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Math Help - Stability and Lyapunov functions

  1. #1
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    Stability and Lyapunov functions

    I've been trying to work out some problems but I haven't been able to... I don't quite understand the concept of stability and Lyapunov functions. This is my problem:

    Consider the equation y''+u(y)y'+h(y)=0 where u and h are continuously differentiable.

    a) Using y_1=y and y_2=y', rewrite the second order equation as a first order system and find conditions on u and h to ensure that the origin is an isolated equilibrium point.

    What I found is...
    y_1'=y_2
    y_2'=-u(y_1)y_2-h(y_1)

    y_2=0
    -u(y_1)y_2-h(y_1)=0
    h(y_1)=0
    u(y_1)= anything

    b) Using V(y_1,y_2)=\int_0^{y_1} h(s)ds + \frac{1}{2}y_2^2 as a possible Lyapunov function, find conditions on u and h to ensure that the origin is asymptotically stable.

    To be honest, I don't even know how to start with this one. The book my course is using is Brauer's "An Introduction The Qualitative Theory of Ordinary Differential Equations" and it doesn't have many examples. Can anyone recommend a friendlier, more example oriented book? Thanks!
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  2. #2
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    Try James Liu's "A first course in the qualitative theory of differential equations" and in particular chapters 5 and 9.
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