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Math Help - Dynamical Systems

  1. #1
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    Dynamical Systems

    I have a continuous dynamical system given by

    dx/dt = x(y-z),
    dy/dt = y(z-x)
    dz/dt = z(x-y)

    and have to show that for any initial condition
    x0 , y0 , z0 such that
    x0 +y0+z0= k for any constant k
    the solution of the above equations will always lie in the plane
    x+y+z=k
    originally i tried to look at the solution, i got
    (x(t),y(t),z(t) )= (e^(y-z)t , e^(z-x)t, e^(x-y)t)
    but couldn't see anything to do from then onwards.
    i think there's a nice and simple answer i'm just not seeing as it's only two percent of the sheet so i'm guessing sides of A4 covered in calculations are out.
    cheers.
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  2. #2
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    hint: statement: dx/dt + dy/dt + dz/dt = 0
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  3. #3
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    Quote Originally Posted by Skalkaz View Post
    hint: statement: dx/dt + dy/dt + dz/dt = 0
    Another hint

    \frac{\dot{x}}{x} + \frac{\dot{y}}{y} + \frac{\dot{z}}{z} = 0

    where

     \dot{x} = \frac{dx}{dt}.
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  4. #4
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    cheers - i feel a bit foolish for not seeing that myself though.
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