Global stability analysis of S-asymptotically ω-periodic oscillation in fractional-order cellular neural networks with time variable delays

Qu, Huizhen; Zhang, Tianwei; Zhou, Jianwen

doi:10.1016/j.neucom.2020.03.005

“…On the other hand, the Poincare inequality and the Dirichlet zero boundary value yields 25) where λ 1 is the smallest positive eigenvalue of the following eigenvalue problem:…”

Section: Next For Any Givenmentioning

confidence: 99%

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

Rao¹

2020

Preprint

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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.

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“…Since the Caputo fractional-order derivative, it is shown that ω-periodic or autonomous fractional-order neural networks cannot generate exactly ω-periodic signals. Thus, during recent years, many researchers have paid increasing attention to deal with the S-asymptotically ω-periodic solution of fractional-order non-autonomous neural networks [41]- [43]. Compared with the previous results, rare results are available for S-asymptotically ω-periodic solutions of fractional-order complex-valued neural networks.…”

Section: Introductionmentioning

confidence: 99%

“…However, some results show that ω-periodic or autonomous fractional-order neural networks cannot generate exactly ω-periodic signals. So, in past decade, many authors studied S-asymptotically ω-periodic oscillations of fractional-order non-autonomous neural networks [41]- [43]. (2) Recently, many literatures [33]- [38] had studied the fractional-order complex-valued neural networks.…”

Section: Introductionmentioning

confidence: 99%

S-Asymptotically ω-Periodic Solutions of Fractional-Order Complex-Valued Recurrent Neural Networks With Delays

Hou

¹

,

Dai

²

2021

IEEE Access

6

0

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In this paper, we consider the problem of the S-asymptotically ω-periodic synchronization of fractional-order complex-valued recurrent neural networks with time delays. Firstly, we can not explicitly decompose the fractional-order complex-valued systems into equivalent fractional-order real-valued systems, by means of the contraction mapping principle and some important features of Mittag-Leffler functions, we obtain some sufficient conditions for the existence and uniqueness of S-asymptotically ω-periodic solutions for this class of neural networks. Then, by constructing an appropriate Lyapunov functional, the theory of fractional differential equation, and some inequality techniques, sufficient conditions are obtained to guarantee the global Mittag-Leffler synchronization of the drive-response systems. Finally, two examples are given to illustrate the effectiveness and feasibility of our main results.

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“…In order to achieve these functions, the cellular neural network must be completely stable, that is, all output trajectories must converge to a stable equilibrium point. So the stability of cellular neural network has become a hot topic ( [24][25][26]). As we all know, time delay may destroy the stability of the system and lead to oscillation, bifurcation, chaos and other phenomena, thus changing the characteristics of the system.…”

Section: Introductionmentioning

confidence: 99%

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

Rao¹

2020

Preprint

View full text Add to dashboard Cite

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.

show abstract

Global stability analysis of S-asymptotically ω-periodic oscillation in fractional-order cellular neural networks with time variable delays

Cited by 18 publications

References 38 publications

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

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

S-Asymptotically ω-Periodic Solutions of Fractional-Order Complex-Valued Recurrent Neural Networks With Delays

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

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