scholarly journals Method of Centre Manifold from Bifurcation Theory

2019 ◽  
Vol 7 (10) ◽  
Author(s):  
Dr. Basher Suleiman Othman
Keyword(s):  
Visual Cortex ◽  
Primary Visual Cortex ◽  
Bifurcation Theory ◽  
Center Manifold ◽  
Hopf Bifurcations ◽  
Center Manifold Theory ◽  
Field Equations ◽  
Centre Manifold ◽  
Manifold Theory ◽  
Neural Field Equations

The aim of this paper is to introduce tools from bifurcation theory is necessary in ways in our life particularly in the study of neural field equations set in the primary visual cortex. So we deal with saddle-node, trans- critical, pitchfork and Hopf. Bifurcations as an elementary bifurcation; directly related to the center manifold theory which is a canonical way to write differential equations. We conclude this paper with an overview of bifurcations with symmetry by solving some problems and giving Branching Lemma as the equivariant result

Dynamical system analysis of a nonminimally coupled scalar field

2019 ◽  
Vol 28 (15) ◽  
pp. 1950173 ◽  
Author(s):  
Subhajyoti Pal ◽  
Sudip Mishra ◽  
Subenoy Chakraborty
Keyword(s):  
Scalar Field ◽  
Critical Points ◽  
Center Manifold ◽  
System Analysis ◽  
Nonlinear Differential Equations ◽  
Center Manifold Theory ◽  
Einstein Field Equations ◽  
Field Equations ◽  
Flat Spacetime ◽  
Manifold Theory

This paper deals with a nonminimally coupled scalar field in the background of homogeneous and isotropic Friedmann–Lemaître–Robertson–Walker (FLRW) flat spacetime. As Einstein field equations are coupled second-order nonlinear differential equations, it is very hard to find exact solutions. By suitable choice of variables, we transform Einstein field equations to an autonomous system and critical points are determined. We use center manifold theory to characterize nonhyperbolic critical points and are found to be saddle in nature. We discuss possible bifurcation scenarios, which indicate the existence of the cosmological bouncing model.

scholarly journals Bifurcation and Patterns Analysis for a Spatiotemporal Discrete Gierer-Meinhardt System

Mathematics ◽  
2022 ◽  
Vol 10 (2) ◽  
pp. 243
Author(s):  
Biao Liu ◽  
Ranchao Wu
Keyword(s):  
Bifurcation Theory ◽  
Center Manifold ◽  
Turing Instability ◽  
Flip Bifurcation ◽  
Center Manifold Theory ◽  
Pattern Dynamics ◽  
Instability Theory ◽  
Pattern Formations ◽  
Manifold Theory ◽  
Theoretical Results

The Gierer-Meinhardt system is one of the prototypical pattern formation models. The bifurcation and pattern dynamics of a spatiotemporal discrete Gierer-Meinhardt system are investigated via the couple map lattice model (CML) method in this paper. The linear stability of the fixed points to such spatiotemporal discrete system is analyzed by stability theory. By using the bifurcation theory, the center manifold theory and the Turing instability theory, the Turing instability conditions in flip bifurcation and Neimark–Sacker bifurcation are considered, respectively. To illustrate the above theoretical results, numerical simulations are carried out, such as bifurcation diagram, maximum Lyapunov exponents, phase orbits, and pattern formations.

STABILITY AND BIFURCATION ANALYSIS IN A DELAYED PREDATOR-PREY SYSTEM

2009 ◽  
Vol 02 (04) ◽  
pp. 483-506 ◽  
Author(s):  
ZHICHAO JIANG ◽  
WENZHI ZHANG ◽  
DONGSHENG HUO
Keyword(s):  
Local Stability ◽  
Center Manifold ◽  
Hopf Bifurcations ◽  
Center Manifold Theory ◽  
Predator Prey ◽  
Prey System ◽  
Stability And Bifurcation Analysis ◽  
Characteristic Values ◽  
The Stability ◽  
Manifold Theory

A delayed ratio-dependent one-predator and two-prey system with Michaelis–Menten type functional response is investigated. We show the existence of nonnegative equilibria under some appropriated conditions. Criteria for local stability, instability of nonnegative equilibria are obtained. The existence of Hopf bifurcations at the endemic equilibrium is established by analyzing the distribution of the characteristic values. An explicit algorithm for determining the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions is derived by using the normal form and the center manifold theory. At last, some numerical simulations to support the analytical conclusions are carried out.

ON HOPF BIFURCATION OF A DELAYED PREDATOR–PREY SYSTEM WITH DIFFUSION

2013 ◽  
Vol 23 (02) ◽  
pp. 1350023 ◽  
Author(s):  
JIANXIN LIU ◽  
JUNJIE WEI
Keyword(s):  
Hopf Bifurcation ◽  
Center Manifold ◽  
Dirichlet Boundary ◽  
Dirichlet Boundary Conditions ◽  
Hopf Bifurcations ◽  
Center Manifold Theory ◽  
Predator Prey ◽  
Prey System ◽  
Normal Form Method ◽  
Manifold Theory

A delayed predator–prey system with diffusion and Dirichlet boundary conditions is considered. By regarding the growth rate a of prey as a main bifurcation parameter, we show that Hopf bifurcation occurs when the parameter a is varied. Then, by using the center manifold theory and normal form method, an explicit algorithm for determining the direction of the Hopf bifurcations and stability of the bifurcating periodic solutions is derived.

Bifurcation Analysis for Sailing Stability of Autonomous Underwater Vehicle

2010 ◽  
Vol 44-47 ◽  
pp. 1682-1686
Author(s):  
Song Shi Shao ◽  
Jiong Sun ◽  
Kai Liu
Keyword(s):  
Hopf Bifurcation ◽  
Bifurcation Theory ◽  
Center Manifold ◽  
Autonomous Underwater Vehicle ◽  
Underwater Vehicle ◽  
Hydrodynamic Parameter ◽  
System State ◽  
Center Manifold Theory ◽  
Proportional Derivative Controller ◽  
Manifold Theory

There are several nonlinear elements in the equations of Autonomous Underwater Vehicle(AUV) movements. It is difficult to deal nonlinear problem with traditional methods. A hydrodynamic parameter interference is chosen as bifurcation parameter at first. Then the sailing stability of AUV with proportional-derivative controller is analysed by bifurcation theory. The center manifold theory is used to get the expression of system state parameters. And the Hopf bifurcation of system is analysed. The result is verified by numerical simulations. It shows that the hydrodynamic parameter’s changing will bring Hopf bifurcation for depthkeeping saiiling. And the range of hydrodynamic parameter value that insures AUV sailing stability is given.

scholarly journals Nonlinear Dynamics of a Nutrient-Phytoplankton Model with Time Delay

2015 ◽  
Vol 2015 ◽  
pp. 1-12 ◽  
Author(s):  
DeBing Mei ◽  
Min Zhao ◽  
Hengguo Yu ◽  
Chuanjun Dai ◽  
Yi Wang
Keyword(s):  
Time Delay ◽  
Center Manifold ◽  
Critical Value ◽  
Hopf Bifurcations ◽  
Type Ii ◽  
Center Manifold Theory ◽  
Existence And Stability ◽  
Manifold Theory ◽  
Theoretical Results ◽  
Type Ii Functional Response

We consider a nutrient-phytoplankton model with a Holling type II functional response and a time delay. By selecting the time delay used as a bifurcation parameter, we prove that the system is stable if the delay value is lower than the critical value but unstable when it is above this value. First, we investigate the existence and stability of the equilibria, as well as the existence of Hopf bifurcations. Second, we consider the direction, stability, and period of the periodic solutions from the steady state based on the normal form and the center manifold theory, thereby deriving explicit formulas. Finally, some numerical simulations are given to illustrate the main theoretical results.

scholarly journals Stability and Hopf bifurcation periodic orbits in delay coupled Lotka-Volterra ring system

2019 ◽  
Vol 17 (1) ◽  
pp. 962-978
Author(s):  
Rina Su ◽  
Chunrui Zhang
Keyword(s):  
Hopf Bifurcation ◽  
Periodic Orbits ◽  
Bifurcation Theory ◽  
Center Manifold ◽  
Ring System ◽  
Center Manifold Theory ◽  
Dihedral Groups ◽  
The Stability ◽  
Delay Differential ◽  
Manifold Theory

Abstract In this paper, we consider a class of delay coupled Lotka-Volterra ring systems. Based on the symmetric bifurcation theory of delay differential equations and representation theory of standard dihedral groups, properties of phase locked periodic solutions are given. Moreover, the direction and the stability of the Hopf bifurcation periodic orbits are obtained by using normal form and center manifold theory. Finally, the research results are verified by numerical simulation.

Bifurcation Analysis for Neural Networks in Neutral Form

2016 ◽  
Vol 26 (06) ◽  
pp. 1650100 ◽  
Author(s):  
Hong-Bing Chen ◽  
Xiao-Ke Sun
Keyword(s):  
Neural Networks ◽  
Hopf Bifurcation ◽  
Center Manifold ◽  
Hopf Bifurcations ◽  
Center Manifold Theory ◽  
Neutral Form ◽  
Theoretical Predictions ◽  
Normal Form Method ◽  
The Stability ◽  
Manifold Theory

In this paper, a system of neural networks in neutral form with time delay is investigated. Further, by introducing delay [Formula: see text] as a bifurcation parameter, it is found that Hopf bifurcation occurs when [Formula: see text] is across some critical values. The direction of the Hopf bifurcations and the stability are determined by using normal form method and center manifold theory. Next, the global existence of periodic solution is established by using a global Hopf bifurcation result. Finally, an example is given to support the theoretical predictions.

STABILITY ANALYSIS OF A PREDATOR-PREY MODEL

2012 ◽  
Vol 05 (01) ◽  
pp. 1250007 ◽  
Author(s):  
ZHICHAO JIANG ◽  
ZHAOZHUANG GUO ◽  
YUEFANG SUN
Keyword(s):  
Center Manifold ◽  
Positive Equilibrium ◽  
Hopf Bifurcations ◽  
Center Manifold Theory ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey System ◽  
Characteristic Values ◽  
The Stability ◽  
Manifold Theory

In this paper, a time-delayed predator-prey system is considered. The existence of Hopf bifurcations at the positive equilibrium is established by analyzing the distribution of the characteristic values. An explicit algorithm for determining the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions are derived by using the normal form and the center manifold theory. Numerical simulations to support the analytical conclusions are carried out.

scholarly journals Bifurcation Analysis in a Two-Dimensional Neutral Differential Equation

2013 ◽  
Vol 2013 ◽  
pp. 1-9
Author(s):  
Ming Liu ◽  
Xiaofeng Xu
Keyword(s):  
Periodic Solutions ◽  
Center Manifold ◽  
Hopf Bifurcations ◽  
Center Manifold Theory ◽  
Neutral Form ◽  
Global Hopf Bifurcation ◽  
Normal Form Method ◽  
Existence Of Periodic Solutions ◽  
The Stability ◽  
Manifold Theory

The dynamics of a 2-dimensional neural network model in neutral form are investigated. We prove that a sequence of Hopf bifurcations occurs at the origin as the delay increases. The direction of the Hopf bifurcations and the stability of the bifurcating periodic solutions are determined by using normal form method and center manifold theory. Global existence of periodic solutions is established using a global Hopf bifurcation result of Krawcewicz et al. Finally, some numerical simulations are carried out to support the analytic results.

Sign in / Sign up

Export Citation Format

Share Document