Preprint Article Version 2 Preserved in Portico This version is not peer-reviewed

Stability Analysis of Nontrivial Stationary Solution and Constant Equilibrium Point of Reaction-Diffusion Neural Networks with Time Delays under Dirichlet Zero Boundary Value

Version 1 : Received: 15 April 2020 / Approved: 16 April 2020 / Online: 16 April 2020 (13:33:43 CEST)
Version 2 : Received: 13 August 2020 / Approved: 20 August 2020 / Online: 20 August 2020 (04:13:36 CEST)
Version 3 : Received: 21 August 2020 / Approved: 22 August 2020 / Online: 22 August 2020 (09:55:22 CEST)
Version 4 : Received: 29 August 2020 / Approved: 31 August 2020 / Online: 31 August 2020 (08:26:12 CEST)
Version 5 : Received: 8 September 2020 / Approved: 11 September 2020 / Online: 11 September 2020 (08:27:42 CEST)

How to cite: Rao, R. Stability Analysis of Nontrivial Stationary Solution and Constant Equilibrium Point of Reaction-Diffusion Neural Networks with Time Delays under Dirichlet Zero Boundary Value. Preprints 2020, 2020040277 (doi: 10.20944/preprints202004.0277.v2). Rao, R. Stability Analysis of Nontrivial Stationary Solution and Constant Equilibrium Point of Reaction-Diffusion Neural Networks with Time Delays under Dirichlet Zero Boundary Value. Preprints 2020, 2020040277 (doi: 10.20944/preprints202004.0277.v2).

Abstract

In this paper, Lyapunov-Razumikhin technique, design of state-dependent switching laws, a fixed point theorem and variational methods are employed to derive the existence and the unique existence of positive stationary solution of delayed reaction-diffusion cell neural networks under Dirichlet zero boundary value. Next, sufficient conditions are proposed to guarantee the global stability invariance of ordinary differential systems under the influence of diffusions. New theorems show that the diffusion is a double-edged sword in judging the stability of diffusion systems. Besides, an example is constructed to illuminate that any non-zero constant equilibrium point must be not in the phase plane of dynamic system under Dirichlet zero boundary value, or it must lead to a contradiction. Next, under Lipschitz assumptions on active function, another example is designed to prove that the small diffusion effect will cause the essential change of the phase plane structure of the dynamic behavior of the delayed neural networks via a Saddle point theorem. Finally, a numerical example illustrates the feasibility of the proposed methods.

Subject Areas

reaction-diffusion; cellular neural networks; exponential stability; stationary solutions; Saddle point theorem

Comments (1)

Comment 1
Received: 20 August 2020
Commenter: Ruofeng Rao
Commenter's Conflict of Interests: Author
Comment: The main revision notes are presented as follows:
(1) The title is changed. Just adding “under Dirichlet zero boundary value” to the original title.
(2) The abstract part are refined and compressed.
(3) Theorem 3.1 is modified, and deepened via adding a new theorem—Theorem 3.2.
(4) The original Theorem 3.2 becomes Theorem 3.3 now.
(5) Other theorems in the original paper are refined, and become Statement 1-2.
(6) Adding a new corollary—Corollary 3.4.
(7) Numerical example is modified and deepened. Besides, two tables are added.
(8) The final section is refined and compressed, too.
(9) Adding a Lemma—Lemma 2.3, and a Definition—Definition 1. And the original Definition 1 becomes Definition 2 now.
(10)Several literature in the reference list are changed. 
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