scholarly journals Bifurcation Study on Fractional-Order Cohen–Grossberg Neural Networks Involving Delays

2020 ◽  
Vol 2020 ◽  
pp. 1-16
Author(s):  
Bingnan Tang
Keyword(s):  
Neural Networks ◽  
Hopf Bifurcation ◽  
Fractional Order ◽  
Sufficient Criterion ◽  
Software Simulation ◽  
Stability Behavior ◽  
Delayed Neural Networks ◽  
Simulation Results ◽  
Set Up ◽  
The Stability

This work is chiefly concerned with the stability behavior and the appearance of Hopf bifurcation of fractional-order delayed Cohen–Grossberg neural networks. Firstly, we study the stability and the appearance of Hopf bifurcation of the involved neural networks with identical delay ϑ 1 = ϑ 2 = ϑ . Secondly, the sufficient criterion to guarantee the stability and the emergence of Hopf bifurcation for given neural networks with the delay ϑ 2 = 0 is set up. Thirdly, we derive the sufficient condition ensuring the stability and the appearance of Hopf bifurcation for given neural networks with the delay ϑ 1 = 0 . The investigation manifests that the delay plays a momentous role in stabilizing networks and controlling the Hopf bifurcation of the addressed fractional-order delayed neural networks. At last, software simulation results successfully verified the rationality of the analytical results. The theoretical findings of this work can be applied to design, control, and optimize neural networks.

scholarly journals Further Study on Dynamics for a Fractional-Order Competitor-Competitor-Mutualist Lotka–Volterra System

Complexity ◽  
2021 ◽  
Vol 2021 ◽  
pp. 1-15
Author(s):  
Bingnan Tang
Keyword(s):  
Hopf Bifurcation ◽  
Fractional Order ◽  
Natural World ◽  
Volterra Model ◽  
Sufficient Criterion ◽  
Volterra System ◽  
Stability Behavior ◽  
Predator Model ◽  
Set Up ◽  
The Stability

On the basis of the previous publications, a new fractional-order prey-predator model is set up. Firstly, we discuss the existence, uniqueness, and nonnegativity for the involved fractional-order prey-predator model. Secondly, by analyzing the characteristic equation of the considered fractional-order Lotka–Volterra model and regarding the delay as bifurcation variable, we set up a new sufficient criterion to guarantee the stability behavior and the appearance of Hopf bifurcation for the addressed fractional-order Lotka–Volterra system. Thirdly, we perform the computer simulations with Matlab software to substantiate the rationalisation of the analysis conclusions. The obtained results play an important role in maintaining the balance of population in natural world.

scholarly journals New Stability Criterion for Fractional-Order Quaternion-Valued Neural Networks Involving Discrete and Leakage Delays

2021 ◽  
Vol 2021 ◽  
pp. 1-20
Author(s):  
Bingjun Li ◽  
Bingnan Tang
Keyword(s):  
Neural Networks ◽  
Computer Simulation ◽  
Fractional Order ◽  
Stability Criterion ◽  
Uniform Stability ◽  
Sufficient Criterion ◽  
Leakage Delays ◽  
Analysis Strategy ◽  
Set Up ◽  
Mapping Theory

In the current work, we are devoted to the issue of uniform stability of fractional-order quaternion-valued neural networks involving discrete and leakage delays. Making use of the contracting mapping theory, we prove that the equilibrium point of the involved fractional-order quaternion-valued neural networks exists and is unique. Taking advantage of mathematical analysis strategy, a sufficient criterion involving delay to verify the global uniform stability for the considered fractional-order quaternion-valued neural networks is set up. Computer simulation figures are displayed to sustain the rationality of the established conclusions. This study generalizes and supplements the research of Xiu et al. (2020).

Bifurcations due to different delays of high-order fractional neural networks

2021 ◽  
pp. 2150075
Author(s):  
Chengdai Huang ◽  
Jinde Cao
Keyword(s):  
Neural Networks ◽  
Hopf Bifurcation ◽  
Fractional Order ◽  
Numerical Experiments ◽  
Critical Values ◽  
Communication Delay ◽  
Leakage Delay ◽  
Bifurcation Parameter ◽  
Stability Performance ◽  
The Stability

This paper expounds the bifurcations of two-delayed fractional-order neural networks (FONNs) with multiple neurons. Leakage delay or communication delay is viewed as a bifurcation parameter, stability zones and bifurcation conditions with respect to them are commendably established, respectively. It declares that both leakage delay and communication delay immensely influence the stability and bifurcation of the developed FONNs. The explored FONNs illustrate superior stability performance if selecting a lesser leakage delay or communication delay, and Hopf bifurcation generates once they overstep their critical values. The verification of the feasibility of the developed analytic results is implemented via numerical experiments.

scholarly journals Periodic Oscillatory Phenomenon in Fractional-Order Neural Networks Involving Different Types of Delays

2021 ◽  
Vol 2021 ◽  
pp. 1-17
Author(s):  
Nengfa Wang ◽  
Changjin Xu ◽  
Zixin Liu
Keyword(s):  
Neural Network ◽  
Neural Networks ◽  
Hopf Bifurcation ◽  
Fractional Order ◽  
Network Systems ◽  
Different Types ◽  
Variable Substitution ◽  
The Stability ◽  
Distribution Of Roots ◽  
Oscillatory Phenomenon

This research is chiefly concerned with the stability and Hopf bifurcation for newly established fractional-order neural networks involving different types of delays. By means of an appropriate variable substitution, equivalent fractional-order neural network systems involving one delay are built. By discussing the distribution of roots of the characteristic equation of the established fractional-order neural network systems and selecting the delay as bifurcation parameter, a novel delay-independent bifurcation condition is derived. The investigation verifies that the delay is a significant parameter which has an important influence on stability nature and Hopf bifurcation behavior of neural network systems. The computer simulation plots and bifurcation graphs effectively illustrate the reasonableness of the theoretical fruits.

scholarly journals Bifurcation dynamics in a fractional-order oregonator model including time delay

2021 ◽  
Vol 87 (2) ◽  
pp. 397-414
Author(s):  
Changjin Xu ◽  
◽  
Wei Zhang ◽  
Chaouki Aouiti ◽  
Zixin Liu ◽  
...  
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Mathematical Models ◽  
Fractional Order ◽  
Vital Role ◽  
Chemical Models ◽  
Set Up ◽  
The Stability ◽  
Oregonator Model ◽  
Theoretical Results

Setting up mathematical models to describe the interaction of chemical variables has been a hot issue in chemical and mathematical areas. Nevertheless, many mathematical models are only involved with the integer-order differential equation case. The fruits on fractional-order chemical models are very scarce. In this present work, on the basis of the previous studies, we set up a novel fractional-order delayed Oregonator model. Selecting the time delay as bifurcation parameter, we obtain novel delay-independent bifurcation conditions that guarantee the stability and the appearance of Hopf bifurcation for the fractional-order delayed Oregonator model. The study shows that time delay plays a vital role in controlling the stability and the appearance of Hopf bifurcation of the considered fractional-order delayed Oregonator model. In order to verify the rationality of theoretical results, computer simulations are carried out.

scholarly journals Active Sliding Mode for Synchronization of a Wide Class of Four-Dimensional Fractional-Order Chaotic Systems

2014 ◽  
Vol 2014 ◽  
pp. 1-11 ◽  
Author(s):  
Bin Wang ◽  
Yuangui Zhou ◽  
Jianyi Xue ◽  
Delan Zhu
Keyword(s):  
Sliding Mode Control ◽  
Fractional Order ◽  
Wide Class ◽  
Stability Theory ◽  
Chaotic Systems ◽  
Sliding Mode ◽  
Order System ◽  
Fractional Order Calculus ◽  
Simulation Results ◽  
The Stability

We focus on the synchronization of a wide class of four-dimensional (4-D) chaotic systems. Firstly, based on the stability theory in fractional-order calculus and sliding mode control, a new method is derived to make the synchronization of a wide class of fractional-order chaotic systems. Furthermore, the method guarantees the synchronization between an integer-order system and a fraction-order system and the synchronization between two fractional-order chaotic systems with different orders. Finally, three examples are presented to illustrate the effectiveness of the proposed scheme and simulation results are given to demonstrate the effectiveness of the proposed method.

Turing Instability and Hopf Bifurcation in Cellular Neural Networks

2021 ◽  
Vol 31 (08) ◽  
pp. 2150143
Author(s):  
Zunxian Li ◽  
Chengyi Xia
Keyword(s):  
Neural Networks ◽  
Hopf Bifurcation ◽  
Turing Instability ◽  
Cellular Neural Networks ◽  
Bifurcation Theorem ◽  
Decoupling Method ◽  
Dynamical Behaviors ◽  
The Stability ◽  
Approximate Expressions ◽  
Zero Equilibrium

In this paper, we explore the dynamical behaviors of the 1D two-grid coupled cellular neural networks. Assuming the boundary conditions of zero-flux type, the stability of the zero equilibrium is discussed by analyzing the relevant eigenvalue problem with the aid of the decoupling method, and the conditions for the occurrence of Turing instability and Hopf bifurcation at the zero equilibrium are derived. Furthermore, the approximate expressions of the bifurcating periodic solutions are also obtained by using the Hopf bifurcation theorem. Finally, numerical simulations are provided to demonstrate the theoretical results.

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