A characterization of k-cycles

作者：Canovas J S^*; Linero A; Soler G

来源：Nonlinear Analysis-Theory Methods & Applications, 2010, 72(1): 364-372.

DOI：10.1016/j.na.2009.06.070

• Difference equations
• Global periodicity
• Conjugacy

We study global periodicity for the difference equation of order l given by x(n+l) = f(x(n+l-1), x(n+l-2),...,X(n)), where f : (0, +infinity)(l) -> (0, +infinity) is a continuous map, l is an element of Z(+). Our main results are the following. We prove that if any solution of the equation is periodic, then there is a minimal k is an element of N such that the period of any solution divides k (and therefore f is called a k-cycle). In addition, if l = 2, then for any k > 2 there are, up to conjugacy, only a k-cycle. Finally, if l > 2 andf gives a (l + 1)-cycle, thenf is topologically conjugate to:
x(n+l) = 1/x(n),x(n+1),...,x(n+l-1), if l is even.
The previous equation or x(n+l) = Pi((l+1)/2)(j=1)x(n+2j-2)/Pi((l-1)/2)(j=1)x(n+2j-1,) if l is odd.
Our results solve some open questions from [J. S. Canovas, A. Linero and G. Soler, On global periodicity of difference equations, Taiwanese J. Math. (in press)] and [M. R. S. Kulenovic and G. Ladas, Dynamics of Second Order Rational Difference Equations. With Open Problems and Conjectures, Chapman & Hall/CRC, Boca Raton, FL, 2002].

• 出版日期2010-1-1