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 Hopf Bifurcation of a Delayed Mutualistic System

2021 ◽  
Vol 31 (14) ◽  
Author(s):  
Ruimin Zhang ◽  
Xiaohui Liu ◽  
Chunjin Wei
Keyword(s):  
Hopf Bifurcation ◽  
Center Manifold ◽  
Stage Structure ◽  
Center Manifold Theory ◽  
Mutualistic Relationship ◽  
Normal Form Method ◽  
Explicit Formulae ◽  
The Stability ◽  
Manifold Theory ◽  
Theoretical Results

In this paper, we study a classic mutualistic relationship between the leaf cutter ants and their fungus garden, establishing a time delay mutualistic system with stage structure. We investigate the stability and Hopf bifurcation by analyzing the distribution of the roots of the associated characteristic equation. By means of the center manifold theory and normal form method, explicit formulae are derived to determine the stability, direction and other properties of bifurcating periodic solutions. Finally, some numerical simulations are carried out for illustrating the theoretical results.

Bifurcation Analysis of Phytoplankton and Zooplankton Interaction System with Two Delays

2020 ◽  
Vol 30 (03) ◽  
pp. 2050039
Author(s):  
Zhichao Jiang ◽  
Jiangtao Dai ◽  
Tongqian Zhang
Keyword(s):  
Periodic Solutions ◽  
Center Manifold ◽  
Threshold Values ◽  
Center Manifold Theory ◽  
Local Hopf Bifurcation ◽  
Two Delays ◽  
The Stability ◽  
Manifold Theory ◽  
Theoretical Results ◽  
Positivity And Boundedness

In this paper, the system of describing the interactions between poisonous phytoplankton and zooplankton is presented. It focuses on the effects of two delays on the dynamic behavior of the system. At first, the properties of solutions including positivity and boundedness are given. Next, the stability of equilibria and the existence of local Hopf bifurcation are established when delays change and cross some threshold values. Especially, the existence of global periodic solutions is discussed when the two delays are equal. Furthermore, the implicit algorithm is derived for deciding the properties of the branching periodic solutions by using center manifold theory. Some numerical simulations are performed for supporting the theoretical results. Finally, some conclusions are given.

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.

scholarly journals Hopf Bifurcation of an SIQR Computer Virus Model with Time Delay

2015 ◽  
Vol 2015 ◽  
pp. 1-8 ◽  
Author(s):  
Zizhen Zhang ◽  
Huizhong Yang
Keyword(s):  
Hopf Bifurcation ◽  
Center Manifold ◽  
Computer Virus ◽  
Center Manifold Theory ◽  
Virus Prevalence ◽  
Virus Model ◽  
Normal Form Method ◽  
The Stability ◽  
Manifold Theory ◽  
Theoretical Results

A delayed SIQR computer virus model is considered. It has been observed that there exists a critical value of delay for the stability of virus prevalence by choosing the delay as a bifurcation parameter. Furthermore, the properties of the Hopf bifurcation such as direction and stability are investigated by using the normal form method and center manifold theory. Finally, some numerical simulations for supporting our theoretical results are also performed.

scholarly journals Dynamics of a Delayed Model for the Transmission of Malicious Objects in Computer Network

2014 ◽  
Vol 2014 ◽  
pp. 1-14
Author(s):  
Zizhen Zhang ◽  
Huizhong Yang
Keyword(s):  
Periodic Solutions ◽  
Center Manifold ◽  
Computer Network ◽  
Center Manifold Theory ◽  
Two Delays ◽  
Normal Form Method ◽  
The Stability ◽  
Manifold Theory ◽  
Delayed Model ◽  
Theoretical Results

An SEIQRS model for the transmission of malicious objects in computer network with two delays is investigated in this paper. We show that possible combination of the two delays can affect the stability of the model and make the model bifurcate periodic solutions under some certain conditions. For further investigation, properties of the periodic solutions are studied by using the normal form method and center manifold theory. Finally, some numerical simulations are given to justify the theoretical results.

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

Application of Center Manifold Theory to Regulation of a Flexible Beam

1997 ◽  
Vol 119 (2) ◽  
pp. 158-165 ◽  
Author(s):  
Amir Khajepour ◽  
M. Farid Golnaraghi ◽  
Kirsten A. Morris
Keyword(s):  
Control Strategy ◽  
Center Manifold ◽  
Flexible Beam ◽  
Trial And Error ◽  
Center Manifold Theory ◽  
Control Techniques ◽  
Lumped Parameter ◽  
The Stability ◽  
Frequency Relationship ◽  
Manifold Theory

In this paper we consider the problem of regulation of a flexible lumped parameter beam. The controller is an active/passive mass-spring-dashpot mechanism which is free to slide along the beam. In this problem the plant/controller equations are coupled and nonlinear, and the linearized equations of the system have two uncontrollable modes associated with a pair of pure imaginary eigenvalues. As a result, linear control techniques as well as most conventional nonlinear control techniques can not be applied. In earlier studies Golnaraghi (1991) and Golnaraghi et al. (1994) a control strategy based on Internal resonance was developed to transfer the oscillatory energy from the beam to the slider, where it was dissipated through controller damping. Although these studies provided very good understanding of the control strategy, the analytical method was based on perturbation techniques and had many limitations. Most of the work was based on numerical techniques and trial and error. In this paper we use center manifold theory to address the shortcomings of the previous studies, and extend the work to a more general control law. The technique is based on reducing the dimension of system and simplifying the nonlinearities using center manifold and normal forms techniques, respectively. The simplified equations are used to investigate the stability and to develop a relation for the optimal controller/plant natural frequencies at which the maximum transfer of energy occurs. One of the main contributions of this work is the elimination of the trial and error and inclusion of damping in the optimal frequency relationship.

Twelve Limit Cycles in 3D Quadratic Vector Fields with Z3 Symmetry

2018 ◽  
Vol 28 (11) ◽  
pp. 1850139 ◽  
Author(s):  
Laigang Guo ◽  
Pei Yu ◽  
Yufu Chen
Keyword(s):  
Limit Cycles ◽  
Vector Fields ◽  
Center Manifold ◽  
Three Dimensional ◽  
Center Manifold Theory ◽  
Normal Form Theory ◽  
Quadratic Vector Fields ◽  
Fine Focus ◽  
Form Theory ◽  
Manifold Theory

This paper is concerned with the number of limit cycles bifurcating in three-dimensional quadratic vector fields with [Formula: see text] symmetry. The system under consideration has three fine focus points which are symmetric about the [Formula: see text]-axis. Center manifold theory and normal form theory are applied to prove the existence of 12 limit cycles with [Formula: see text]–[Formula: see text]–[Formula: see text] distribution in the neighborhood of three singular points. This is a new lower bound on the number of limit cycles in three-dimensional quadratic systems.

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