Results 1 to 3 of 3

Math Help - Critical Points and Linearization

  1. #1
    Newbie
    Joined
    Sep 2009
    From
    Michigan
    Posts
    20

    Critical Points and Linearization

    Problem:

    Determine the location and type of all critical points by linearization.

    y1' = -y1 + y2 - (y2)**2
    y2' = -y1 - y2

    From the above, I set both equal to zero and determined the critical points to be (0,0) and (-2, 2).

    This is where I get a little bit lost as to what to do next. It seems to me that I need to drop non-linear terms and find the eigenvalues.

    y1' = -y1 + y2
    y2' = -y1 - y2

    In matrix form:

    | -1-lambda 1 |
    | -1 -1-lambda |

    I compute lambda to be (-1 +/- i)

    This is where I get completely lost and could use some help. Thanks in advance.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member
    Joined
    Aug 2008
    Posts
    903
    You have to linearize the system at each equilibrium point so it's not just dropping the non-linear terms. The linearized system is the Jacobian at each equilibrium point. You can calculate the Jacobian right? It's just the partials of the right side:

    \left(\begin{array}{cc}\displaystyle\frac{\partial f}{\partial x}(x_0,y_0) & \displaystyle\frac{\partial f}{\partial y}(x_0,y_0) \\ \displaystyle\frac{\partial g}{\partial x}(x_0,y_0) & \displaystyle\frac{\partial f}{\partial y}(x_0,y_0)\end{array}\right)

    At the point (0,0), the eigenvalues are (-1\pm i). That's a spirial sink according to the standard naming convention.

    Find "Differential Equations" by Blanchard, Devaney, and Hall. It's the best intro to the subject.

    Now, use Mathematica to plot the phase portrait:

    Code:
    StreamPlot[{-x + y - y^2, -x - y}, {x, -2, 2}, {y, -5, 5}]
    The origin looks like a spirial sink to me.
    Attached Thumbnails Attached Thumbnails Critical Points and Linearization-phaseportrait6.jpg  
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Sep 2009
    From
    Michigan
    Posts
    20

    Re: Critical Points and Linearization

    Got it! Thanks! Took me a while to learn more about how and when to determine the Jacobian, but now it looks fairly obvious to me what the process steps are in this type of problem. Wish my text book was as clear!
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 16
    Last Post: June 10th 2011, 07:49 AM
  2. Non-Linear First Order ODE: Critical Point Linearization
    Posted in the Differential Equations Forum
    Replies: 13
    Last Post: October 9th 2010, 07:59 AM
  3. Replies: 1
    Last Post: April 20th 2010, 08:36 PM
  4. Replies: 2
    Last Post: October 29th 2009, 09:02 PM
  5. Critical Points, Linearization
    Posted in the Advanced Applied Math Forum
    Replies: 0
    Last Post: January 27th 2007, 05:37 PM

Search Tags


/mathhelpforum @mathhelpforum