Further Improvement on Delay-Dependent Global Robust Exponential Stability for Delayed Cellular Neural Networks with Time-Varying Delays

Liu, Pin‐Lin

doi:10.1007/s11063-017-9683-6

Cited by 4 publications

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References 23 publications

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“…This means that we should study the reaction diffusion model for neural networks, rather than ordinary differential equation model. given in many previous related literature ( [31,[34][35][36][37][38][39] and the references therein). Limited to the length of the article, we can't point out one by one.…”

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confidence: 99%

“…Usually, if the activation function is strictly monotonous, the number the equilibrium points of the ordinary differential equation model for cellular neural networks will be not more than three in the case of n = 1. Particularly in many literature ( [31,[34][35][36][37][38][39]), under the Lipschitz conditions on activation functions, the equilibrium point is always unique (see,e.g. [31,Theorem 1]).…”

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confidence: 99%

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Stability Analysis of Nontrivial Stationary Solution and Constant Equilibrium Point of Reaction-Diffusion Neural Networks With Time Delays

Rao¹

2020

Preprint

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Firstly, the existence of asymptotically stable nontrivial stationary solution is derived by the comprehensive applications of Lyapunov-Razumikhin technique, design of state-dependent switching laws, a fixed point theorem, variational methods, and construction of compact operators on a convex set. This new theorem shows that the diffusion is a double-edged sword to the stability, refuting the views in previous literature that the greater the diffusion effect, the more stable the system will be. Next, a series of new theorems are presented one by one, which illustrates that the globally asymptotical stability of ordinary differential equations model for delayed neural networks may be locally stable in actual operation due to the inevitable diffusion. Besides, the non-zero constant equilibrium point is pointed out to be not the solution of delayed reaction diffusion system so that the stability of the non-zero constant equilibrium point of reaction diffusion system must lead to a contradiction. That is, non-zero constant equilibrium points are not in the phase plane of dynamic system. In addition, new theorems are further presented 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, and thereby one equilibrium solution may become several stationary solutions, even infinitely many positive stationary solutions. Finally, a numerical example illustrates the feasibility of the proposed methods.

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