Abstract
We consider a three-term nonlinear recurrence relation involving a nonlinear filtering function with a positive threshold . We work out a complete asymptotic analysis for all solutions of this equation when the threshold varies from to . It is found that all solutions either tends to 0, a limit 1-cycle, or a limit 2-cycle, depending on whether the parameter is smaller than, equal to, or greater than a critical value. It is hoped that techniques in this paper may be useful in explaining natural bifurcation phenomena and in the investigation of neural networks in which each neural unit is inherently governed by our nonlinear relation.
Highlights
In 1, Zhu and Huang discussed the periodic solutions of the following difference equation: xn axn−1 1 − a fλ xn−k, n ∈ N, 1.1 where a ∈ 0, 1, k is a positive integer, and f : R → R is a nonlinear signal filtering function of the form
We first note that our equation is autonomous time invariant, and if {xn}∞n −2 is a solution of 1.3, for any k ∈ N, the sequence {yn}∞n −2, defined by yn xn k for n −2, −1, 0, . . . , is a solution
X2k 1 ak 1x−1 > λ, which implies lim x2k 0 lim x2k 1. This is contrary to the fact that xn ∈ λ, ∞ for n ∈ N
Summary
⎨1, x ∈ 0, λ , fλ x ⎩0, x ∈ −∞, 0 ∪ λ, ∞ , 1.2 in which the positive number λ can be regarded as a threshold parameter
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