Hopf bifurcation of a tumor immune model with time delay

In this paper, a tumor immune model with time delay is studied. First, the stability of nonnegative equilibria is analyzed. Then the time delay τ is selected as a bifurcation parameter and the existence of Hopf bifurcation is proved. Finally, by using the canonical method and the central manifold theory, the criteria for judging the direction and stability of Hopf bifurcation are given.     

Topological Aspects of Triplet Superconductors

In this paper, using the Φ-mapping theory, it is shown that two kinds of topological defects, i.e., the vortex lines and the monopoles exist in the helical configuration of magnetic field in triplet superconductors. And the inner topological structure of these defects is studied. Because the knot solitons in the triplet superconductors are characterized by the Hopf invariant, we also establish a relationship between the Hopf invariant and the linking number of knots family, and reveal the inner topological structure of the Hopf invariant.     

Complex Behaviors of a Simple Traffic Model

In this paper, we propose a modified traffic model in which a single car moves through a sequence of traffic lights controlled by a step function instead of a sine function. In contrast to the previous work [Phys. Rev. E 70 (2004)016107], we have investigated in detail the dependence of the behavior on four parameters, ω, α, η, and a1, and given three kinds of bifurcation diagrams, which show three kinds of complex behaviors. We have found that in this model there are chaotic and complex periodic motions, as well as special singularities. We have also analyzed the characteristic of the complex period motion and the essential feature of the singularity.     

Theoretical and experimental study of Chen chaotic system with notch filter feedback control

Since the past two decades, the time delay feedback control method has attracted more and more attention in chaos control studies because of its simplicity and efficiency compared with other chaos control schemes. Recently, it has been proposed to suppress low-dimensional chaos with the notch filter feedback control method, which can be implemented in a laser system. In this work, we have analytically determined the controllable conditions for notch filter feedback controlling of Chen chaotic system in terms of the Hopf bifurcation theory. The conditions for notch filter feedback controlled Chen chaoitc system having a stable limit cycle solution are given. Meanwhile, we also analysed the Hopf bifurcation direction, which is very important for parameter settings in notch filter feedback control applications. Finally, we apply the notch filter feedback control methods to the electronic circuit experiments and numerical simulations based on the theoretical analysis. The controlling results of notch filter feedback control method well prove the feasibility and reliability of the theoretical analysis.     

非即变相位共轭反馈对半导体激光器动态特性的影响

从四波混频产生相位共轭的物理原因出发,定义了相位共轭镜（PCM）的响应时间．建立起非即变相位共轭反馈条件下半导体激光器的外腔模型。以响应时间及频率失调为参变量,对其分岔及噪声等动态行为进行数值分析。结果表明,不考虑噪声影响时,增加相位共轭镜响应时间会使混沌带出现的次数和范围得到较大的抑制,当响应时间增大到1．5ns时,混沌带消失,半导体激光器保持稳定的单周期状态;考虑噪声影响后,随若响应时间的相对强度噪声（RIN）可减小几dB甚至十几dB,产生突变需要的反馈量也增大一个数量级以上,且其频谱的峰值向高频方向移动;另外,由于共轭反馈引起的频率失调低于半导体激光器激射频率3个数量级以上．它只对分岔特性有影响．对相对强度噪声的影响几乎为零。     

Hopf Bifurcation Control of a Hyperchaotic Circuit System

This paper is concerned with the Hopf bifurcation control of a new hyperchaotic circuit system. The stability of the hyperchaotie circuit system depends on a selected control parameter is studied, and the critical value of the system parameter at which Hopf bifurcation occurs is investigated. Theoretical analysis give the stability of the Hopf bifurcation. In particular, washout filter aided feedback controllers are designed for delaying the bifurcation point and ensuring the stability of the bifurcated limit cycles. An important feature of the control laws is that they do not result in any change in the set of equilibria. Computer simulation results are presented to confirm the analytical predictions.     

Hopf bifurcation control via a dynamic state-feedback control

To relocate two Hopf bifurcation points, simultaneously, to any desired locations in n-dimensional nonlinear systems, a novel dynamic state-feedback control law is proposed. Analytical schemes to determine the control gains according to the conditions for the emergence of Hopf bifurcation are derived. To verify the effectiveness of the proposed control law, numerical examples are provided.     

Dynamic response of a growing inclusion in a discrete system

The propagation of a semi-infinite line defect, contained in an infinite square-cell lattice is considered. The defect is composed of particles lighter than those in the ambient lattice and it is assumed this defect propagates with constant speed. Dispersion properties of the lattice are related to waves generated by the propagating defect. In order to determine these properties, the Wiener–Hopf technique is applied. Additional features, related to localisation along the defect are also identified. Analysis of the dispersion relations for this lattice, from the kernel function inside the Wiener–Hopf equation, is carried out. The solution of the Wiener–Hopf equation is presented for the case when an external load is applied corresponding to an energy flux at infinity.