- Research Article
- Open Access
Complete Asymptotic Analysis of a Nonlinear Recurrence Relation with Threshold Control
© Qi Ge et al. 2010
Received: 23 November 2009
Accepted: 10 January 2010
Published: 3 February 2010
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.
Let In , Zhu and Huang discussed the periodic solutions of the following difference equation:
In this paper, we consider the following delay difference equation:
where and . Besides the obvious and complementary differences between (1.1) and our equation, a good reason for studying (1.3) is that the study of its behavior is preparatory to better understanding of more general (neural) network models. Another one is that there are only limited materials on basic asymptotic behavior of discrete time dynamical systems with piecewise smooth nonlinearities! (Besides , see [2–6]. In particular, in , Chen considers the equation
in which is a constant which acts as a threshold. In , convergence and periodicity of solutions of a discrete time network model of two neurons with Heaviside type nonlinearity are considered, while "polymodal" discrete systems in  are discussed in general settings.) Therefore, a complete asymptotic analysis of our equation is essential to further development of polymodal discrete time dynamical systems.
We need to be more precise about the statements to be made later. To this end, we first note that given we may compute from (1.3) the numbers in a unique manner. The corresponding sequence is called the solution of (1.3) determined by the initial vector For better description of latter results, we consider initial vectors in different regions in the plane. In particular, we set
Recall also that a positive integer is a period of the sequence if for all and that is the least or prime period of if is the least among all periods of The sequence is said to be -periodic if is the least period of The sequence is said to be asymptotically periodic if there exist real numbers where is a positive integer, such that
In case is an -periodic sequence, we say that is an asymptotically -periodic 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.) In particular, an asymptotically -periodic sequence is a convergent sequence and conversely.
Note that (1.3) is equivalent to the following two-dimensional autonomous dynamical system:
by means of the identification for Therefore our subsequent results can be interpreted in terms of the dynamics of plane vector sequences defined by (1.10). For the sake of simplicity, such interpretations will be left in the concluding section of this paper.
To obtain complete asymptotic behavior of (1.3), we need to derive results for solutions of (1.3) determined by vectors in the entire plane. The following easy result can help us to concentrate on solutions determined by vectors in
which has exactly two solutions , when and has the unique solution when However, since is a discontinuous function, the standard theories that employ continuous arguments cannot be applied to our equilibrium solutions or to yield a set of complete asymptotic criteria. Fortunately, we may resort to elementary arguments as to be seen below.
To this end, we first note that our equation is autonomous (time invariant), and hence if is a solution of (1.3), then for any the sequence defined by for is also a solution. For the sake of convenience, we need to let
We also let
We first show the following.
The proof is complete.
We first show that following result.
As in Case 6 of Lemma 2.1, if then by (1.3), we have We may thus apply the conclusion of Case 5 to deduce our assertion. If where is an arbitrary nonnegative integer, then by (1.3), we have We may thus use induction to conclude our assertion.
As in Case 8 of Lemma 2.1, if then by (1.3), we have We may now apply the assertion in Case 5 to conclude our proof. If where is an arbitrary nonnegative integer, then by (1.3), we have We may thus complete our proof by induction.
The proof is the same as the discussions in Cases 5 through Case 8 in the proof of Lemma 2.1, and hence is skipped.
The proof is complete.
5. Concluding Remarks
The results in the previous sections can be stated in terms of the two-dimensional dynamical system (1.10). Indeed, a solution of (1.10) is a vector sequence of the form that renders (1.10) into an identity for each It is uniquely determined by
Since we have obtained a complete set of asymptotic criteria, we may deduce (bifurcation) results such as the following.
If then all solutions originated from the positive orthant approach the limit -cycle if then all solutions originated from the positive orthant tend to if then all solutions originated from the positive orthant tend to if and approach the limit cycle otherwise.
Roughly the above statements show that when the threshold parameter is a relatively small positive parameter, all solutions from the positive orthant tend to a limit -cycle; when it reaches the critical value some of these solutions (those from ) switch away and tend to a limit -cycle, and when drifts beyond the critical value, all solutions tend to the limit -cycle. Such an observation seems to appear in many natural processes and hence our model may be used to explain such phenomena. It is also expected that when a group of neural units interact with each other in a network where each unit is governed by evolutionary laws of the form (1.3), complex but manageable analytical results can be obtained. These will be left to other studies in the future.
This project was supported by the National Natural Science Foundation of China (10661011).
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