scholarly journals Stability and Bifurcation Analysis for a Class of Generalized Reaction-Diffusion Neural Networks with Time Delay

2016 ◽  
Vol 2016 ◽  
pp. 1-9 ◽  
Author(s):  
Tianshi Lv ◽  
Qintao Gan ◽  
Qikai Zhu
Keyword(s):  
Neural Networks ◽  
Local Field ◽  
Neumann Boundary ◽  
Reaction Diffusion ◽  
Normal Form Theory ◽  
Center Manifold Reduction ◽  
Static Neural Networks ◽  
Stability And Bifurcation ◽  
Uniform Steady State ◽  
The Stability

Considering the fact that results for static neural networks are much more scare than results for local field neural networks and our purpose letting the problem researched be more general in many aspects, in this paper, a generalized neural networks model which includes reaction-diffusion local field neural networks and reaction-diffusion static neural networks is built and the stability and bifurcation problems for it are investigated under Neumann boundary conditions. First, by discussing the corresponding characteristic equations, the local stability of the trivial uniform steady state is discussed and the existence of Hopf bifurcations is shown. By using the normal form theory and the center manifold reduction of partial function differential equations, explicit formulae which determine the direction and stability of bifurcating periodic solutions are acquired. Finally, numerical simulations show the results.

scholarly journals Bifurcation Analysis in a Diffusive Mussel-Algae Model with Delay

2019 ◽  
Vol 29 (11) ◽  
pp. 1950144 ◽  
Author(s):  
Zuolin Shen ◽  
Junjie Wei
Keyword(s):  
Hopf Bifurcation ◽  
Functional Differential Equations ◽  
Neumann Boundary ◽  
Reaction Diffusion ◽  
Delayed Reaction ◽  
Center Manifold Reduction ◽  
Existence Of Positive Solutions ◽  
Boundary Equilibrium ◽  
Functional Differential ◽  
The Stability

In this paper, we consider the dynamics of a delayed reaction–diffusion mussel-algae system subject to Neumann boundary conditions. When the delay is zero, we show the existence of positive solutions and the global stability of the boundary equilibrium. When the delay is not zero, we obtain the stability of the positive constant steady state and the existence of Hopf bifurcation by analyzing the distribution of characteristic values. By using the theory of normal form and center manifold reduction for partial functional differential equations, we derive an algorithm that determines the direction of Hopf bifurcation and the stability of bifurcating periodic solutions. Finally, some numerical simulations are carried out to support our theoretical results.

scholarly journals Hopf bifurcation in a delayed reaction–diffusion–advection equation with ideal free dispersal

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Yunfeng Liu ◽  
Yuanxian Hui
Keyword(s):  
Hopf Bifurcation ◽  
Principal Eigenvalue ◽  
Reaction Diffusion ◽  
Advection Equation ◽  
Delayed Reaction ◽  
Normal Form Theory ◽  
Center Manifold Reduction ◽  
Form Theory ◽  
The Stability ◽  
Ideal Free

AbstractIn this paper, we investigate a delay reaction–diffusion–advection model with ideal free dispersal. The stability of positive steady-state solutions and the existence of the associated Hopf bifurcation are obtained by analyzing the principal eigenvalue of an elliptic operator. By the normal form theory and the center manifold reduction, the stability and bifurcation direction of Hopf bifurcating periodic solutions are obtained. Moreover, numerical simulations and a brief discussion are presented to illustrate our theoretical results.

scholarly journals Stability and bifurcation in a simplified four-neuron BAM neural network with multiple delays

2006 ◽  
Vol 2006 ◽  
pp. 1-29 ◽  
Author(s):  
Xiang-Ping Yan ◽  
Wan-Tong Li
Keyword(s):  
Neural Network ◽  
Critical Values ◽  
Multiple Delays ◽  
Multiple Time ◽  
Normal Form Theory ◽  
Center Manifold Reduction ◽  
Fourth Degree ◽  
Form Theory ◽  
Bam Neural Network ◽  
The Stability

We first study the distribution of the zeros of a fourth-degree exponential polynomial. Then we apply the obtained results to a simplified bidirectional associated memory (BAM) neural network with four neurons and multiple time delays. By taking the sum of the delays as the bifurcation parameter, it is shown that under certain assumptions the steady state is absolutely stable. Under another set of conditions, there are some critical values of the delay, when the delay crosses these critical values, the Hopf bifurcation occurs. Furthermore, some explicit formulae determining the stability and the direction of periodic solutions bifurcating from Hopf bifurcations are obtained by applying the normal form theory and center manifold reduction. Numerical simulations supporting the theoretical analysis are also included.

scholarly journals Exponential Stability and Periodicity of Fuzzy Delayed Reaction-Diffusion Cellular Neural Networks with Impulsive Effect

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
Guowei Yang ◽  
Yonggui Kao ◽  
Changhong Wang
Keyword(s):  
Neural Networks ◽  
Exponential Stability ◽  
Matrix Theory ◽  
Sufficient Conditions ◽  
Neumann Boundary ◽  
Reaction Diffusion ◽  
Cellular Neural Networks ◽  
Impulsive Effect ◽  
Time Varying ◽  
Delayed Reaction

This paper considers dynamical behaviors of a class of fuzzy impulsive reaction-diffusion delayed cellular neural networks (FIRDDCNNs) with time-varying periodic self-inhibitions, interconnection weights, and inputs. By using delay differential inequality,M-matrix theory, and analytic methods, some new sufficient conditions ensuring global exponential stability of the periodic FIRDDCNN model with Neumann boundary conditions are established, and the exponential convergence rate index is estimated. The differentiability of the time-varying delays is not needed. An example is presented to demonstrate the efficiency and effectiveness of the obtained results.

scholarly journals Stability and Hopf Bifurcation in a Three-Component Planktonic Model with Spatial Diffusion and Time Delay

Complexity ◽  
2019 ◽  
Vol 2019 ◽  
pp. 1-17 ◽  
Author(s):  
Kejun Zhuang ◽  
Gao Jia ◽  
Dezhi Liu
Keyword(s):  
Hopf Bifurcation ◽  
Equilibrium Solution ◽  
Neumann Boundary ◽  
Reaction Diffusion ◽  
Time Behavior ◽  
Delayed Reaction ◽  
Constant Equilibrium ◽  
Homogeneous Neumann Boundary Condition ◽  
Long Time ◽  
The Stability

Due to the different roles that nontoxic phytoplankton and toxin-producing phytoplankton play in the whole aquatic system, a delayed reaction-diffusion planktonic model under homogeneous Neumann boundary condition is investigated theoretically and numerically. This model describes the interactions between the zooplankton and two kinds of phytoplanktons. The long-time behavior of the model and existence of positive constant equilibrium solution are first discussed. Then, the stability of constant equilibrium solution and occurrence of Hopf bifurcation are detailed and analyzed by using the bifurcation theory. Moreover, the formulas for determining the bifurcation direction and stability of spatially bifurcating solutions are derived. Finally, some numerical simulations are performed to verify the appearance of the spatially homogeneous and nonhomogeneous periodic solutions.

Spatiotemporal Dynamics in Reaction–Diffusion Neural Networks Near a Turing–Hopf Bifurcation Point

2019 ◽  
Vol 29 (11) ◽  
pp. 1950154 ◽  
Author(s):  
Jiazhe Lin ◽  
Rui Xu ◽  
Xiaohong Tian
Keyword(s):  
Neural Networks ◽  
Hopf Bifurcation ◽  
Normal Form ◽  
Bifurcation Point ◽  
Reaction Diffusion ◽  
Spatiotemporal Dynamics ◽  
Normal Form Theory ◽  
Hopf Bifurcation Point ◽  
Codimension Two ◽  
Form Theory

Since the electromagnetic field of neural networks is heterogeneous, the diffusion phenomenon of electrons exists inevitably. In this paper, we investigate the existence of Turing–Hopf bifurcation in a reaction–diffusion neural network. By the normal form theory for partial differential equations, we calculate the normal form on the center manifold associated with codimension-two Turing–Hopf bifurcation, which helps us understand and classify the spatiotemporal dynamics close to the Turing–Hopf bifurcation point. Numerical simulations show that the spatiotemporal dynamics in the neighborhood of the bifurcation point can be divided into six cases and spatially inhomogeneous periodic solution appears in one of them.

scholarly journals The Hopf Bifurcation for a Predator-Prey System with -Logistic Growth and Prey Refuge

2013 ◽  
Vol 2013 ◽  
pp. 1-13 ◽  
Author(s):  
Shaoli Wang ◽  
Zhihao Ge
Keyword(s):  
Hopf Bifurcation ◽  
Logistic Growth ◽  
Positive Equilibrium ◽  
Prey Refuge ◽  
Normal Form Theory ◽  
Center Manifold Reduction ◽  
Predator Prey ◽  
Prey System ◽  
Form Theory ◽  
The Stability

The Hopf bifurcation for a predator-prey system with -logistic growth and prey refuge is studied. It is shown that the ODEs undergo a Hopf bifurcation at the positive equilibrium when the prey refuge rate or the index- passed through some critical values. Time delay could be considered as a bifurcation parameter for DDEs, and using the normal form theory and the center manifold reduction, explicit formulae are derived to determine the direction of bifurcations and the stability and other properties of bifurcating periodic solutions. Numerical simulations are carried out to illustrate the main results.

Stability, Bifurcation and Jumping Phenomenon in a 2-D Model of Supersonic Lifting Surfaces

2005 ◽  
Author(s):  
Pei Yu ◽  
Zhen Chen ◽  
Liviu Librescu ◽  
Piergiovanni Marzocca
Keyword(s):  
Nonlinear Feedback ◽  
Feedback Controls ◽  
Normal Form Theory ◽  
Center Manifold Reduction ◽  
Form Theory ◽  
Lifting Surfaces ◽  
The Stability ◽  
Parameter Values ◽  
Manifold Reduction ◽  
D Model

This paper is concerned with the linear/nonlinear aeroelastic control of 2-D supersonic lifting surfaces. Its goal is to provide the feedback control mechanism enabling one to enlarge the flight envelope by increasing the flutter speed, and also to control the character, benign/catastrophic of the flutter instability boundary. Structural and aerodynamic nonlinearities are included in the aeroelastic governing equations, and linear and nonlinear feedback controls in both plunging and pitching are employed in conjunction with proportional velocity feedback controls. The attention of the paper is focused on multiple Hopf bifurcations. In particular, the jumping phenomenon found in our previous work will be further investigated to reveal the physical implications. It is found that such a jumping occurs when the system has multiple families of limit cycles bifurcating from a same set of parameter values with multiple solutions for frequencies. The case investigated in this paper is restricted to zero structure damping. Center manifold reduction and normal form theory are applied to consider the stability of post-flutter solutions and the associated jumping phenomenon. Numerical simulations are presented to show the implications of time delay in the considered controls.

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