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Math Help - Lyapunov function-Differential equation prob

  1. #1
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    Lyapunov function-Differential equation prob

    For dx/dt = y^3 - 2*x^3

    dy/dt = -3*x - y^3

    Show that the origin is nonlinearly stable by nding the appropriate Lyapunov function.
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  2. #2
    Super Member
    Joined
    Aug 2008
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    Hey, I don't have it but let me try to rough it in ok:

    So we have:

    x'=y^3-2x^3

    y'=-3x-y^3

    In these, sometimes we can find a Lyapunov function of the form ax^2+by^2 for suitable choices of a and b such that L(p_0)=0 at the critical point p_0, L(x)>0 if x\neq p_0 in some region about p_0 and F\cdot\nabla L \leq 0 in that same region. If so, then the point p_0 is stable. Well the critical point is the origin so let a>0 and b>0 and then:

    F\cdot\nabla L =(y^3-2x^3,-3x-y^3)\cdot(2ax,2bx)=2axy^3-4ax^4-6bxy-2by^4

    =2axy^3-6bxy-(4ax^4+2by^4)

    The last term is always negative. However, I can't show that for suitable choices of a and b, the entire expression is less than or equal zero in some deleted neighborhood of the origin.
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