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Math Help - Liapunov stability exercise

  1. #1
    Senior Member
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    Liapunov stability exercise

    What can you say about the (Liapunov) stability of the zero solution of the following system?

    \left\{\begin{array}{l}x'_1=e^{-t}x_1^3-x_2\\\\\displaystyle x'_2=\frac{t^2}{1+t^2}x_1+\alpha x_2\end{array}\right\}

    ???

    First of all, what the heck is the "zero solution"?

    Second, does anyone know how to answer this? I don't even know where to start. If it were a linear system then I'd convert to a first-order vector form, but it's not, so I can't.

    Any help would be much appreciated!
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  2. #2
    Senior Member
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    When
    <br />
x_1=0 \; \; x_2=0<br />
    then
    <br />
x'_1=0 \; \; x'_2=0<br />
    so point (0,0) is stable or unstable point.
    One of the methods to discover this is to linearise this system
    and find eigenvalues. If
    <br />
| \; \lambda_i \; | < 0<br />
    it is stable.

    Please look here
    www.personal.rdg.ac.uk/~shs99vmb/notes/anc/lecture2.pdf

    page 12.
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