ExercisesSolutions

Classifying the fixed points using the eigenvalues 3

  • Notes
  • 32
  • 100% (1) 1 out of 1 people found this document helpful

Info icon This preview shows pages 14–19. Sign up to view the full content.

Classifying the fixed points using the eigenvalues, ([3 , 7 , 11 . . . ] π/ 2 , 2 ) : bracketleftbigg 0 1 1 0 bracketrightbigg . Eigenvalues are 1 and - 1, and thus the fixed point(s) is a saddle point . ([1 , 5 , 9 . . . ] π/ 2 , 2 ) : bracketleftbigg 0 1 - 1 0 bracketrightbigg . Eigenvalues are I and - I , and thus the fixed point(s) is a center . (f) ˙ x = xy - 1 , ˙ y = x - y 3 The real fixed points are (1 , 1), ( - 1 , - 1) and ( I, - I ). The Jacobian of the system is bracketleftbigg y x 1 - 3 y 2 bracketrightbigg . Classifying the fixed points using the eigenvalues, (1 , 1) : bracketleftbigg 1 1 1 - 3 bracketrightbigg . Eigenvalues are - 3 . 2361 and 1 . 2361, and thus (1 , 1) is a saddle point . ( - 1 , - 1) : bracketleftbigg - 1 - 1 1 - 3 bracketrightbigg . Eigenvalues are - 2 and - 2, and thus ( - 1 , - 1) is a stable node . 14
Image of page 14

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

3. Purpose: To compare computer generated phase portratis with initial sketches. Exercise: For each of the nonlinear systems in Exercise 2, plot a computer-generated phase portrait and compare to your approximate sketch. Solution: The required computer-generated phase portrait for each system was created in Maple and are shown below. –4 –2 0 2 4 y –4 –2 2 4 x Figure 12: Phase portrait for Exercise 3a –8 –6 –4 –2 0 2 4 6 8 y –2 –1 1 2 x Figure 13: Phase portrait for Exercise 3b 15
Image of page 15
–4 –2 0 2 4 y –4 –2 2 4 x Figure 14: Phase portrait for Exercise 3c –4 –2 0 2 4 y –4 –2 2 4 x Figure 15: Phase portrait for Exercise 3d –4 –2 0 2 4 y –4 –2 2 4 x Figure 16: Phase portrait for Exercise 3e 16
Image of page 16

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

–4 –2 0 2 4 y –4 –2 2 4 x Figure 17: Phase portrait for Exercise 3f 17
Image of page 17
4. Purpose: To use qualitative arguments to deduce the phase portrait of a system. From [Str94], Problem 6.6.6 Exercise: Consider the reversible system ˙ x = y (1 - x 2 ) , ˙ y = 1 - y 2 . (a) Plot the nullclines ˙ x = 0 and ˙ y = 0. (b) Find the sign of ˙ x, ˙ y in different regions of the plane. (c) Calculate the eigenvalues and eigenvectors of the saddle points at ( - 1 , ± 1). (d) Consider the unstable manifold of ( - 1 , - 1). By making an argument about the signs of ˙ x, ˙ y , prove that this unstable manifold intersects the negative x -axis. Then use reversibility to prove the existence of a heteroclini trajectory connecting ( - 1 , - 1) to ( - 1 , 1). (e) Using similar arguments, prove that another heteroclinis trajectory exists, and sketch several other trajectories to fill in thephase portrait. Solution: (a) The nullcline plot of ˙ x = 0 and ˙ y = 0 can be seen in Figure 18. –2 –1 0 1 2 y –2 –1 1 2 x Figure 18: Nullcline plot for Exercise 4a (b) Figure 19 shows the signs of ˙ x, ˙ y in the different regions of the plane. dy/dt<0 dx/dt>0 dy/dt>0 dx/dt>0 dy/dt>0 dx/dt<0 dy/dt<0 dx/dt<0 dy/dt<0 dx/dt<0 dy/dt>0 dx/dt<0 dy/dt>0 dx/dt>0 dy/dt<0 dx/dt>0 dy/dt<0 dx/dt>0 dy/dt>0 dx/dt>0 dy/dt>0 dx/dt<0 dy/dt<0 dx/dt<0 –2 –1 0 1 2 y –2 –1 1 2 x Figure 19: The sign of ˙ x, ˙ y in different regions of the plane.
Image of page 18

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

Image of page 19
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern