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Math Help - Problems with Continuous Dynamical Systems

  1. #1
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    Problems with Continuous Dynamical Systems

    Hi there, I'm new to this but I was wondering if you could help me with my understanding of Dynamical Systems.

    I have a continuous dynamical system given by the equations:

    dx1/dt = x1(x2-x3)
    dx2/dt = x2(x3-x1)
    dx3/dt = x3(x1-x3)

    Which I need to analyse. There are a few problems I have but the main ones are centred around orbits.

    My question is why can an orbit starting in the region x1 >= 0, x2 >= 0 , x3 >= 0 never cross one of the lines x_i = 0?

    Any help on the matter would be greatly appreciated I'm drawing a complete blank with this and could really do with some help.
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  2. #2
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    Proceed as you would with a differential equation: For example in your first equation isolating x_1 we get  \ln(x_1)=\int (x_2-x_3)dt+C_1. Applying this to the other two, adding them up, condensing the constants and using that the logarithm opens products we arrive at \ln (x_1x_2x_3)=\int (x_2-x_3)dt+C or equivalently x_1x_2x_3=e^{C\int (x_2-x_3)dt} and since the exponential is always positive, none of the x_i can be zero or change sign because of continuity.

    PS. I'm not too familiar with dynamical systems so check everything carefully.
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  3. #3
    MHF Contributor FernandoRevilla's Avatar
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    Quote Originally Posted by wattsup View Post
    dx1/dt = x1(x2-x3)
    dx2/dt = x2(x3-x1)
    dx3/dt = x3(x1-x3)
    My question is why can an orbit starting in the region x1 >= 0, x2 >= 0 , x3 >= 0 never cross one of the lines x_i = 0?
    An alternative. For the given system x'=v(x) we have:

    \begin{Bmatrix}v(\alpha,0,0)=(0,0,0)\\ v(0,\beta,0)=(0,0,0) \\ v(0,0,\gamma)=(0,0,-\gamma^2)\end{matrix}

    This means that all points of the x_1 and x_2 axes are equilibrium points and the x_3 axis decomposes into three orbits. You can conclude.

    Fernando Revilla
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