Kernel methods for center manifold approximation and a weak data-based version of the Center Manifold Theorem

Abstract We investigate a delayed epidemic model for the propagation of worm in wireless sensor network with two latent periods. We derive sufficient conditions for local stability of the worm-induced equilibrium of the system and the existence of Hopf bifurcation by regarding different combination of two latent time delays as the bifurcation parameter and analyzing the distribution of roots of the associated characteristic equation. In particular, we investigate the direction and stability of the Hopf bifurcation by means of the normal form theory and center manifold theorem. To verify analytical results, we present numerical simulations. Also, the effect of some influential parameters of sensor network is properly executed so that the oscillations can be reduced and removed from the network.

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Dynamical analysis of a giving up smoking model with time delay

Advances in Difference Equations ◽

10.1186/s13662-019-2450-4 ◽

2019 ◽

Vol 2019 (1) ◽

Cited By ~ 4

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.

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Stability and Hopf Bifurcation Analysis of Coupled Optoelectronic Feedback Loops

Abstract and Applied Analysis ◽

10.1155/2013/918943 ◽

2013 ◽

Vol 2013 ◽

pp. 1-11

Author(s):

Gang Zhu ◽

Junjie Wei

Keyword(s):

Hopf Bifurcation ◽

Normal Form ◽

Center Manifold ◽

Stable Equilibrium ◽

Feedback Loops ◽

Asymptotically Stable ◽

Center Manifold Theorem ◽

Feedback Strength ◽

Coupling Parameters ◽

Normal Form Method

The dynamics of a coupled optoelectronic feedback loops are investigated. Depending on the coupling parameters and the feedback strength, the system exhibits synchronized asymptotically stable equilibrium and Hopf bifurcation. Employing the center manifold theorem and normal form method introduced by Hassard et al. (1981), we give an algorithm for determining the Hopf bifurcation properties.

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Hopf Bifurcation of a Delayed Ecoepidemic Model with Ratio-Dependent Transmission Rate

Journal of Function Spaces ◽

10.1155/2018/5626039 ◽

2018 ◽

Vol 2018 ◽

pp. 1-10

Author(s):

Wanjun Xia ◽

Soumen Kundu ◽

Sarit Maitra

Keyword(s):

Hopf Bifurcation ◽

Time Delay ◽

Center Manifold ◽

Transmission Rate ◽

Sufficient Conditions ◽

Normal Form Theory ◽

Center Manifold Theorem ◽

Form Theory ◽

Ratio Dependent ◽

Theoretical Results

A delayed ecoepidemic model with ratio-dependent transmission rate has been proposed in this paper. Effects of the time delay due to the gestation of the predator are the main focus of our work. Sufficient conditions for local stability and existence of a Hopf bifurcation of the model are derived by regarding the time delay as the bifurcation parameter. Furthermore, properties of the Hopf bifurcation are investigated by using the normal form theory and the center manifold theorem. Finally, numerical simulations are carried out in order to validate our obtained theoretical results.

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Bifurcation Analysis in a Delayed Diffusive Leslie-Gower Model

Discrete Dynamics in Nature and Society ◽

10.1155/2013/170501 ◽

2013 ◽

Vol 2013 ◽

pp. 1-11 ◽

Cited By ~ 1

Author(s):

Shuling Yan ◽

Xinze Lian ◽

Weiming Wang ◽

Youbin Wang

Keyword(s):

Stability Analysis ◽

Hopf Bifurcation ◽

Normal Form ◽

Bifurcation Analysis ◽

Center Manifold ◽

Neumann Boundary ◽

Positive Equilibrium ◽

Center Manifold Theorem ◽

Normal Form Theorem ◽

The Stability

We investigate a modified delayed Leslie-Gower model under homogeneous Neumann boundary conditions. We give the stability analysis of the equilibria of the model and show the existence of Hopf bifurcation at the positive equilibrium under some conditions. Furthermore, we investigate the stability and direction of bifurcating periodic orbits by using normal form theorem and the center manifold theorem.

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STABILITY AND BIFURCATION ANALYSIS ON A DELAYED NEURAL NETWORK MODEL

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127405013757 ◽

2005 ◽

Vol 15 (09) ◽

pp. 2883-2893 ◽

Cited By ~ 12

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.

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Hopf Bifurcation and Chaos of a Delayed Finance System

Complexity ◽

10.1155/2019/6715036 ◽

2019 ◽

Vol 2019 ◽

pp. 1-18 ◽

Cited By ~ 6

Author(s):

Xuebing Zhang ◽

Honglan Zhu

Keyword(s):

Numerical Simulation ◽

Hopf Bifurcation ◽

Center Manifold ◽

Chaotic State ◽

Bifurcation And Chaos ◽

Finance System ◽

Center Manifold Theorem ◽

Normal Form Method ◽

Simulation Results ◽

The Stability

In this paper, a finance system with delay is considered. By analyzing the corresponding characteristic equations, the local stability of equilibrium is established. The existence of Hopf bifurcations at the equilibrium is also discussed. Furthermore, formulas for determining the direction of Hopf bifurcation and the stability of the bifurcating periodic solutions are derived by applying the normal form method and center manifold theorem. Finally, numerical simulation results are presented to validate the theoretical analysis. Numerical simulation results show that delay can lead a stable system into a chaotic state.

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

Complexity ◽

10.1155/2019/3492589 ◽

2019 ◽

Vol 2019 ◽

pp. 1-17 ◽

Cited By ~ 1

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.

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

Abstract and Applied Analysis ◽

10.1155/2014/624162 ◽

2014 ◽

Vol 2014 ◽

pp. 1-12 ◽

Cited By ~ 1

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.

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Center Manifold Reduction and Perturbation Method in a Delayed Model with a Mound-Shaped Cobb-Douglas Production Function

Abstract and Applied Analysis ◽

10.1155/2013/738460 ◽

2013 ◽

Vol 2013 ◽

pp. 1-6 ◽

Cited By ~ 8

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.

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