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Stability of nonlinear system

  1. Mar 31, 2013 #1
    i need show that at the following system the zero solution is nominally stable, using some change of variable that transforme in a linear system

    [tex] \frac{dx}{dt}=-x + \beta (x^2+ y^2) [/tex]

    [tex] \frac{dy}{dt}=-2y + \gamma x y [/tex]

    i tried with the eigenvalues of the Jacobian matrix at (0,0), but one of them is positive , then the system is unstable...
     
    Last edited: Mar 31, 2013
  2. jcsd
  3. Apr 1, 2013 #2
    What does the linearized matrix look like?
     
  4. Apr 1, 2013 #3

    HallsofIvy

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    No, it is pretty obvious just looking at this system what the eigenvalues are and they are both negative.
     
  5. Apr 1, 2013 #4

    epenguin

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    What point are you linearising about?

    OK now I see you say only (0, 0) is required. Doesn't seem to me you need any transformation for that point.

    If I am not mistaken there is the possibility of three 'equilibrium' points - the other two may be more interesting than (0, 0).
     
    Last edited: Apr 1, 2013
  6. Apr 1, 2013 #5
    the problem says:
    "show that the zero solution is nonlinear stable. For this, find the change of variable that transforms this system in a linear system"....

    i dont understand
     
  7. Apr 3, 2013 #6

    epenguin

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    Can anyone tell me what 'nominally stable' means? I know what 'locally stable' is, which would usually be the question.

    To transform the whole system in which there are in general three different stationary points into a linear one would seem on the face of it impossible, isn't it?:uhh:
     
    Last edited: Apr 3, 2013
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