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.

STABILITY AND BIFURCATION ANALYSIS ON A DELAYED NEURAL NETWORK MODEL

2005 ◽  
Vol 15 (09) ◽  
pp. 2883-2893 ◽  
Author(s):  
XIULING LI ◽  
JUNJIE WEI
Keyword(s):  
Neural Network ◽  
Hopf Bifurcation ◽  
Network Model ◽  
Neural Network Model ◽  
Center Manifold ◽  
Normal Form Theory ◽  
Center Manifold Theorem ◽  
Form Theory ◽  
Delayed Neural Network ◽  
The Stability

A simple delayed neural network model with four neurons is considered. Linear stability of the model is investigated by analyzing the associated characteristic equation. It is found that Hopf bifurcation occurs when the sum of four delays varies and passes a sequence of critical values. The stability and direction of the Hopf bifurcation are determined by applying the normal form theory and the center manifold theorem. An example is given and numerical simulations are performed to illustrate the obtained results. Meanwhile, the bifurcation set is provided in the appropriate parameter plane.

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.

scholarly journals Center Manifold Reduction and Perturbation Method in a Delayed Model with a Mound-Shaped Cobb-Douglas Production Function

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Massimiliano Ferrara ◽  
Luca Guerrini ◽  
Giovanni Molica Bisci
Keyword(s):  
Hopf Bifurcation ◽  
Perturbation Method ◽  
Production Function ◽  
Center Manifold ◽  
Normal Form Theory ◽  
Center Manifold Reduction ◽  
Center Manifold Theorem ◽  
Form Theory ◽  
The Stability ◽  
Manifold Reduction

Matsumoto and Szidarovszky (2011) examined a delayed continuous-time growth model with a special mound-shaped production function and showed a Hopf bifurcation that occurs when time delay passes through a critical value. In this paper, by applying the center manifold theorem and the normal form theory, we obtain formulas for determining the direction of the Hopf bifurcation and the stability of bifurcating periodic solutions. Moreover, Lindstedt’s perturbation method is used to calculate the bifurcated periodic solution, the direction of the bifurcation, and the stability of the periodic motion resulting from the bifurcation.

Bifurcations of Time-Delay-Induced Multiple Transitions Between In-Phase and Anti-Phase Synchronizations in Neurons with Excitatory or Inhibitory Synapses

2019 ◽  
Vol 29 (11) ◽  
pp. 1950147 ◽  
Author(s):  
Li Li ◽  
Zhiguo Zhao ◽  
Huaguang Gu
Keyword(s):  
Time Delay ◽  
Inhibitory Synapses ◽  
Normal Form Theory ◽  
Center Manifold Reduction ◽  
Subthreshold Oscillations ◽  
Dynamical Behaviors ◽  
Stable Periodic ◽  
Form Theory ◽  
The Stability ◽  
Manifold Reduction

Time-delay-induced synchronous behaviors and synchronization transitions have been widely investigated for coupled neurons, and they play important roles for physiological functions. In the present study, time-delay-induced synchronized subthreshold oscillations were simulated, and the bifurcations underlying the synchronized behaviors were identified for a pair of coupled FitzHugh–Nagumo neurons. Multiple transitions between in-phase and anti-phase synchronizations induced by the time delay were simulated for the inhibitory and excitatory couplings. Subcritical or supercritical Hopf bifurcations and the stability of the Hopf-bifurcating periodic subthreshold oscillations were acquired using center manifold reduction and normal form theory. The in-phase or anti-phase synchronizations of the stable periodic subthreshold oscillations, which appear for multiple values of the time delay, were interpreted with the related eigenspace. The distributions of the different dynamical behaviors, including the synchronizations and bifurcations in the two-parameter plane of the time delay and coupling strength, were acquired for both types of synapses, and the different roles of the inhibitory and excitatory couplings on the synchronization transitions were compared.

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 Hopf Bifurcation Analysis for a Gause-Type Predator-Prey System with Multiple Delays

2013 ◽  
Vol 2013 ◽  
pp. 1-12 ◽  
Author(s):  
Juan Liu ◽  
Changwei Sun ◽  
Yimin Li
Keyword(s):  
Hopf Bifurcation ◽  
Sufficient Conditions ◽  
Multiple Delays ◽  
Normal Form Theory ◽  
Predator Prey ◽  
Prey System ◽  
Form Theory ◽  
Two Delays ◽  
The Stability ◽  
Theoretical Results

This paper is concerned with a Gause-type predator-prey system with two delays. Firstly, we study the stability and the existence of Hopf bifurcation at the coexistence equilibrium by analyzing the distribution of the roots of the associated characteristic equation. A group of sufficient conditions for the existence of Hopf bifurcation is obtained. Secondly, an explicit formula for determining the stability and the direction of periodic solutions that bifurcate from Hopf bifurcation is derived by using the normal form theory and center manifold argument. Finally, some numerical simulations are carried out to illustrate the main theoretical results.

scholarly journals Stability and Sensitive Analysis of a Model with Delay Quorum Sensing

2015 ◽  
Vol 2015 ◽  
pp. 1-11 ◽  
Author(s):  
Zhonghua Zhang ◽  
Yaohong Suo ◽  
Juan Zhang
Keyword(s):  
Normal Form ◽  
Center Manifold ◽  
Reduction Method ◽  
Delay Model ◽  
Threshold Parameter ◽  
Normal Form Theory ◽  
Center Manifold Reduction ◽  
Form Theory ◽  
The Stability ◽  
Manifold Reduction

This paper formulates a delay model characterizing the competition between bacteria and immune system. The center manifold reduction method and the normal form theory due to Faria and Magalhaes are used to compute the normal form of the model, and the stability of two nonhyperbolic equilibria is discussed. Sensitivity analysis suggests that the growth rate of bacteria is the most sensitive parameter of the threshold parameterR0and should be targeted in the controlling strategies.

STABILITY OF BIFURCATED PERIODIC SOLUTIONS IN A DELAYED COMPETITION SYSTEM WITH DIFFUSION EFFECTS

2009 ◽  
Vol 19 (03) ◽  
pp. 857-871 ◽  
Author(s):  
XIANG-PING YAN ◽  
WAN-TONG LI
Keyword(s):  
Periodic Solutions ◽  
Center Manifold ◽  
Functional Differential Equations ◽  
Dirichlet Boundary Conditions ◽  
Normal Form Theory ◽  
Center Manifold Reduction ◽  
Theoretical Predictions ◽  
Form Theory ◽  
Functional Differential ◽  
The Stability

In this paper, a delayed Lotka—Volterra two species competition diffusion system with a single discrete delay and subject to the homogeneous Dirichlet boundary conditions is considered. By applying the normal form theory and the center manifold reduction for partial functional differential equations (PFDEs), the stability of bifurcated periodic solutions occurring through Hopf bifurcations is studied. It is shown that the bifurcated periodic solution occurring at the first bifurcation point is orbitally asymptotically stable on the center manifold while those occurring at other bifurcation points are unstable. Finally, some numerical simulations to a special example are included to verify our theoretical predictions.

scholarly journals Bifurcation Analysis for a Two-Dimensional Discrete-Time Hopfield Neural Network with Delays

2007 ◽  
Vol 2007 ◽  
pp. 1-8
Author(s):  
Yaping Ren ◽  
Yongkun Li
Keyword(s):  
Neural Network ◽  
Discrete Time ◽  
Bifurcation Analysis ◽  
Hopfield Neural Network ◽  
Discrete Time System ◽  
Normal Form Theory ◽  
Center Manifold Theorem ◽  
Form Theory ◽  
The Stability ◽  
Two Parameters

A bifurcation analysis is undertaken for a discrete-time Hopfield neural network with four delays. Conditions ensuring the asymptotic stability of the null solution are obtained with respect to two parameters of the system. Using techniques developed by Kuznetsov to a discrete-time system, we study the Neimark-Sacker bifurcation (also called Hopf bifurcation for maps) of the system. The direction and the stability of the Neimark-Sacker bifurcation are investigated by applying the normal form theory and the center manifold theorem.

Sign in / Sign up

Export Citation Format

Share Document