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MathGroup Archive 1998

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Re: ODEs and phase portraits

  • To: mathgroup at smc.vnet.net
  • Subject: [mg14728] Re: ODEs and phase portraits
  • From: Selwyn Hollis <shollis at peachnet.campus.mci.net>
  • Date: Wed, 11 Nov 1998 17:53:34 -0500
  • Organization: fair
  • References: <728kp1$ehc@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

Dear 'lord,

This should help:

    f[epsilon_][x_,y_]:= -epsilon(x^2-1)y-x

    system[epsilon_,{x0_,y0_}]=
        {x'[t]==y[t], y'[t]==f[epsilon][x[t],y[t]],  x[0]==x0, y[0]==y0}

    <<Graphics`PlotField`

    field=PlotVectorField[{y,f[1][x,y]},{x,-3,3},{y,-3,3},
        ScaleFunction->(1&),ColorFunction->(Hue[0]&)];

    soln={x[t],y[t]}/.NDSolve[system[1,{0,.1}],{x,y},{t,0,10}]

    curve=ParametricPlot[Evaluate[soln],{t,0,10},
        PlotStyle->Thickness[.008]];

    Show[field,curve];

--sh

phantomlord at my-dejanews.com wrote:

> 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

--
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Dr. Selwyn Hollis
Associate Professor of Mathematics
Armstrong Atlantic State University
Savannah, GA 31419 USA
<http://www.math.armstrong.edu/faculty/hollis/>
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~



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