Periodic solutions of a bang bang recurrence equation with least periods 1 through 8

Abstract

In this paper, we study a very simple three term recurrence relation involving the discontinuous Heaviside step function. One reason for studying such an relation is that solutions of our recurrence relation are steady state distributions in some basic neural network models. Since analytic tools cannot be used to handle discontinuous models such as ours, existence of periodic solutions is investigated by combining combinatorial elimination technique as well as existence arguments for linear systems. By such means, we are able to obtain all periodic solutions with least periods 1 through 8. Some periodic solutions with periods 12, 20 and 36 can also be found, but exhaustive results are not yet available.

Periodic solutions of a bang bang recurrence equation with least periods 1 through 9

“…By induction and Lemma 5 we may then see that u has the period 72, since 1,2,3,4,6,8,9,12,18,24,36, as can be checked easily, are not its periods. Thus u is a 72-periodic sequence and u(a, 72) is given by (17).…”

“…[1] and other results in the literature for recurrence and difference equations with continuous functions). In [2], we studied a prototype discontinuous recurrence relation of the form:…”

6-Periodic travelling waves in an artificial neural network with bang–bang control

“…It is pointed out in [8] that the above equation arises in the study of 'standing wave solutions' of artificial neural networks with bang -bang controls. Besides these standing wave solutions, more general 'travelling wave solutions' are also of great interest.…”

“…Before describing such equations, we quickly sketch the introductory parts of [2,8]. Let Z ¼ {0;^1;^2; .…”

“…[5] and other results in the literature for recurrence relations involving continuous functions). For this reason, in [8], a very simple prototype equation where H is the Heaviside step function (or bang -bang control) defined by…”