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MathGroup Archive 1998

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Re: ODEs and phase portraits

  • To: mathgroup at smc.vnet.net
  • Subject: [mg14743] Re: ODEs and phase portraits
  • From: "Kevin J. McCann" <kevinmccann at Home.com>
  • Date: Thu, 12 Nov 1998 02:17:42 -0500
  • Organization: @Home Network
  • References: <728kp1$ehc@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

Try this one:

Remove[de]
de[epsilon_, x0_, v0_, T_] :=
  Module[{eqn, x, y, t},
   eqn = {Derivative[1][Derivative[1][x]][t] +
        epsilon*(x[t]^2 - 1)*
         Derivative[1][x][t] + x[t] == 0,
      x[0] == x0, Derivative[1][x][0] == v0};
    y = x /. NDSolve[eqn, x, {t, 0, T}][[1]];
    ParametricPlot[{y[t], Derivative[1][y][t]},
     {t, 0, T}, PlotStyle -> RGBColor[1, 0, 0],
     Epilog ->
      {RGBColor[0, 0, 1], AbsolutePointSize[5],
       Point[{y[0], Derivative[1][y][0]}]}]; ]

Sample usage:

de[-.01,1,0,100]

If you copy the above into a NB, highlight the bracket, and type
Ctl-Shif-N, it will look a lot better.  Anyway, try it.

Kevin

phantomlord at my-dejanews.com wrote in message <728kp1$ehc at smc.vnet.net>...
>I am trying to write a function such that I have the following ODE:
>x''(t)+epsilon*(x(t)^2-1)*x'(t)+x(t)==0  [1] where epsilon is to be one
>of the parameters in the function. I want to beable to draw out the
>phase portraits for the equation for different values of epsilon.
>
>To deduce the phase portraits in a mathematical procedure I multiply
>equation[1] by dx(t)/dt and integrate w.r.t. t.  To do this in
>Mathematica is trivial, so I'll skip past this - it is the next step
>that I would like assistance with:
>
>Q:Is there a way that I can decompose the result of the above (I'll call
>it [2]) into the corresponding pair (below) of ODEs to deduce the
>trajectory of the phase portrait?
>
>x'(t) = y(t)
>y'(t) = f(x,y)
>some f(x,y) function of x and y.
>
>Perhaps there is a built in function that will allow me to do this? If
>not do, can anybody suggest another way to do this?
>
>Also after plotting the trajectory is there any way to determine the
>direction of it in Mathematica?
>
>thanks for you time.
>Paul
>
>-----------== Posted via Deja News, The Discussion Network ==----------
>http://www.dejanews.com/       Search, Read, Discuss, or Start Your Own
>
>



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