Eventually Periodic Solutions for Difference Equations with Periodic Coefficients and Nonlinear Control Functions

Abstract

For nonlinear difference equations of the formxn=F(n,xn−1,…,xn−m), it is usually difficult to find periodic solutions. In this paper, we consider a class of difference equations of the formxn=anxn−1+bnf(xn−k), where{an}, {bn}are periodic sequences andfis a nonlinear filtering function, and show how periodic solutions can be constructed. Several examples are also included to illustrate our results.

Bifurcation in a Discrete-Time Piecewise Constant Dynamical System

“…Then, −1 ∈ ( ) ( ) for some , ∈ N. By Table 12 if 0 ≤ ≤ , then ( 2 , 2 +1 ) ∈ − − × ( − −1) − . By the last entry in Table 9, we see that there is some such that 1) . By the last entry in Table 9, we see that there is some such that ( , +1 ) ∈ − − .…”

“…By the third entry in Table 9, we see that there is some such that , +1 ∈ − − . Suppose that A6 holds, that is, − 1 < min{ , − 1}, then by Table 13, 1) . If − ≤ − − 1, then by Table 11,…”

“…In this paper, we are concerned with the asymptotic behavior of (3) when the nonnegative parameter is treated as a bifurcation variable and allowed to vary from 0 to ∞ (cf. [1,2]). What is interesting is that a complete asymptotic and bifurcation analysis can be obtained.…”

“…The sequence {V } ∞ = is said to be -periodic if is the least period of {V } ∞ = . The sequence {V } ∞ = is said to be asymptotically periodic if there exist real numbers V (0) , V (1) , . .…”

“….} is also an -periodic sequence, we say that V is an asymptoticallyperiodic sequence tending to the limit -cycle (this term is introduced since the underlying concept is similar to that of the limit cycle in the theory of ordinary differential equations) (1) , . .…”