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Re: Dynamics of z = sin(z)
Posted:
Aug 17, 2007 2:20 AM


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 .)



