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MathGroup Archive 2007

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Re: help plot log[f[t]] vs a parameter using an ODE

  • To: mathgroup at smc.vnet.net
  • Subject: [mg81172] Re: help plot log[f[t]] vs a parameter using an ODE
  • From: john <johnboy98105 at yahoo.com>
  • Date: Fri, 14 Sep 2007 03:44:17 -0400 (EDT)
  • References: <fc85qt$liu$1@smc.vnet.net><fcb42d$fmp$1@smc.vnet.net>

On Sep 13, 3:40 am, Jens-Peer Kuska <ku... at informatik.uni-leipzig.de>
wrote:
> Hi,
>
> and
>
> LogPlot[y[t] /. ksol[[1]], {t, 0, 40}, PlotRange -> All]
>
> will not do that ? but in you post, it is not clear
> what do you mean by "while varying a over a range
>  > of values ( 0-40)" ..
>
> Regards
>    Jens
>
>
>
> john wrote:
> > I've solved the following equation for y[t].
>
> > eqn = y'[t] == a (y[t]/((y[t])^2 + b )) - c y[t] + d
>
> > keqn = eqn /. {a -> 10 , d -> 0.1 , c -> 1 , b -> 10 }
>
> > ksol = NDSolve[{keqn, y[0] == 0}, y, {t, 0, 200}]
>
> > Plot[y[t] /. ksol, {t, 0, 200}, PlotRange -> All];
>
> > Now I would like to plot the log of y[t] while varying a over a range
> > of values ( 0-40)
>
> > How will I accomplish this?  It seems like I would have to solve the
> > DE while varying a a little at a time then take the log of it?
>
> > At first I thought the following will do the trick, but it didn't. I
> > was trygin to make varying parameter a into a function x[t] by using
> > interpolation.
>
> > {{x -> Interpolation[Range[0, 40]]}}
>
> > then
>
> > ParametricPlot[y[t], x[t], {t, 0, 200}]
>
> > generates
>
> > ParametricPlot::pllim: Range specification x[t] is not of the form {x,
> > xmin, xmax}
>
> > I'm trying to find the value of y[t] as a parameter a incerases from
> > 0  to 40.
>
> > When the Log[y[t]] is plotted against parameter over the range of 0 to
> > 40, that should show bistability.
>
> > Except y[t] will change as you vary a. and Log[y[t]] will change along
> > with it.
>
> > Thanks for any input.- Hide quoted text -
>
> - Show quoted text -



I've been told that what I am tryign to accomplish is "continuation"

"The study of the parametric dependence of differential equations is
called continuation and the analysis of how solutions appear and
disappear as parameters vary is called bifurcation theory from the
Latin word bifurkus for branching. "

That is from xppaut website under "auto" section.  I'm bacisally
tryign to see what happens to the Log[f[t] as you vary parameter a
conbtinuosly.  using the parameters above I would get a bifurcation
diagram.

http://www.math.pitt.edu/~bard/bardware/tut/xpptut2.html#auto


i want to use Mathematica to do the similar thing.





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