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# 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)
Version 6 : Received: 1 October 2020 / Approved: 2 October 2020 / Online: 2 October 2020 (13:57:27 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.v4). 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.v4).

## 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 results of (globally) exponentially stable positive stationary solution of delayed reaction-diffusion cell neural networks under Dirichlet zero boundary value, including the global stability criteria \textbf{in the classical meaning}. 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.

## Keywords

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

Comment 1
Commenter: Ruofeng Rao
Commenter's Conflict of Interests: Author
Comment: Dear editors of Preprint,
Based on my last manuscript "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.v3), I only add the following two changes:

(1) Adding the sentence "On the other hand, according to the Introduction in [51], the function $f$ defined by (3.39) satisfies the conditions [51, (7)] and [51, (8)],
and hence, the zero solution is actually the unique equilibrium point of the ordinary differential system (3.37). Thereby, Statement 2 has actually
verified that under the Lipschitz assumption on the function f,
the small diffusion can truly make one equilibrium point become multiple stationary solutions (three stationary solutions, even infinitely many stationary solutions)." to Remark 10.

(2) Adding a literatrue ( [51]  Daoyi Xu, Zhiguo Yang, Yumei Huang. Existence-uniqueness and continuation theorems for stochastic functional differential equations. J. Differential Equations, 245 (2008) 1681-1703.)
to the references list.

Thank you for your patience and responsibility in the editorial work， and sorry for the trouble.

Best wishes,

Ruofeng Rao,

Professor at Chengdu Normal University, China.

Email: ruofengrao@163.com

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