一类自治系统Hopf分叉及极限环幅值的时滞反馈控制

对含有非线性时滞位移的van der Pol—Duffing方程进行了研究，着重研究了时滞参数对van der Pol—Duffing系统Hopf分叉及极限环幅值的控制．首先采用摄动法从理论上推导出极限环幅值与时滞参数之间的关系，分析时滞参数对幅值大小的影响，并着重讨论了不改变振动频率情况下对幅值的控制．通过对零解的稳定性分析，得出Hopf分叉产生的条件．最后用数值计算的方法验证了理论计算结果，数值计算结果与理论结果相当吻合．

Control of a kind of autonomovs system to Hopf bifurcation and amplitude of limit cycle based on time delays feedback

A typical dynamics system, van der Pol-Duffing equation with two time delays, was studied. The key aim was to study amplitude control of limit cycle in van der Pol-Duffing system by using time delays feedback controller. Perturbation method was used to obtain the relation equation between amplitude and time-delays. Based on the equation, the controlling of time delays to amplitude were discussed. The stability of normal solution was analyzed, and the Hopf bifurcation condition of the controlled system was obtained. Numerical method was utilized to testify the theoretic results, which indicated the numerical results were in good a greement with the theoretical results.