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Help ODE system

  1. Jun 11, 2009 #1
    1. The problem statement, all variables and given/known data

    Hello.


    I want to study the stability of the origin of the following problem:


    dx/dt = -2y


    dy/dt = x + 2y


    dz/dt = -2z


    So the eigenvalues of this 3 x 3 matrix are -2, 1 + i, 1-i.


    The eigenvectors are (0,0,1) , (2,-1-i,0), (-2,-1+i,0).


    The solution (confirmed with Mathematica) is given by:


    x(t) = 2*exp(t)cos(t) * C_1 + 2*exp(t)*sin(t) * C_2 - 2*exp(t)*sin(t) *C_3


    y(t) = C_1 * (-exp(t)*¨sin(t) - exp(t)*cos(t)) + C_2 * (exp(t)cos(t) -exp(t)sin(t)) +
    C_3 * (-exp(t)cos(t)+exp(t)sin(t) )


    z(t) = 2*exp(2*t) *C_3


    Where C_1,C_2,C_3 are constants.


    How can I find (analytically, not by plotting) if the origin (0,0,0) is stable, asymptotically stable? unstable, a node, a center?


    I'm having trouble figuring this out. Can you please help?
     
  2. jcsd
  3. Jun 11, 2009 #2

    Mark44

    Staff: Mentor

    Is there a typo in z(t)? One of your eigenvalues is -2, so I would expect to see e^(-2t) in one of your solution functions.

    Because x(t) and y(t) both have terms with e^t, I would expect orbits that move away from the origin over time, which would make the origin unstable or a node (I don't recall exactly what these terms mean in the context of phase diagrams. And because z(t) is a decaying exponential function, whatever the orbits are doing, they are going to be heading down to the x-y plane over time. I'm just going off the top of my head here, so take what I'm saying with a grain of salt.
     
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