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From: junoexpress on 14 Apr 2010 11:01
Hi,

I am trying to understand the behavior of one dimensional dynamical
system which has the peculiar property that the only periodic orbits
is possesses are two cycles. I am wondering if anyone knows of any
general resuts about these type of systems (for example, 1D systems
having period 3 have periods of all orders, are chaotic, etc).


The specific system I am looking at is a one-dimensional autonomous
dynamical system which has the form:
y = x + f(x)
where x is a real number, f maps R1 to R1 and f has a period of pi and
is an odd function.

The points +/- pi/2 are repelling fixed points of the dynamical system
and in the open interval (-pi/2,pi/2), the dynamical system has a
single attractive fixed point at x=0 with a basin of attraction whose
border is the points of the two cycle, which I'll denote as +/- x2.
Outside the two cycle, ( i.e. in the intervals [-pi/2,-x2) and (x2,+pi/
2] ), there are a known number of repelling fixed points.

Thanks for any insights or experience anyone can share with me,

Matt

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