Hopf Bifurcation

1. INTRODUCTION p53 is a transcription factor that regulates cell cycle and functions as a tumor supressor. The concentration of p53 increases in response to stress signal, such as DNA damage or oncogene activation. p53 induces transcription of several hundred genes ...

We analyze a second-order, nonlinear delay-differential equation with negative feedback. The characteristic equation for the linear stability of the equilibrium is completely solved, as a function of two parameters describing the strength of the feedback and the damping in the autonomous system. The bifurcations occurring as the linear stability is lost are investigated by the construction of a center manifold:

In this paper, a three‐dimensional eco‐epidemiological model with delay is considered. The stability of the two equilibria, the existence of Hopf bifurcation and the permanence are investigated. It is found that Hopf bifurcation occurs when the delay τ passes a sequence of critical values. Moreover, by applying Nyquist criterion, the length of delay is estimated for which the stability continues to hold. Numerical simulation with a hypothetical set of data has been done to support the analytical results. This work is supported by the National Natural Science Foundation of China (No. 10771104 and No.10471117), Program for Innovative Research Team (in Science and Technology) in University of Henan Province (No. 2010IRTSTHN006) and Program for Key Laboratory of Simulation and Control for Population Ecology in Xinyang Normal University (No. 201004) and Natural Science Foundation of the Education Department of Henan Province (No. 2009B1100200 and No. 2010A110017)

We study the dynamics near a symmetric Hopf-zero (also known as saddle-node Hopf or fold-Hopf) bifurcation in a reversible vector field in R3R3, with involutory an reversing symmetry whose fixed point subspace is one-dimensional. We focus on the case in which the normal form for this bifurcation displays a degenerate family of heteroclinics between two asymmetric saddle-foci. We study local perturbations of this degenerate family of heteroclinics within the class of reversible vector fields and establish the generic existence of hyperbolic basic sets (horseshoes), independent of the eigenvalues of the saddle-foci, as well as cascades of bifurcations of periodic, heteroclinic and homoclinic orbits.Finally, we discuss the application of our results to the Michelson system, describing stationary states and travelling waves of the Kuramoto–Sivashinsky PDE.

This work concentrates on the lateral oscillations in vehicles, also called shimmy, with a particular empha- sis on aircraft. A mathematical model of a nose landing gear is discussed with geometric detail that has been mostly neglected in the past research. Stability criteria for the shimmy-free operation of the landing gear are derived using linear stability analysis. Nonlinear analy- sis is used not only to study the qualitative behaviour of the Hopf bifurcation but also to analyze the system be- yond the Hopf bifurcation. The manuscript concludes with suggestions for future research.

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The modified Leslie-Gower and Holling-type II predator-prey model is generalized in the context of ecoepidemiology, with disease spreading only among the prey species. A new feature is introduced, the intraspecific competition of infected prey. All the equilibria are characterized and the existence of a Hopf bifurcation at the coexistence equilibrium is shown.

In this paper, we investigate the problem of bifurcation control for a delayed logistic growth model. By choosing the timedelay as the bifurcation parameter, we present a Proportional - Derivative (PD) Controller to control Hopf bifurcation. We show that the onset of Hopf bifurcation can be delayed or advanced via a PD Controller by setting proper controlling parameter. Under consideration model as operator Equation, apply orthogonal decomposition, compute the center manifold and normal form we determined the direction and stability of bifurcating periodic solutions. Therefore the Hopf bifurcation of the model became controllable to achieve desirable behaviour which are applicable in certain circumstances.

This work deals with the analysis of a predator–prey model derived from the Leslie–Gower type model, where the most common mathematical form to express the Allee effect in the prey growth function is considered. It is well-known that the Leslie–Gower model has a ...