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Sharkovsky orderings of higher order difference equations

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  • Given an arbitrary continuous function $f:I\rightarrow I$, $I$ some interval, the well known Sharkovsky ordering

    $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.

    Mathematics Subject Classification: 39A11, 39A12, 37F20.


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