and its validity in controlling the bifurcation behavior at the equilibrium point was tested through numerical calculation. The results show that translation is invalid to control the Flip and Hopf bifurcations at the equilibrium point in mechanical system with periodic coefficients. When a 2-periodic point is generated by the period-doubling Flip bifurcation at the unstable equilibrium point,either of the linear controller and the parameter method can be used to control the 2-periodic point back to a 1-periodic point. When a Hopf circle is generated

考虑含参数激励的广义Van der Pol方程的Hopf分岔与控制问题。通过设计线性位移和速度时滞反馈控制器构造了受控系统,着重研究了控制器对该类参数激励系统的1/2亚谐共振的分岔响应控制。采用多尺度法从理论上推导出时滞动力系统的分岔响应方程,并进一步得到Hopf分岔的存在条件。通过数值模拟,验证了所设计的控制器不仅能控制极限环的幅值,也能控制Hopf分岔的产生。

Hopf bifurcation and controlling of a generalized Van der Pol equation with parameter excitation are studied .The delayed feedback controller of linear displacement and velocity is designed to control the system .The paper is devoted to the study of the bifurcation response controlling of 1/2 harmonic resonance of the system .Bifurcation response equation of the time-delays dynamic system is obtained in theory by multiple scale method .Moreover , condition of the existence of Hopf bifurca-tion is also obtained .Numerical simulations show that the controller can control the bifurcation as well as the amplitude of limit cycle.

为进一步了解一个复杂的有不稳定奇点的三维动力系统在Hopf分岔点附近的非线性特性,采用非线性控制器,提出了相应的控制系统,使得受控系统可能发生余维一、余维二和余维三的Hopf分岔.通过严格的数学推导给出了受控系统发生分岔的参数条件,证明了可控制系统在指定区域内发生退化分岔和可调控分岔的稳定性.

In order to understand the complex three-dimensional dynamical system with the unstable nodes, we propose a nonlinear controller. The corresponding controlling system makes the codimension one, two, and three Hopf bifurcations happen. The mathematical deduction demonstrates that the system can be controlled to produce the degenerate Hopf bifurcation at desired location and stability of controllable bifurcation.