scholarly journals Flip and Neimark–Sacker Bifurcations in a Coupled Logistic Map System

2020 ◽  
Vol 2020 ◽  
pp. 1-14
Author(s):  
A. Mareno ◽  
L. Q. English
Keyword(s):  
Bifurcation Analysis ◽  
Center Manifold ◽  
Logistic Map ◽  
Local Bifurcation ◽  
Center Manifold Theory ◽  
Strongly Coupled ◽  
Local Bifurcation Analysis ◽  
The Stability ◽  
Two Parameters ◽  
Manifold Theory

In this paper, we consider a system of strongly coupled logistic maps involving two parameters. We classify and investigate the stability of its fixed points. A local bifurcation analysis of the system using center manifold theory is undertaken and then supported by numerical computations. This reveals the existence of a flip and Neimark–Sacker bifurcations.

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.

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.

scholarly journals Stability and Hopf Bifurcation Analysis of a Nutrient-Phytoplankton Model with Delay Effect

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Xinhong Pan ◽  
Min Zhao ◽  
Chuanjun Dai ◽  
Yapei Wang
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Center Manifold ◽  
Center Manifold Theory ◽  
Delay Effect ◽  
Delay Differential System ◽  
The Stability ◽  
Delay Differential ◽  
Manifold Theory ◽  
Two Equilibria

A delay differential system is investigated based on a previously proposed nutrient-phytoplankton model. The time delay is regarded as a bifurcation parameter. Our aim is to determine how the time delay affects the system. First, we study the existence and local stability of two equilibria using the characteristic equation and identify the condition where a Hopf bifurcation can occur. Second, the formulae that determine the direction of the Hopf bifurcation and the stability of periodic solutions are obtained using the normal form and the center manifold theory. Furthermore, our main results are illustrated using numerical simulations.

Vibration Suppression of a Flexible Beam Using Center Manifold Theory

1995 ◽  
Author(s):  
Amir Khajepour ◽  
Farid Golnaraghi ◽  
K. A. Morris
Keyword(s):  
Nonlinear Control ◽  
Center Manifold ◽  
Vibration Suppression ◽  
Flexible Beam ◽  
Center Manifold Theory ◽  
Linear Control Theory ◽  
Modal Coupling ◽  
The Stability ◽  
Manifold Theory ◽  
Mass Spring

Abstract In this paper we develop a nonlinear control strategy based on modal coupling using the center manifold theory. As an example we use the technique for vibration suppression of a flexible beam. The controller in this case is a mass-spring-dashpot mechanism which is free to slide along the beam. The equations of the plant/controller are coupled and nonlinear, and the linearized equations of the system have two uncontrollable modes. As a result, the performance of the system can not be improved by linear control theory or by most conventional nonlinear control techniques. We use the normal forms method to simplify the center manifold equations and derive a relation which includes all system parameters. We then show that there exists a set of optimal controller parameters (feedback gains, controller damping and frequency) which maximized the energy dissipation. Finally we consider the stability and design issues, and use numerical simulation to verify the 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.

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.

Stability and Bifurcation Analysis in a Food Chain Model with Delay

2011 ◽  
Vol 110-116 ◽  
pp. 3382-3388
Author(s):  
Zhang Li
Keyword(s):  
Hopf Bifurcation ◽  
Food Chain ◽  
Center Manifold ◽  
Chain Model ◽  
Center Manifold Theory ◽  
Stability And Bifurcation Analysis ◽  
Existence And Stability ◽  
The Stability ◽  
Manifold Theory ◽  
Food Chain Model

In this paper, we investigate a delayed three-species food chain model. The existence and stability of equilibria are obtained. A 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.

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.

scholarly journals Neimark-Sacker, flip and transcritical bifurcation in a symmetric system of difference equations with exponential terms

2021 ◽  
Author(s):  
Chrysoula Mylona ◽  
Garyfalos Papaschinopoulos ◽  
Christos Schinas
Keyword(s):  
Normal Form ◽  
Difference Equations ◽  
Bifurcation Analysis ◽  
Center Manifold ◽  
Transcritical Bifurcation ◽  
Symmetric System ◽  
Center Manifold Theory ◽  
Real Numbers ◽  
System Of Difference Equations ◽  
Manifold Theory

In this paper, we study the conditions under which the following symmetric system of difference equations with exponential terms: \[ x_{n+1} =a_1\frac{y_n}{b_1+y_n} +c_1\frac{x_ne^{k_1-d_1x_n}}{1+e^{k_1-d_1x_n}},\] \[ y_{n+1} =a_2\frac{x_n}{b_2+x_n} +c_2\frac{y_ne^{k_2-d_2y_n}}{1+e^{k_2-d_2y_n}}\] where $a_i$, $b_i$, $c_i$, $d_i$, $k_i$, for $i=1,2$, are real constants and the initial values $x_0$, $y_0$ are real numbers, undergoes Neimark-Sacker, flip and transcritical bifurcation. The analysis is conducted applying center manifold theory and the normal form bifurcation analysis.

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