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Birkhoff cones, symbolic dynamics and spectrum of transfer operators
Sharkovsky orderings of higher order difference equations
| 1. | Institute of Mathematics, University of Witten/Herdecke, 58448 Witten, Germany, Germany |
$3\vartriangleright 5\rhd \cdots \rhd 2\cdot 3\rhd 2\cdot 5 \triangleright \cdots \rhd 2^2\cdot 3\rhd 2^2\cdot 5\rhd \cdots \rhd \cdots \rhd 2^2 \rhd 2\rhd 1 $
tells that if the difference equation
$x_n=f(x_{n-1}),\quad n=1,2,\ldots $ (1)
has a periodic solution with period $p$ then it has also periodic solutions of period $p'$ for all $p'$ to the right of $p$ in the Sharkovsky ordering. Here we generalize this result to the difference equation of $k-$th order
$x_n=f(x_{n-k}),\quad n=1,2,\ldots $ (2)
for arbitrary $k\in \mathbb N$. It turns out that for each $k$ there is an individual ordering. In these orderings the prime number decomposition of $k$ plays an important role. In particular each number in the set
$S_k(p')=${ $l\cdot p'$ where $l$ divides $k$ and the pair $(k/l,p')$ is coprime}
is a period of (2) if $p\trianglerighteq p'$ and $p$ is a period
of (1). Thus, for different values of $k$ there are generally
different bifurcation schemes.
We also prove theorems about the number of periodic solutions and of attractive cycles of $x_n=f(x_{n-k})$.
We suggest that the $k-$th order difference equation (2) may give
important insight to the behavior of delay--differential equations
of the type $\varepsilon \dot
x(t)+x(t)=f(x(t-1))$ by considering the parameter
$\varepsilon \rightarrow 0$ in a singular perturbation problem.
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