scholarly journals Bifurcation Analysis for a Kind of Nonlinear Finance System with Delayed Feedback and Its Application to Control of Chaos

2012 ◽  
Vol 2012 ◽  
pp. 1-18 ◽  
Author(s):  
Rongyan Zhang
Keyword(s):  
Center Manifold ◽  
Chaotic Oscillation ◽  
Delayed Feedback ◽  
Stable Steady State ◽  
Normal Form Theory ◽  
Finance System ◽  
Stable Periodic ◽  
Form Theory ◽  
The Stability ◽  
Time Delayed Feedback

A kind of nonlinear finance system with time-delayed feedback is considered. Firstly, by employing the polynomial theorem to analyze the distribution of the roots to the associate characteristic equation, the conditions of ensuring the existence of Hopf bifurcation are given. Secondly, by using the normal form theory and center manifold argument, we derive the explicit formulas determining the stability, direction, and other properties of bifurcating periodic solutions. Finally, we give several numerical simulations, which indicate that when the delay passes through certain critical values, chaotic oscillation is converted into a stable steady state or a stable periodic orbit.

Delayed Feedback Control and Bifurcation Analysis in a Chaotic Chemostat System

2015 ◽  
Vol 25 (06) ◽  
pp. 1550087 ◽  
Author(s):  
Zhichao Jiang ◽  
Wanbiao Ma
Keyword(s):  
Center Manifold ◽  
Chaotic Oscillation ◽  
Delayed Feedback ◽  
Delayed Feedback Control ◽  
Stable Steady State ◽  
Positive Equilibrium ◽  
Normal Form Theory ◽  
Stable Periodic ◽  
Form Theory ◽  
The Stability

In this paper, the effect of delay on a nonlinear chaotic chemostat system with delayed feedback is investigated by regarding delay as a parameter. At first, the stability of the positive equilibrium and the existence of Hopf bifurcations are obtained. Then an explicit algorithm for determining the direction and the stability of the bifurcating periodic solutions is derived by using the normal form theory and center manifold argument. Finally, some numerical simulation examples are given, which indicate that the chaotic oscillation can be converted into a stable steady state or a stable periodic orbit when delay passes through certain critical values.

BIFURCATION ANALYSIS AND CHAOS CONTROL FOR LÜ SYSTEM WITH DELAYED FEEDBACK

2007 ◽  
Vol 17 (12) ◽  
pp. 4309-4322 ◽  
Author(s):  
MIN XIAO ◽  
JINDE CAO
Keyword(s):  
Chaotic Oscillation ◽  
Delayed Feedback ◽  
Steady States ◽  
Stable Steady State ◽  
Unstable Periodic Orbits ◽  
Normal Form Theory ◽  
Center Manifold Theorem ◽  
Lü System ◽  
Stable Periodic ◽  
Lu System

Time-delayed feedback has been introduced as a powerful tool to control unstable periodic orbits or control unstable steady states. In the present paper, regarding the delay as a parameter, we investigate the effect of delay on the dynamics of Lü system with delayed feedback. After the effect of delay on the steady states is analyzed, Hopf bifurcation is studied, where the direction, stability and other properties of the bifurcating periodic solutions are determined by using the normal form theory and the center manifold theorem. Finally, we provide several numerical simulations, which indicate that when the delay passes through certain critical values, chaotic oscillation is converted into a stable steady state or a stable periodic orbit.

Bifurcation Analysis and Chaos Control in a Modified Finance System with Delayed Feedback

2016 ◽  
Vol 26 (06) ◽  
pp. 1650105 ◽  
Author(s):  
Jihua Yang ◽  
Erli Zhang ◽  
Mei Liu
Keyword(s):  
Center Manifold ◽  
Numerical Solutions ◽  
Chaotic Oscillation ◽  
Delayed Feedback ◽  
Finance System ◽  
Normal Form Method ◽  
The Interest Rate ◽  
The Stability ◽  
Manifold Theory ◽  
Bifurcating Periodic Solution

We investigate the effect of delayed feedback on the finance system, which describes the time variation of the interest rate, for establishing the fiscal policy. By local stability analysis, we theoretically prove the existences of Hopf bifurcation and Hopf-zero bifurcation. By using the normal form method and center manifold theory, we determine the stability and direction of a bifurcating periodic solution. Finally, we give some numerical solutions, which indicate that when the delay passes through certain critical values, chaotic oscillation is converted into a stable equilibrium or periodic orbit.

Bifurcation Analysis of a Diffusive Activator–Inhibitor Model in Vascular Mesenchymal Cells

2015 ◽  
Vol 25 (10) ◽  
pp. 1530026 ◽  
Author(s):  
Rui Yang ◽  
Yongli Song
Keyword(s):  
Steady State ◽  
Center Manifold ◽  
Normal Forms ◽  
Mesenchymal Cells ◽  
Positive Equilibrium ◽  
Normal Form Theory ◽  
Form Theory ◽  
Spatially Homogeneous ◽  
The Stability ◽  
Activator Inhibitor

In this paper, a diffusive activator–inhibitor model in vascular mesenchymal cells is considered. On one hand, we investigate the stability of the equilibria of the system without diffusion. On the other hand, for the unique positive equilibrium of the system with diffusion the conditions ensuring stability, existence of Hopf and steady state bifurcations are given. By applying the center manifold and normal form theory, the normal forms corresponding to Hopf bifurcation and steady state bifurcation are derived explicitly. Numerical simulations are employed to illustrate where the spatially homogeneous and nonhomogeneous periodic solutions and the steady states can emerge. The numerical results verify the obtained theoretical conclusions.

scholarly journals Stability and Hopf Bifurcation Analysis of a Vector-Borne Disease with Time Delay

2014 ◽  
Vol 2014 ◽  
pp. 1-8
Author(s):  
Yuanyuan Chen ◽  
Ya-Qing Bi
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Bifurcation Theory ◽  
Center Manifold ◽  
Local Analysis ◽  
Normal Form Theory ◽  
Form Theory ◽  
Vector Borne ◽  
The Stability ◽  
Delay Differential

A delay-differential modelling of vector-borne is investigated. Its dynamics are studied in terms of local analysis and Hopf bifurcation theory, and its linear stability and Hopf bifurcation are demonstrated by studying the characteristic equation. The stability and direction of Hopf bifurcation are determined by applying the normal form theory and the center manifold argument.

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 Stability and Bifurcation Analysis of a Modified Epidemic Model for Computer Viruses

2014 ◽  
Vol 2014 ◽  
pp. 1-14 ◽  
Author(s):  
Chuandong Li ◽  
Wenfeng Hu ◽  
Tingwen Huang
Keyword(s):  
Hopf Bifurcation ◽  
Epidemic Model ◽  
Center Manifold ◽  
Dynamical Behavior ◽  
Three Dimensional ◽  
Normal Form Theory ◽  
Computer Viruses ◽  
Stability And Bifurcation ◽  
Form Theory ◽  
The Stability

We extend the three-dimensional SIR model to four-dimensional case and then analyze its dynamical behavior including stability and bifurcation. It is shown that the new model makes a significant improvement to the epidemic model for computer viruses, which is more reasonable than the most existing SIR models. Furthermore, we investigate the stability of the possible equilibrium point and the existence of the Hopf bifurcation with respect to the delay. By analyzing the associated characteristic equation, it is found that Hopf bifurcation occurs when the delay passes through a sequence of critical values. An analytical condition for determining the direction, stability, and other properties of bifurcating periodic solutions is obtained by using the normal form theory and center manifold argument. The obtained results may provide a theoretical foundation to understand the spread of computer viruses and then to minimize virus risks.

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