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Topic: Dynamics of z = sin(z)
Replies: 6   Last Post: Aug 17, 2007 9:48 PM

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stephane.frion@gmail.com

Posts: 14
Registered: 8/15/07
Re: Dynamics of z = sin(z)
Posted: Aug 17, 2007 2:20 AM
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On Aug 16, 7:52 pm, Robert Israel
<isr...@math.MyUniversitysInitials.ca> wrote:
> mike3 <mike4...@yahoo.com> writes:
> > Has anyone ever examined the dynamics of the iterative map z = sin(z)
> > on the complex plane? It seems this converges for some values, wanders
> > chaotically for others, and just flat out explodes violently to
> > infinity
> > for yet more.

>
> Yes, and yes.
>
> The obvious fixed point z=0 is marginally unstable, but the iterations
> converge to 0 on the real axis, while from all points but 0 on the imaginary
> axis they
> go to infinity. There are infinitely many other fixed points, all unstable.
> I don't know if there are stable periodic points.
>
> Here's my best attempt at plotting part of the plane that's attracted to z=0
> (shown in black). The set is symmetric about x=pi/2 and y = 0 and periodic
> in the x direction with period pi.
>
> <http://www.math.ubc.ca/~israel/sindyn.gif>
> --
> Robert Israel isr...@math.MyUniversitysInitials.ca
> Department of Mathematics http://www.math.ubc.ca/~israel
> University of British Columbia Vancouver, BC, Canada


it looks familiar ... You can almost see ( graphically speaking )
similitude with the Mandelbrot set ... ( Just a thought .)




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