Im looking for examples of periodic orbits on the real line given by piecewise analytic functions $g$. More specific at most 2 pieces. Im talking about integer iterations starting at $f(0)=0$ and with $f(n+1) = g(f(n))$.

The period needs to be irrational.

For example there is a piecewise analytic function function $g$ mapping $sin(n)$ to $sin(n+1)$ and then $f(n) = sin(n)$ which has the irrational Period $2 \pi$.

$f(n) = sin(n \space \mod \space 2\pi)$.

$g_n(x) = \cos(1) x + \sin(1) sign(cos(n)) \sqrt (1 - x^2).$

Strongly related is chaos theory and fractals I assume.

I note that it is hard to distinguish a periodic orbit from an orbit that converges to a periodic orbit , or is asymptotically periodic for small imput , but diverges or converges to a nonperiodic orbit.

The sine example above is probably one of the simplest possible and it suggests the related equation

$g(x) = q^{-1}( \sin(\arcsin(q(x)) +1))$

What becomes

$q(g(x)) = \sin(\arcsin(q(x)) + 1)$

Of which I know no easy ways to solve , simplify or special cases.

Truncated fourier series with rational coëfficiënts are also intresting.

But I am very intrested in cases that do not have their " origin " from sine or cosine.

Maybe more like a logistic map or mandelbrot.

So , how to get such examples with irrational Period ? How to decide if a given $g$ gives a periodic function , and what that Period is ?

And how does this all relate to fixpoints ?

Edit (For clarity)

Let $a_n$ be a real sequence.

Assume there exists a continuous real-periodic function $f(x)$ such that

$f(n) = a_n$

And $f(x)$ has the period $t$ , where $t$ is An irrational real number.

Then we say $a_n$ is periodic and has An irrational period. Or we say $a_n$ Has An irrational periodic orbit.

  • 3
    $\begingroup$ What does it mean for a sequence to be periodic? If it's $x_{n+p} = x_n$, then $p$ must be an integer... $\endgroup$ – lhf Nov 30 '15 at 12:33
  • $\begingroup$ Yes the Op was not so Well stated. I had to run :). I edited the OP. Should be much clearer now. At everyone. @lhf $\endgroup$ – mick Nov 30 '15 at 20:06
  • $\begingroup$ Made a new edit to clarify what is meant by periodic here. Hope this helps. $\endgroup$ – mick Dec 2 '15 at 21:16
  • $\begingroup$ $g_n(x) = \cos(1) x + \sin(1) sign(cos(n)) \sqrt (1 - x^2).$ Added as example for f = sine. $\endgroup$ – mick Dec 4 '15 at 12:27

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