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What we need to find out the periods of a periodic difference equation. (English) Zbl 1180.39021
The author studies a set of periods of a \(p\)-periodic difference equation satisfying a generic condition.
Consider a set \(X\), a sequence \((f_n)_{n\in\mathbb{Z}^+}\) of functions \(f_n: X\to X\), \(n\in\mathbb{Z}^+\), a sequence \((x_n)_{n\in\mathbb{Z}^+}\) of elements of \(X\) and a difference equation
\[ (\forall n\in\mathbb{Z}^+)\;(x_{n+1}= f_n(x_n)).\tag{1} \] A sequence \((s_n)_{n\in\mathbb{Z}^+}\) is called a solution of (1) if
\[ (\forall n\in\mathbb{Z}^+)\;(s_{n+1}= f_n(s_n)). \] If \(q\in\mathbb{Z}^+\), a solution \((s_n)_{n\in\mathbb{Z}^+}\) of (1) is called \(q\)-periodic if \(q\) is the smallest positive integer satisfying
\[ (\forall i\in\mathbb{Z}^+)\;(s_{i+q}= s_i). \] A positive integer \(q\) is called a period of (1) if there exists a \(q\)-periodic solution of (1). Denote by \({\mathcal P}\) the set of periods of equation (1). If \(p\in{\mathcal P}\), the equation (1) is called \(p\)-periodic if \(p\) is the smallest positive integer satisfying
\[ (\forall n\in\mathbb{Z}^+)\;(f_{n+p}= f_n). \] Denote by \(Q= \{n\in{\mathcal P}; p\nmid n\}\) the set of periods of \(p\)-periodic equation (1) which are not divisible by \(p\).
If (1) is a \(p\)-periodic difference equation suppose
\[ (\forall(i,j)\in\mathbb{Z}^+\times \mathbb{Z}^+)(i\neq j\pmod p)(\text{card}\{x\i X; f_i(x)= f_j(x)\}<+\infty).\tag{2} \] The author studies the set \(Q\) of the \(p\)-periodic difference equation (1) satisfying the generic condition (2).
Define \(g_0= \text{id}_X\) and
\((\forall n\in \mathbb{Z}^+)\) \((g_n= f_{n-1}\circ g_{n-1})\),
\((\forall m\in\mathbb{Z}^+)\) \((\forall(i,j)\in \mathbb{Z}^+\times \mathbb{Z}^+)(j= i\pmod m)(X_m(i)=\{x\in X; f_i(x)= f_j(x)\})\),
\((\forall m\in\mathbb{Z}^+)\) \((X_m= \bigcap^{m-1}_{i= 0} g^{-1}_i(X_m(i)))\)
and the functions \(h_m: X_m\to X\), \(m\in\mathbb{Z}^+\), the restrictions of \(g_m\) to \(X_m\), \(m\in\mathbb{Z}^+\). Let \(d\) be a divisor of \(p\) satisfying \(1\leq d< p\).
Define the matrix \(M_d\) putting \(M_d= 0\) if \(X_d=\emptyset\) and
\[ M_d(i,j)= \begin{cases} 0, &h_d(x_j)\neq x_i,\\ 1, &h_d(x_j)= x_i\end{cases} \] for \((i,j)\in \mathbb{Z}^+\times \mathbb{Z}^+\), if \(X_d= \{x_1,\dots, x_k\}\).
If \(I\) denotes the identity matrix define the polynomial \(P_d(z)= \text{det}(I- zM_d)\). Define
\[ {\mathcal R}_d= \{q\in \mathbb{Z}^+;\;\text{gcd}(p, q)= d\wedge(1- z^{qd^{-1}})\mid P_d(z)\}, \]
\[ {\mathcal R}= \bigcup{\mathcal R}_d, \]
\[ (1\leq i\leq k)\,(d_i= \text{gcd}(p, r_i)\wedge b_i= \text{tr\,}M_{d_i}^{r_i d^{-1}_i}) \] if \(\{r_1,\dots, r_k\}\subset{\mathcal R}\) and consider the linear system
\[ A\begin{pmatrix} \alpha_1\\ \vdots\\ \alpha_k\end{pmatrix}= \begin{pmatrix} b_1\\ \vdots\\ b_k\end{pmatrix},\tag{3} \] where \(A\) is a \(k\times k\) matrix defined by
\[ A(i,j)= \begin{cases} 1,\quad & r_j|r_i,\\ 0,\quad & \text{otherwise},\end{cases}\qquad (i,j)\in \mathbb{Z}^+\times \mathbb{Z}^+. \] The author proves the following results:
the set \(Q\) is finite,
if \(q\in Q\) and \(d= \text{gcd}(p,q)< p\) then the polynomial \(1- z^{qd^{-1}}\) divides \(P_d(z)\),
if \((\alpha_1,\dots, \alpha_k)\) is a unique solution of (3) then the number of \(r_i\)-periodic solutions of (1) is \(\alpha_i\) and \(Q= \{r_i\in{\mathcal R}\); \((1\leq i\leq k)\) \((\alpha_i\neq 0)\}\).

39A23 Periodic solutions of difference equations
Full Text: DOI
[1] DOI: 10.1016/j.jmaa.2005.04.059 · Zbl 1125.39001 · doi:10.1016/j.jmaa.2005.04.059
[2] J.F. Alves, Odd periods of 2-periodic nonautonomous dynamical systems, to appear in Grazer Math. Ber
[3] DOI: 10.2307/1970384 · Zbl 0127.13401 · doi:10.2307/1970384
[4] DOI: 10.1080/10236190600772515 · Zbl 1099.37028 · doi:10.1080/10236190600772515
[5] DOI: 10.1016/j.jde.2003.10.024 · Zbl 1067.39003 · doi:10.1016/j.jde.2003.10.024
[6] Robinson C., Dynamical Systems: Stability, Symbolic Dynamics, and Chaos (1995) · Zbl 0853.58001
[7] Sharkovsky A.N., Ukrain. Mat. Zh. 16 pp 61– (1964)
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