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2 2010

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Bifurcation in a nonlinear steady state system

Wang G. Q., Cheng S. S.

Opuscula Mathematica

2010

Vol. 30, no. 3

349-360

The steady state solutions of a nonlinear digital cellular neural network with ω neural units and a nonnegative variable parameter λ are sought. We show that λ = 1 is a critical value such that the qualitative behavior of our network changes. More specifically, when ω is odd, then for λ ∈ [0, 1), there is one positive and one negative steady state, and for λ ∈ [1, ∞), steady states cannot exist; while when ω is even, then for λ ∈ [0, 1), there is one positive and one negative steady state, and for λ = 1, there are no nontrivial steady states, and for λ ∈ (1, ∞), there are two fully oscillatory steady states. Furthermore, the number of existing nontrivial solutions cannot be improved. It is hoped that our results are of interest to digital neural network designers.

Oscillations of a higher order nonlinear differential equations with impulses

Wen -wen K., Wang G. Q., Cheng S. S.

Demonstratio Mathematica

2010

Vol. 43, nr 3

575-603

Impulsive differential equations are important in simulation of processes with jump conditions. Although there are quite a few studies on the oscillation properties of low order equations, there are not too many studies of higher order equations. In this paper, we derive several oscillation criteria which are either new or improve several recent results in the literature. In addition, we provide several examples to illustrate the use of our results.