非線性分析——第二次作業(yè)

1、The Second HomeworkQuestion 1For the following dynamical systems1） 2） a) Find all fixed points and classify them.b) Sketch the phase space portrait.Solution:(1) Letting, the corresponding state-space equations is The method of Poincare sections is used in the question.Letting , and C is constant, th

2、en we can obtain thatTherefore, the fixed points are given as (0,0). Jacobian matrix is The eigenvalues calculated asTherefore, there is an unstable fixed point, the saddle point.The phase space portrait is present in Fig. 1.Fig. 1(2) Equations LettingThen, the fixed points are obtained as (0,0).Jac

3、obian matrix Eigenvalues are Therefore, the stability cant be determined by the eigenvalues directly.A Lyapunov function is constructed asThe derivative of isLetting , then , so examined fixed point (0,0) is stable.The phase space portrait is present in Fig. 2.Fig. 2Question 2Given the system determ

4、ine the stability at (0,0,0).Solution:To find the fixed points, we have to solve equationsOne fixed point is obtained as (0,0,0).Jacobian matrixEigenvalues are Therefore, the stability cant be determined by the eigenvalues directly.A Lyapunov function is constructed asThe derivative of isLetting , t

5、hen , so examined fixed point (0,0,0) is stable.Question 3For the system a) Find all fixed points and discuss the stability.b) Analyze the bifurcation and sketch the bifurcation diagram.Solution:Letting , the corresponding state-space equations is Solve the equationsWe find two fixed points:(0, 0) a

6、nd ( , 0) Jacobian matrix the system is a) At the fixed point (0,0), the Jacobian matrix isWhen , the eigenvalues are and , there is an unstable fixed point, the saddle point. The phase space portrait is shown in Fig. 3.When , the eigenvalues are i and i, there is neutral fixed point, the node point

7、. The phase space portrait is shown in Fig. 4. Fig. 3 Fig. 4 b) At the fixed point ( ,0), the Jacobian matrix is the eigenvalues are and , there is neutral fixed point, the node point. The phase space portrait is shown in Fig. 5.Fig. 5The bifurcation diagram is shown in Fig. 6.Fig. 6Question 4Show t

8、hat the one parameter system undergoes a Hopf bifurcation at = 0. Plot phase portraits and sketch the bifurcation diagram.Solution:Letting , the corresponding state-space equations is Solve the equationsThen, the fixed points are obtained as (0,0).Jacobian matrix the system is Eigenvalues are When ,

9、 there is an stable fixed node. The phase space portrait is shown in Fig. 7.Fig. 7When , there is an stable star. The phase space portrait is shown in Fig. 8. Fig. 8When , there is an stable spiral. The phase space portrait is shown in Fig. 9. Fig. 9When , there is an node center. The phase space po

10、rtrait is shown in Fig. 10. Fig. 10When , there is an unstable spiral. The phase space portrait is shown in Fig. 11. Fig. 11When , there is an unstable star. The phase space portrait is shown in Fig. 12. Fig. 12When , there is an unstable fixed node. The phase space portrait is shown in Fig. 13. Fig

11、. 13The bifurcation diagram is pictured in Fig. 14.Fig. 14Question 5For Hénon map 1) Find the points of period-1 and period-2 for the Hénon map.2) Investigate the bifurcation diagrams for the Hénon map by plotting the values as a function of when = 0.38.Solution:For period-1,Letting , and then So