On 20.10.2012 23:20, Jaime Villate wrote:
> On 10/20/2012 08:41 PM, Adam wrote:
>> On 20.10.2012 21:00, Jaime Villate wrote:
>>> On 10/20/2012 10:55 AM, Adam wrote:
>>>> Here are good images
>>>> http://www.clarku.edu/~djoyce/complex/powers.html
>>>> http://facstaff.unca.edu/mcmcclur/Mathematica/ComplexDynamics/HTMLLinks/ComplexQuadAndCubicIteration_1.html
>>>>
>>>>
>>>>
>>>> The problem is that vector draws a straight line, but I need a curve (
>>>> complex square is a rotation ).
>>>>
>>>> I'm working to get better result.
>>>> Any help is welcome.
>>> Hi Adam,
>>> If your objective is just to draw the first figure in that reference,
>>> with the arrows, it can be done using plotdf (see the attached figure).
>>> If your objective is something else, please remind us.
>>> Cheers,
>>> Jaime
>> Wow, that's it !
>>
>> How did you do it ?
> I used a dynamical system that I know it has a circular repulsive limit
> cycle:
> dx/dt = -y+x*(x^2+y^2-1)
> dy/dt = x+y*(x^2+y^2-1)
>
> (%i1) dx: -y+x*(x^2+y^2-1)$
> (%i2) dy: x+y*(x^2+y^2-1)$
> (%i3) plotdf([dx,dy],[x,-2,2],[y,-2,2],[vectors,""],[trajectory_at,0,1]);
>
> That will show you only the limit cycle (currently plotdf accepts only
> one "trajectory_at"). I then did the blue and green trajectories by hand:
> * Click on the tools icon in plotdf's menu
> * Change the fieldlines color to blue (or gray or whatever)
> * Change nsteps to 50.
> * Click on OK and then click on some points inside the circle.
> I may then change the fieldlines color again and click on some points
> outside of the circle. When you are happy with the plot, click on the
> disk-drive icon and save your plot in Postscript format.
>
> Regards,
> Jaime (who wishes he had some free time to translate his books
> http://def.fe.up.pt into English)
So you are finding continous dynamical system which has the same
dynamics an can be used as model of discrete dynamical system.
For parabolic system with 2 petals ( eggbeater dynamics)
http://en.wikibooks.org/wiki/Fractals/Iterations_in_the_complex_plane/parabolic
one can use :
a 2D vector field with 2 centers ( second-order degenerate points )
http://commons.wikimedia.org/wiki/File:VFPt_ringcurrent.svg
Is there a general method of finding such model for given c ?
Regards
Adam