Re: Problem in plotting Bifurcation Diagram (ListPlot with

• To: mathgroup at smc.vnet.net
• Subject: [mg102504] Re: Problem in plotting Bifurcation Diagram (ListPlot with
• From: juan flores <juanfie at gmail.com>
• Date: Tue, 11 Aug 2009 04:06:42 -0400 (EDT)
• References: <200907290910.FAA19700@smc.vnet.net> <h4rpgo\$lbf\$1@smc.vnet.net>

```On Aug 9, 5:15 pm, A <creativesolution... at gmail.com> wrote:
> On Aug 1, 4:59 pm, juan flores <juan... at gmail.com> wrote:
>
>
>
> > On Jul 31, 4:52 am, AH <creativesolution... at gmail.com> wrote:
>
> > > On Jul 30, 6:35 pm, DrMajorBob <btre... at austin.rr.com> wrote:
>
> > > > MakeMapFunction is undefined, so of course the code doesn't work...
> > > > although I get a different error.
>
> > > > Also note, your semicolon after ListPlot will prevent display of th=
e =
> plot=
> > > .
>
> > > > Bobby
>
> > > > On Wed, 29 Jul 2009 04:10:48 -0500, AH <creativesolution... at gmail.c=
om=
> > =
> > > > wrote:
>
> > > > > Hi
> > > > > I have following piece of code:
> > > > > -----------------------------------------------------------------=
--=
> ----=
>
> > > ---------------------------------
>
> > > > > In[1]:=BifurcationDiagram[f_, {r_, rmin_, rmax_, rstep_}, {x_, =
x0=
> _},
> > > > > start_,
> > > > >          combine_] :=
> > > > >    Module[{R, temp, MapFunction},
> > > > >            R = Table[r, {r, rmin, rmax, rstep}]; (*=
T=
> he r=
> > > ange
> > > > >     of values of the parameter *)
> > > > >            MapFunction = MakeMapFunction[{r, x}, f]=
;(=
> *
> > > > >         Construct the iterating function *)
> > > > >            temp = Nest[MapFunction[R, #] &, x0 + 0.=
R,=
>  sta=
> > > rt + 1];(* Starting
> > > > > iterates \
> > > > > *)
> > > > >            temp = NestList[MapFunction[R, #] &, tem=
p,=
>  com=
> > > bine - 1];(*
> > > > > Subsequent \
> > > > > iterates *)
> > > > >            temp = Map[ Union, Transpose[
> > > > >           temp] ]; (* Remove duplicate values from cycles *)
> > > > >            Flatten[ MapThread[Thread[{#1, #2}] &, {R, temp}], 1]
> > > > >    ];
>
> > > > > In[2]:=ListPlot[BifurcationDiagram[(1 - r) x + (r(2858.16)/(x -=
5=
> 00)
> > > > > ^0.82) - 30000r, \
> > > > > {r, 0.1, .2, .0001}, {x, 600}, 10000, 100], PlotStyle ->
> > > > >         AbsolutePointSize[0.0001]];
> > > > > ------------------------------------------------------
> > > > > The following errors are produced:
> > > > > Graphics:: gptn : Coordiantes -30000.5+0.0169522 i {0.1,
> > > > > -30000.5+0.0169522 i }....is not a floating pont. Is there any
> > > > > possible solution to this problem ?
> > > > > Best regards.
> > > > > --------------------------------------------------------
>
> > > > --
> > > > DrMajor... at bigfoot.com
>
> > > You can ignore MakeMapFunction.
> > > In my case the only error is what that has been reported.
> > > May be you can list the errors that you have got.
>
> > > The main problem is that how to display deal with complex numbers whi=
le
> > > plotting bifurcation diagrams.....Any idea will be very much
> > > appreciated.
>
> > > Also, the following link tells that ListPlot cannot be used for
> > > complex numbers.http://www.stephenwolfram.com/publications/articles/c=
om=
> puting/92-desi...
> > > ---------------------------------------------------------------------=
--=
> ----=
> > > -------------------------------------------------------------
>
> > I find this post rather incomplete.  Eventhough, I know what you are
>
> > Complex roots of your ODE(s) correspond to double-period (or higher)
> > bifurcation points.  Read Strogatz.  Now, ploting real roots, you g=
et
> > the usual bifurcation diagrams, just like the ones XppAuto produces.
> > I recently plotted these bifurcation diagrams for one and two-
> > parameter systems.  You'll find the code at the end of the message
> > (not showing the results.  Evaluted and running in M 7.0.0 for Mac).
>
> > I hope it helps.
>
> > Cheers,
>
> > Juan Flores
> > -----
> > one-parameter ODE
> > -----
> > f[x_, r_] := r x + x^3 - x^5
>
> > Clear[NotComplexQ];
> > NotComplexQ[c_Complex] := False;
> > NotComplexQ[c_] := True
>
> > CartProd[l_] := Outer[List, l[[1]], l[[2]]]
>
> > ArreglaLista[l_] := Select[Map[(x /. #) &, Flatten[l]], NotComplexQ]
>
> > Points = Flatten[
> >   Map[CartProd,
> >    Table[{{r}, ArreglaLista[NSolve[f[x, r] == 0, x]]}, {r, -1, =
2,
> >      0.05}]], 2]
>
> > ListPlot[Points]
> > -----
> > two-parameter ODE
>
> > Since the bifurcation diagram is not a funciont, ListPlot3D does not
> > work, so I produce a little sphere for each point, and plot those.
> > -----
> > cubic ODE
> > f[x_, r_, h_] := h + r x - x^3
>
> > Clear[NotComplexQ];
> > NotComplexQ[c_Complex] := False;
> > NotComplexQ[c_] := True
>
> > CartProd[l_] := Outer[List, l[[1]], l[[2]]]
>
> > ArreglaLista[l_] := Select[Map[(x /. #) &, Flatten[l]], NotComplexQ]
>
> > Arregla2[{l1_, l2_}] := Map[Join[l1, {#}] &, l2]
>
> > kk = Flatten[
> >   Map[Arregla2,
> >    Map[{#[[1]], ArreglaLista[Flatten[#[[2]]]]} &,
> >     Flatten[Table[{{r, h}, NSolve[f[x, r, h] == 0, x]}, {r, -3,=
3=
> ,
> >        0.1}, {h, -3, 3, 0.1}], 1]]], 1]
>
> > makeSphere[l_] := Sphere[l, 0.03]
>
> > spheres = Map[makeSphere, kk]
>
> > Graphics3D[spheres]
>
> Thanks a lot Juan for the code.
> I have tried and both of above scripts worked fine.
> Then I have tried to replace the f[x_, r_] to my case, but no
> bifurcation diagram
> was produced. I am pasting the modification (copied as plain text):
>
> f[x_,r_]:=(1-r)x+(r (2858.16)/(x-500)^0.82)-30000r
>
> Clear[NotComplexQ];
> NotComplexQ[c_Complex]:=False;
> NotComplexQ[c_]:=True
>
> CartProd[l_]:=Outer[List,l[[1]],l[[2]]]
>
> ArreglaLista[l_]:=Select[Map[(x/.#)&,Flatten[l]],NotComplexQ]
>
> Points=Flatten[Map[CartProd,Table[{{r},ArreglaLista[NSolve[f[x,r]
> ==0,x]]},{r,0.1, 0.2, 0.0001}]],600]
>
> ListPlot[Points]
>
> I will appreciate your help for fixing this problem ?
> Best regards.
> -------------------------------------------------------------------------=
--=
> ----------

Ok, problem is:

your f has a singularity at x=500.  It becomes infinity.  f does not
have any real roots for -infinity <= x < 500.  f>0 for x>500.

So no fixed points for your function, therefore, the bifurcation
diagram is empty.

What is your field of application?

Regards,

Juan Flores

```

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