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Overdetermined SDE and other problems

  1. Mar 4, 2010 #1
    Hello all,

    I am working with an economic problem and have ended up with the following dynamical system which is overdetermined, Jacobian has zero determinant (so one zero eigenvalue?) and probably other problems. The system looks something like this:

    dot{x1}=c1 x_2+c2 x_4/(x_2 x_1)
    dot{x2}=x_3*(1-x_1)+c3
    dot{x3}=c4 x_1
    dot{x4}=c5 x_1

    where c are parameters or constants. For overall stability first condition requires an x_1 s.t. dot{x_3}=dot{x_4}=0. Next even if that is not the case I still want to say something about how fast the system diverges. In any case if I assume the condition and get a fixed point for x_1, it means I can solve for x_3 from dot{x_2}=0 but then I end up with one equation dot{x_1} and two unknowns x_4 and x_2. Mathematically that can not be solved without further assumptions. Should/Can I do that numerically? Is there a formal people approach this problems?

    I have been playing around with some data and I can get a feel about how fast the model diverges based on the choices for the parameters. My question is, can I can analyze in a formal way any qualitative features of such model, or is it simply badly specified and I should just forget about it? Any references you can direct me to would be very much appreciated. Finally I tried playing with this model in Mathematica with NDSolve but it gives me "overdetermined" error. Any way to get around that?
    Many thanks!
     
  2. jcsd
  3. Mar 5, 2010 #2
    It is overdetermined because third and fourth equations are proportional. Repeat the analysis removing the third equation, and setting [tex]x_{3}=c_{6}+\frac{c_{4}}{c_{5}} x_{4}[/tex]
    By the way, the equation is singular at [tex]x_{1}=0[/tex]
     
  4. Mar 5, 2010 #3
    Here is how you make the local analysis, after eliminating x3 from there. As x1=0 is singular, we will try [tex]x_{1}=\epsilon\rightarrow 0[/tex]. The equilibrium points are
    [tex]x_{1}=\epsilon , x_{4}=-c_{5}(c_{3}+c_{6})/c_{4} , x_{2}=\gamma/\epsilon^{1/2}[/tex]

    Where [tex]\gamma=\pm \sqrt(\frac{c_{2} c_{5}}{c_{1} c_{4}}(c_{3}+c_{6}))[/tex]. After computing the Jacobian, the eigenvalue equation is given by:

    [tex]\epsilon^{3/2}\Lambda^{3}-\gamma \Lambda^{2}-[\epsilon c_{2}c_{5}/(c_{1}\gamma)+2c_{1}c_{3}\epsilon^{3/2}]\Lambda-2c_{1}c_{4}\epsilon^{3/2}=0[/tex]

    For small values of [tex]\epsilon[/tex], you get the following approximation:

    [tex]\Lambda_{1}\sim \gamma\epsilon^{-3/2}+O(\epsilon^{1/2})[/tex]

    [tex]\Lambda_{2,3}\sim \frac{c_{2}c_{5}}{2\gamma^{2}c_{1}}(-1\pm i)\epsilon^{1/2}[/tex]

    Showing instability in the direction corresponding to the first eigenvalue, for [tex]\gamma>0[/tex], and stability otherwise.

    This needs some revising...
     
    Last edited: Mar 5, 2010
  5. Mar 8, 2010 #4
    Gato, thanks a lot for taking the time to answer my post. I will work on your suggestions.
     
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