CPAA
Quasilinear retarded differential equations with functional dependence on piecewise constant argument
Marat Akhmet
We introduce a new class of differential equations, retarded differential equations with functional dependence on piecewise constant argument, $RFDEPCA$ and focus on quasilinear systems. Formulation of the initial value problem, bounded solutions, periodic and almost periodic solutions, their stability are under investigation. Illustrating examples are provided.
keywords: Bohr almost periodic solutions periodic solutions exponential stability. Retarded differential equations alternate constancy of argument functional dependence on piecewise constant argument
DCDS
Lyapunov-Razumikhin method for differential equations with piecewise constant argument
Marat Akhmet Duygu Aruğaslan
At the first time, Razumikhin technique is applied for differential equations with piecewise constant argument of generalized type [1, 2]. Sufficient conditions are established for stability, uniform stability and uniform asymptotic stability of the trivial solution of such equations. We also provide appropriate examples to illustrate our results.
keywords: Lyapunov's second method Razumikhin technique logistic equation. Differential equations with piecewise constant argument of generalized type
DCDS-B
Impulsive SICNNs with chaotic postsynaptic currents
Mehmet Onur Fen Marat Akhmet
In the present study, we investigate the dynamics of shunting inhibitory cellular neural networks (SICNNs) with impulsive effects. We give a mathematical description of the chaos for the multidimensional dynamics of impulsive SICNNs, and prove its existence rigorously by taking advantage of the external inputs. The Li-Yorke definition of chaos is used in our theoretical discussions. In the considered model, the impacts satisfy the cell and shunting principles. This enriches the applications of SICNNs and makes the analysis of impulsive neural networks deeper. The technique is exceptionally useful for SICNNs with arbitrary number of cells. We make benefit of unidirectionally coupled SICNNs to exemplify our results. Moreover, the appearance of cyclic irregular behavior observed in neuroscience is numerically demonstrated for discontinuous dynamics of impulsive SICNNs.
keywords: impacts subject to cell and shunting principles chaotification of impulsive SICNNs. chaotic inputs and outputs discontinuous Li-Yorke chaos Shunting inhibitory cellular neural networks near-periodic discontinuous chaos

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