Hopf Bifurcation Analysis for a Novel Hyperchaotic System

2012 ◽  
Vol 8 (1) ◽  
Author(s):  
Kejun Zhuang
Keyword(s):  
Numerical Simulation ◽  
Hopf Bifurcation ◽  
Normal Form ◽  
Local Stability ◽  
Center Manifold ◽  
Hyperchaotic System ◽  
Normal Form Theory ◽  
Center Manifold Theorem ◽  
Form Theory ◽  
The Stability

The paper mainly focuses on a novel hyperchaotic system. The local stability of equilibrium is analyzed and existence of Hopf bifurcation is established. Moreover, formulas for determining the stability and direction of bifurcating periodic solutions are derived by center manifold theorem and normal form theory. Finally, numerical simulation is given to illustrate the theoretical analysis.

scholarly journals Stability and Hopf Bifurcation for a Delayed Computer Virus Model with Antidote in Vulnerable System

2017 ◽  
Vol 2017 ◽  
pp. 1-9
Author(s):  
Zizhen Zhang ◽  
Yougang Wang ◽  
Massimiliano Ferrara
Keyword(s):  
Hopf Bifurcation ◽  
Normal Form ◽  
Local Stability ◽  
Center Manifold ◽  
Computer Virus ◽  
Normal Form Theory ◽  
Center Manifold Theorem ◽  
Form Theory ◽  
Virus Model ◽  
Bifurcation And Stability

A delayed computer virus model with antidote in vulnerable system is investigated. Local stability of the endemic equilibrium and existence of Hopf bifurcation are discussed by analyzing the associated characteristic equation. Further, direction of the Hopf bifurcation and stability of the bifurcating periodic solutions are investigated by using the normal form theory and the center manifold theorem. Finally, numerical simulations are presented to show consistency with the obtained results.

scholarly journals Dynamical analysis of a giving up smoking model with time delay

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Zizhen Zhang ◽  
Ruibin Wei ◽  
Wanjun Xia
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Local Stability ◽  
Center Manifold ◽  
Dynamical Behavior ◽  
Sufficient Conditions ◽  
Dynamical Analysis ◽  
Normal Form Theory ◽  
Center Manifold Theorem ◽  
Form Theory

AbstractIn this paper, we are concerned with a delayed smoking model in which the population is divided into five classes. Sufficient conditions guaranteeing the local stability and existence of Hopf bifurcation for the model are established by taking the time delay as a bifurcation parameter and employing the Routh–Hurwitz criteria. Furthermore, direction and stability of the Hopf bifurcation are investigated by applying the center manifold theorem and normal form theory. Finally, computer simulations are implemented to support the analytic results and to analyze the effects of some parameters on the dynamical behavior of the model.

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 Hopf Bifurcation Analysis of a Synthetic Drug Transmission Model with Time Delays

Complexity ◽  
2019 ◽  
Vol 2019 ◽  
pp. 1-17 ◽  
Author(s):  
Zizhen Zhang ◽  
Fangfang Yang ◽  
Wanjun Xia
Keyword(s):  
Hopf Bifurcation ◽  
Center Manifold ◽  
Sufficient Conditions ◽  
Transmission Model ◽  
Normal Form Theory ◽  
Center Manifold Theorem ◽  
Form Theory ◽  
Two Delays ◽  
Synthetic Drug ◽  
The Stability

This paper is concerned with the Hopf bifurcation of a synthetic drug transmission model with two delays. Firstly, some sufficient conditions of delay-induced bifurcation for such a model are captured by using different combinations of the two delays as the bifurcation parameter. Secondly, based on the center manifold theorem and normal form theory, some sufficient conditions determining properties of the Hopf bifurcation such as the direction and the stability are established. Finally, to underline the effectiveness of the obtained results, some numerical simulations are ultimately addressed.

scholarly journals Stability and Hopf Bifurcation of Delayed Predator-Prey System Incorporating Harvesting

2014 ◽  
Vol 2014 ◽  
pp. 1-12 ◽  
Author(s):  
Fengying Wei ◽  
Lanqi Wu ◽  
Yuzhi Fang
Keyword(s):  
Hopf Bifurcation ◽  
Center Manifold ◽  
Hopf Bifurcations ◽  
Normal Form Theory ◽  
Predator Prey ◽  
Center Manifold Theorem ◽  
Prey System ◽  
Theoretical Predictions ◽  
Form Theory ◽  
The Stability

A kind of delayed predator-prey system with harvesting is considered in this paper. The influence of harvesting and delay is investigated. Our results show that Hopf bifurcations occur as the delayτpasses through critical values. By using of normal form theory and center manifold theorem, the direction of Hopf bifurcation and the stability of the bifurcating periodic solutions are obtained. Finally, numerical simulations are given to support our theoretical predictions.

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.

scholarly journals Local Hopf Bifurcation in a Competitive Model of Market Structure with Consumptive Delays

2014 ◽  
Vol 2014 ◽  
pp. 1-10
Author(s):  
Xuhui Li
Keyword(s):  
Hopf Bifurcation ◽  
Market Structure ◽  
Local Stability ◽  
Center Manifold ◽  
Positive Equilibrium ◽  
Normal Form Theory ◽  
Local Hopf Bifurcation ◽  
Competitive Model ◽  
Form Theory ◽  
The Stability

A competitive model of market structure with consumptive delays is considered. The local stability of the positive equilibrium and the existence of local Hopf bifurcation are investigated by analyzing the distribution of the roots of the associated characteristic equation. The explicit formulas determining the stability and other properties of bifurcating periodic solutions are derived by using normal form theory and center manifold argument. Finally, numerical simulations are given to support the analytical results.

scholarly journals Delay Induced Hopf Bifurcation of an Epidemic Model with Graded Infection Rates for Internet Worms

2017 ◽  
Vol 2017 ◽  
pp. 1-10
Author(s):  
Tao Zhao ◽  
Dianjie Bi
Keyword(s):  
Hopf Bifurcation ◽  
Local Stability ◽  
Sufficient Conditions ◽  
Propagation Model ◽  
Infection Rates ◽  
Normal Form Theory ◽  
Center Manifold Theorem ◽  
Internet Worms ◽  
Form Theory ◽  
The Stability

A delayed SEIQRS worm propagation model with different infection rates for the exposed computers and the infectious computers is investigated in this paper. The results are given in terms of the local stability and Hopf bifurcation. Sufficient conditions for the local stability and the existence of Hopf bifurcation are obtained by using eigenvalue method and choosing the delay as the bifurcation parameter. In particular, the direction and the stability of the Hopf bifurcation are investigated by means of the normal form theory and center manifold theorem. Finally, a numerical example is also presented to support the obtained theoretical results.

scholarly journals Control of Hopf Bifurcation in Autonomous System Based on Washout Filter

2013 ◽  
Vol 2013 ◽  
pp. 1-16 ◽  
Author(s):  
Wenju Du ◽  
Yandong Chu ◽  
Jiangang Zhang ◽  
Yingxiang Chang ◽  
Jianning Yu ◽  
...  
Keyword(s):  
Hopf Bifurcation ◽  
Center Manifold ◽  
Equilibrium Points ◽  
Normal Form Theory ◽  
Center Manifold Theorem ◽  
Bifurcation Phenomenon ◽  
Controlled Systems ◽  
Washout Filter ◽  
Form Theory ◽  
The Stability

In order to further understand a Lorenz-like system, we study the stability of the equilibrium points and the existence of Hopf bifurcation by center manifold theorem and normal form theory. More precisely, we designed a washout controller such that the equilibriumE0undergoes a controllable Hopf bifurcation, and by adjusting the controller parameters, we delayed Hopf bifurcation phenomenon of the equilibriumE+. Besides, numerical simulation is given to illustrate the theoretical analysis. Finally, two possible electronic circuits are given to realize the uncontrolled and the controlled systems.

Analysis of a delayed diffusive model with Beddington–DeAngelis functional response

2019 ◽  
Vol 12 (04) ◽  
pp. 1950047 ◽  
Author(s):  
Xin-You Meng ◽  
Jiao-Guo Wang
Keyword(s):  
Hopf Bifurcation ◽  
Asymptotic Stability ◽  
Normal Form ◽  
Functional Response ◽  
Local Stability ◽  
Center Manifold ◽  
Normal Form Theory ◽  
Diffusive Model ◽  
Form Theory ◽  
Existence Of Equilibria

In this paper, a delayed diffusive phytoplankton-zooplankton model with Beddington–DeAngelis functional response and toxins is investigated. Existence of equilibria of the system are solved. The global asymptotic stability of the zooplankton-free equilibrium is obtained. The local stability of the coexistent equilibrium and existence of Hopf bifurcation are discussed. In addition, the properties of the Hopf bifurcation are studied based on the center manifold and normal form theory for partial differential equations. Finally, some numerical simulations are also carried out to confirm our theoretical analysis.

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