scholarly journals Stability analysis of nontrivial stationary solution and constant equilibrium point of reaction-diffusion neural networks with time delays under Dirichlet zero boundary value

2021 ◽  
Author(s):  
Ruofeng Rao ◽  
Jialin Huang ◽  
Xiaodi Li
Keyword(s):  
Neural Networks ◽  
Stability Analysis ◽  
Equilibrium Point ◽  
Stationary Solution ◽  
Time Delays ◽  
Boundary Value ◽  
Reaction Diffusion ◽  
Constant Equilibrium

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

2020 ◽  
Author(s):  
Ruofeng Rao
Keyword(s):  
Neural Networks ◽  
Global Stability ◽  
Equilibrium Point ◽  
Stationary Solution ◽  
Point Theorem ◽  
Phase Plane ◽  
Boundary Value ◽  
Reaction Diffusion ◽  
Constant Equilibrium ◽  
Essential Change

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.

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

2020 ◽  
Author(s):  
Ruofeng Rao
Keyword(s):  
Neural Networks ◽  
Global Stability ◽  
Equilibrium Point ◽  
Stationary Solution ◽  
Point Theorem ◽  
Phase Plane ◽  
Boundary Value ◽  
Reaction Diffusion ◽  
Constant Equilibrium ◽  
Essential Change

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.

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

2020 ◽  
Author(s):  
Ruofeng Rao
Keyword(s):  
Neural Networks ◽  
Global Stability ◽  
Equilibrium Point ◽  
Stationary Solution ◽  
Point Theorem ◽  
Phase Plane ◽  
Boundary Value ◽  
Reaction Diffusion ◽  
Constant Equilibrium ◽  
Essential Change

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.

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

2020 ◽  
Author(s):  
Ruofeng Rao
Keyword(s):  
Neural Networks ◽  
Equilibrium Point ◽  
Stationary Solution ◽  
Point Theorem ◽  
Phase Plane ◽  
Boundary Value ◽  
Reaction Diffusion ◽  
Small Diffusion ◽  
Constant Equilibrium ◽  
Essential Change

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.

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

2020 ◽  
Author(s):  
Ruofeng Rao
Keyword(s):  
Neural Networks ◽  
Global Stability ◽  
Equilibrium Point ◽  
Stationary Solution ◽  
Point Theorem ◽  
Phase Plane ◽  
Boundary Value ◽  
Reaction Diffusion ◽  
Constant Equilibrium ◽  
Essential Change

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.

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

2020 ◽  
Author(s):  
Ruofeng Rao
Keyword(s):  
Neural Networks ◽  
Equilibrium Point ◽  
Stationary Solution ◽  
Phase Plane ◽  
Reaction Diffusion ◽  
Stationary Solutions ◽  
Diffusion Effect ◽  
Delayed Neural Networks ◽  
Constant Equilibrium ◽  
The Stability

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.

Stability analysis of fractional-order Hopfield neural networks with time delays

2014 ◽  
Vol 55 ◽  
pp. 98-109 ◽  
Author(s):  
Hu Wang ◽  
Yongguang Yu ◽  
Guoguang Wen
Keyword(s):  
Neural Networks ◽  
Stability Analysis ◽  
Fractional Order ◽  
Time Delays ◽  
Hopfield Neural Networks
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