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All solutions of a class of difference equations are truncated periodic. (English) Zbl 1029.39003

Author’s summary: We propose the difference equation \(x_{n+1}=x_n-f(x_{n-k})\) as a model for a single neuron with no internal decay, where \(f\) satisfies the McCulloch-Pitts nonlinearity. It is shown that every solution is truncated periodic with the minimal period \(2(2l+1)\) for some \(l\geq 0\) such that \((k-l)/2l+1)\) is a nonnegative even integer. The potential application of our results to neural networks is obvious.

MSC:

39A11 Stability of difference equations (MSC2000)
92B20 Neural networks for/in biological studies, artificial life and related topics
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[1] Chow, S.-N.; Walther, H.-O., Characteristic multipliers and stability of symmetric periodic solutions of \(ẋ (t) = g(x(t - 1))\), Trans. Amer. Math. Soc., 307, 127-142 (1988) · Zbl 0672.34069
[2] Nussbaum, R. D., Uniqueness and nonuniqueness for periodic solutions of \(x\)′\((t)\) = −\(g(x(t\) − 1)), J. Differential Equations, 34, 25-54 (1979) · Zbl 0404.34057
[3] Walther, H.-O., Homoclinic solutions and chaos in \(ẋ (t) = f(x(t - 1))\), Nonlinear Anal., 5, 775-788 (1981) · Zbl 0459.34040
[4] Busenberg, S.; Cooke, K. L., Models of vertically transmitted diseases with sequential-continuous dynamics, (Lakshmikanthan, V., Nonlinear Phenomena in Mathematical Science (1982), Academic Press: Academic Press New York), 179-187 · Zbl 0512.92018
[5] Cooke, K. L.; Wiener, J., Retarded differential equations with piecewise constant delays, J. Math. Anal. Appl., 99, 265-297 (1984) · Zbl 0557.34059
[6] Shah, S. M.; Wiener, J., Advanced differential equations with piecewise constant argument deviations, Internat. J. Math. Math. Sci., 6, 671-703 (1983) · Zbl 0534.34067
[7] Cooke, K. L.; Wiener, J., A survey of differential equations with piecewise constant arguments, (Busenberg, S.; Martelli, M., Delay Differential Equations and Dynamical Systems (1991), Springer: Springer Berlin) · Zbl 0737.34045
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