Complete Asymptotic and Bifurcation Analysis for a Difference Equation with Piecewise Constant Control
© Chengmin Hou et al. 2010
Received: 2 June 2010
Accepted: 14 November 2010
Published: 1 December 2010
We consider a difference equation involving three parameters and a piecewise constant control function with an additional positive threshold . Treating the threshold as a bifurcation parameter that varies between 0 and , we work out a complete asymptotic and bifurcation analysis. Among other things, we show that all solutions either tend to a limit 1-cycle or to a limit 2-cycle and, we find the exact regions of attraction for these cycles depending on the size of the threshold. In particular, we show that when the threshold is either small or large, there is only one corresponding limit 1-cycle which is globally attractive. It is hoped that the results obtained here will be useful in understanding interacting network models involving piecewise constant control functions.
Since can be an arbitrary small positive number, (1.1) may be regarded as a limiting case of (1.3). Therefore, it would appear that the qualitative behavior of (1.3) will "degenerate into" that of (1.1) when tends to 0. However, it is our intention to derive a complete asymptotic and bifurcation analysis for our new equation and show that, among other things, our expectation is not quite true and perhaps such discrepancy is due to the nonlinear nature of our model at hand.
Indeed, we are dealing with a dynamical system with piecewise constant nonlinearlities (see e.g., [2–6]), and the usual linear and continuity arguments cannot be applied to our (1.3). Fortunately, we are able to achieve our goal by means of completely elementary considerations.
To this end, we first recall a few concepts. Note that given , we may compute from (1.3) the numbers in a unique manner. The corresponding sequence is called the solution of (1.1) determined by or originated from the initial vector .
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 1-periodic sequence is a convergent sequence and conversely.
The proof is complete.
Furthermore, in each case, we find the exact regions of attraction of the limit cycles. Then we describe our results in terms of our phase plane model (1.21) and compare them with what we have obtained for the phase plane model of (1.1).
We remark that since we need to find the exact regions of attraction, we need to consider initial vectors belonging to (up to 9) different parts of the plane. Therefore the following derivations will seem to be repetitive. Fortunately, the principles behind our derivations are quite similar, and therefore some of the repetitive arguments can be simplified.
Repeating the argument we either find such that , or one has that the subsequence lies in whereas . This would mean that the subsequence satisfies (1.6) or (1.7) for , and hence , a contradiction. The proof is complete.
Therefore, in the last four cases, we may apply the first two cases to conclude our proof. The proof is complete.
Therefore, in the last three cases, we may apply the conclusions in the first four cases to conclude our proof. The proof is complete.
Therefore, in the fourth, sixth, seventh, and the eigth cases, we may use the conclusions in the other cases to conclude our proof. The proof is complete.
If for all , then by (1.3), for all . In view of (1.12) and (1.13), this is impossible, and hence, there must exist a such that . Then . By induction, we see that the subsequence lies in , and hence for all . In view of (1.11), . The proof is complete.
7. Phase Plane Interpretation and Comparison Remarks
- (v)For comparison purposes, let us now recall the asymptotic results in . Let us set
In view of these statements, we see that for a small positive , all solutions of (7.2) tend to a unique "lower" state vector, and for large , to another unique "higher" state vector. On the other hand, for a small positive , there are always solutions of (7.1) which tend to a limit 2-cycle, and solutions which tend to the limit 1-cycle , and for a large , there are solutions of (7.1) which tend to the limit 1-cycle and solutions to the limit 1-cycle . These observations show that it is probably not appropriate to call (1.1) the limiting case of (1.3)!
can be used to describe competing dynamics and it is hoped that our techniques, and results here will be useful in these studies.
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