Complete set of periodic solutions of a discontinuous recurrence equation

Nonnegative Periodic Solutions of a Three-Term Recurrence Relation Depending on Two Real Parameters

“…By [1][2][3], we have observed the importance of nonnegative solutions of (7) with = 0, 1, and −1, respectively. However, we also realize the difficulties of analyzing the solution of (7) with ∉ {−1, 0, 1} since the breaking of into companion and error sequences is not applicable for all the solutions anymore.…”

“…Some of these behaviors can be explained (see [1][2][3]) but some not (at least to the best of our knowledge). Similar models of piecewise constant dynamic systems which exhibit similar behaviors with parameters can be found in many recent investigations; see for examples [4][5][6][7][8][9][10] and the references therein.…”

“…An example has been given in several of our previous studies [1] and a slightly modified model of which is described as follows. Let neurons be placed on the vertices of a regular -gon and let the time dependent state values of the neurons be denoted by ( ), for = 1, 2, .…”

“…In [1][2][3], we are lucky to obtain complete information about the periodic solutions of (4) when = 1 and = 1, 0, or −1 by breaking the solutions into two sequences, a companion and an error (more specifically, for a solution = { } ∈N of (4) with = 1 and ∈ {−1, 0, 1}, the companion sequence is = { } ∈N which is an even integral sequence so that = 2[(1/2) ], where 2[ /2] is the greatest even integer that is less than or equal to ∈ R, and the error sequence = { } ∈N is defined by = − , ∈ Z).…”

“…In [1][2][3], we conducted complete analyses on the periodic solutions (7) with = 1 and ∈ {−1, 0, 1}. By means of breaking a solution into two subsequences, the companion sequence and the error sequence, we successfully show that both the companion sequence and error sequence are periodic so that is periodic.…”