一类周期系数力学系统分岔控制

Bifurcation Control of Mechanical System with Periodic Coefficients

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 by Hopf bifurcation after the equilibrium point loses its stability,the linear controller and the parameter method are all effective for controlling the Hopf circle to a 1-periodic point.