A nonlinear three-term recurrence relation arising from seeking the steady states of a cellular neural network with bang bang control is studied. A complete analysis of its periodic behavior is given. In particular, we show that each solution is periodic and its prime period can be determined by two of its consecutive terms. By means of our periodicity analysis, we may then solve the steady state problem which to our knowledge is not solved by other means.
Simple dynamic systems representing time varying states of interconnected neurons may exhibit extremely complex behaviors when bifurcation parameters are switched from one set of values to another. In this paper, motivated by simulation results, we examine the steady states of one such system with bang-bang control and two real parameters. We found that nonnegative and negative periodic states are of special interests since these states are solutions of linear nonhomogeneous three-term recurrence relations. Although the standard approach to analyse such recurrence relations is the method of finding the general solutions by means of variation of parameters, we find novel alternate geometric methods that offer the tracking of solution trajectories in the plane. By means of this geometric approach, we are then able, without much tedious computation, to completely characterize the nonnegative and negative periodic solutions in terms of the bifurcation parameters.
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