Contingent solutions to the center manifold equation

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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Reduction principle at work

Nonautonomous Dynamical Systems ◽

10.1515/msds-2020-0124 ◽

2021 ◽

Vol 8 (1) ◽

pp. 46-74

Author(s):

Christian Pötzsche ◽

Evamaria Russ

Keyword(s):

Center Manifold ◽

Reduction Principle ◽

Critical Stability ◽

Center Manifold Reduction ◽

Infinite Dimensional ◽

Linearized Stability ◽

Dichotomy Spectrum ◽

Nonautonomous Case ◽

Necessary And Sufficient ◽

Manifold Reduction

Abstract The purpose of this informal paper is three-fold: First, filling a gap in the literature, we provide a (necessary and sufficient) principle of linearized stability for nonautonomous difference equations in Banach spaces based on the dichotomy spectrum. Second, complementing the above, we survey and exemplify an ambient nonautonomous and infinite-dimensional center manifold reduction, that is Pliss’s reduction principle suitable for critical stability situations. Third, these results are applied to integrodifference equations of Hammerstein- and Urysohn-type both in C- and Lp -spaces. Specific features of the nonautonomous case are underlined. Yet, for the simpler situation of periodic time-dependence even explicit computations are feasible.

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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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Dynamical Behavior in a Four-Dimensional Neural Network Model with Delay

Advances in Artificial Neural Systems ◽

10.1155/2012/397146 ◽

2012 ◽

Vol 2012 ◽

pp. 1-11

Author(s):

Changjin Xu ◽

Peiluan Li

Keyword(s):

Neural Network ◽

Hopf Bifurcation ◽

Network Model ◽

Neural Network Model ◽

Center Manifold ◽

Dynamical Behavior ◽

Normal Form Theory ◽

Form Theory ◽

Explicit Formulae ◽

Delay Differential

A four-dimensional neural network model with delay is investigated. With the help of the theory of delay differential equation and Hopf bifurcation, the conditions of the equilibrium undergoing Hopf bifurcation are worked out by choosing the delay as parameter. Applying the normal form theory and the center manifold argument, we derive the explicit formulae for determining the properties of the bifurcating periodic solutions. Numerical simulations are performed to illustrate the analytical results.

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

Physica D Nonlinear Phenomena ◽

10.1016/j.physd.2021.133007 ◽

2021 ◽

Vol 427 ◽

pp. 133007

Author(s):

B. Haasdonk ◽

B. Hamzi ◽

G. Santin ◽

D. Wittwar

Keyword(s):

Kernel Methods ◽

Center Manifold ◽

Center Manifold Theorem

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Numerical Center Manifold Methods

Patterns of Dynamics - Springer Proceedings in Mathematics & Statistics ◽

10.1007/978-3-319-64173-7_15 ◽

2017 ◽

pp. 242-268

Author(s):

Klaus Böhmer

Keyword(s):

Center Manifold

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Application of Center Manifold Theory to Regulation of a Flexible Beam

Journal of Vibration and Acoustics ◽

10.1115/1.2889697 ◽

1997 ◽

Vol 119 (2) ◽

pp. 158-165 ◽

Cited By ~ 6

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.

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Coexistence stability in a four-member hypercycle with error tail through center manifold analysis

Nonlinear Dynamics ◽

10.1007/s11071-017-3769-6 ◽

2017 ◽

Vol 90 (3) ◽

pp. 1873-1883 ◽

Cited By ~ 2

Author(s):

Gerard Farré ◽

Josep Sardanyés ◽

Antoni Guillamon ◽

Ernest Fontich

Keyword(s):

Center Manifold

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Hopf Bifurcation Induced by Delay Effect in a Diffusive Tumor-Immune System

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127418501365 ◽

2018 ◽

Vol 28 (11) ◽

pp. 1850136 ◽

Cited By ~ 1

Author(s):

Ben Niu ◽

Yuxiao Guo ◽

Yanfei Du

Keyword(s):

Immune System ◽

Hopf Bifurcation ◽

Periodic Solutions ◽

Center Manifold ◽

Immune Cell ◽

Tumor Treatment ◽

Delay Effect ◽

Stable Periodic ◽

The Stability ◽

Bifurcation And Stability

Tumor-immune interaction plays an important role in the tumor treatment. We analyze the stability of steady states in a diffusive tumor-immune model with response and proliferation delay [Formula: see text] of immune system where the immune cell has a probability [Formula: see text] in killing tumor cells. We find increasing time delay [Formula: see text] destabilizes the positive steady state and induces Hopf bifurcations. The criticality of Hopf bifurcation is investigated by deriving normal forms on the center manifold, then the direction of bifurcation and stability of bifurcating periodic solutions are determined. Using a group of parameters to simulate the system, stable periodic solutions are found near the Hopf bifurcation. The effect of killing probability [Formula: see text] on Hopf bifurcation values is also discussed.

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