scholarly journals Global Stability of Integral Manifolds for Reaction–Diffusion Delayed Neural Networks of Cohen–Grossberg-Type under Variable Impulsive Perturbations

Mathematics ◽  
2020 ◽  
Vol 8 (7) ◽  
pp. 1082
Author(s):  
Gani Stamov ◽  
Ivanka Stamova ◽  
George Venkov ◽  
Trayan Stamov ◽  
Cvetelina Spirova
Keyword(s):  
Neural Networks ◽  
Global Stability ◽  
Lyapunov Functions ◽  
Type Inequality ◽  
Reaction Diffusion ◽  
Stability Criteria ◽  
Delayed Neural Networks ◽  
Integral Manifolds ◽  
Impulsive Perturbations ◽  
Impulsive Jumps

The present paper introduces the concept of integral manifolds for a class of delayed impulsive neural networks of Cohen–Grossberg-type with reaction–diffusion terms. We establish new existence and boundedness results for general types of integral manifolds with respect to the system under consideration. Based on the Lyapunov functions technique and Poincarѐ-type inequality some new global stability criteria are also proposed in our research. In addition, we consider the case when the impulsive jumps are not realized at fixed instants. Instead, we investigate a system under variable impulsive perturbations. Finally, examples are given to demonstrate the efficiency and applicability of the obtained results.

scholarly journals Global Exponential Stability of Discrete-Time Neural Networks with Time-Varying Delays

2013 ◽  
Vol 2013 ◽  
pp. 1-4 ◽  
Author(s):  
S. Udpin ◽  
P. Niamsup
Keyword(s):  
Neural Networks ◽  
Global Stability ◽  
Discrete Time ◽  
Type Inequality ◽  
Discrete Systems ◽  
Nonlinear Difference Equation ◽  
Time Varying ◽  
Stability Criteria ◽  
Nonlinear Discrete Systems ◽  
Theoretical Results

This paper presents some global stability criteria of discrete-time neural networks with time-varying delays. Based on a discrete-type inequality, a new global stability condition for nonlinear difference equation is derived. We consider nonlinear discrete systems with time-varying delays and independence of delay time. Numerical examples are given to illustrate the effectiveness of our theoretical results.

scholarly journals Impulsive Reaction-Diffusion Delayed Models in Biology: Integral Manifolds Approach

Entropy ◽  
2021 ◽  
Vol 23 (12) ◽  
pp. 1631
Author(s):  
Gani Stamov ◽  
Ivanka Stamova ◽  
Cvetelina Spirova
Keyword(s):  
Lyapunov Functions ◽  
Lyapunov Method ◽  
Impulsive Control ◽  
Type Inequality ◽  
Reaction Diffusion ◽  
Epidemic Models ◽  
Therapy Control ◽  
Delayed Reaction ◽  
Reaction Diffusion Model ◽  
Integral Manifolds

In this paper we study an impulsive delayed reaction-diffusion model applied in biology. The introduced model generalizes existing reaction-diffusion delayed epidemic models to the impulsive case. The integral manifolds notion has been introduced to the model under consideration. This notion extends the single state notion and has important applications in the study of multi-stable systems. By means of an extension of the Lyapunov method integral manifolds’ existence, results are established. Based on the Lyapunov functions technique combined with a Poincarè-type inequality qualitative criteria related to boundedness, permanence, and stability of the integral manifolds are also presented. The application of the proposed impulsive control model is closely related to a most important problems in the mathematical biology—the problem of optimal control of epidemic models. The considered impulsive effects can be used by epidemiologists as a very effective therapy control strategy. In addition, since the integral manifolds approach is relevant in various contexts, our results can be applied in the qualitative investigations of many problems in the epidemiology of diverse interest.

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.

IMPULSIVE EFFECT OF CONTINUOUS-TIME NEURAL NETWORKS UNDER PURE STRUCTURAL VARIATIONS

2007 ◽  
Vol 17 (06) ◽  
pp. 2127-2139 ◽  
Author(s):  
ZHANJI GUI ◽  
WEIGAO GE
Keyword(s):  
Neural Networks ◽  
Continuous Time ◽  
Lyapunov Functions ◽  
Global Exponential Stability ◽  
Coincidence Degree ◽  
Degree Theory ◽  
Simulation Method ◽  
Impulsive Effect ◽  
Structural Variations ◽  
Impulsive Perturbations

By using the continuation theorem of coincidence degree theory and constructing suitable Lyapunov functions, we study the existence, uniqueness and global exponential stability of periodic solution for continuous-time neural networks under pure structural variations with impulsive perturbations: [Formula: see text] The results extend earlier ones where impulses are absent. Further, using numerical simulation method the influences of the impulsive perturbations on the inherent oscillation are investigated.

Sign in / Sign up

Export Citation Format

Share Document