When a severely infectious disease takes place in a region, it will cause a large number of illness. If the suspected people know nothing about the disease, they will be lack of protecting measures, and so the disease will spread quickly. In fact, media coverage and education can help people to take preventive measures in time, and so reduce the contact rate of human beings as we have observed during the spreading of severe acute respiratory syndrome(SARS) during 2002 and 2004. How does the media coverage affect the prevalence and control of the epidemic like SARS? Recently, this subject has attracted the attentions of many researchers [3-9]. Liu et al.first emphasized media impact in an EIH model, where H denotes hospitalized individuals, and assumed a transmission coefficient of the exponential form in [3]. Cui et al. constructed an SEI model with logistic growth, in which another contact transmission rate with exponential form µ e −mI was proposed to describe the media impact on the infectious diseases. When the basic reproduction number R 0 > 1, it was shown that there exists a unique endemic equilibrium and a **Hopf** **bifurcation** can occur under the less media impact (m > 0 is sufficiently small) while the model may have up to three endemic equilibria if the media impact is stronger enough in [6]. It is well known that time delays are inevitable in population interactions and tend to be destabilizing in the sense that longer delays may destroy the stability of positive equilibria. One often introduces time delays in the variables being modeled, this often yields delay differential models [10-14, 17-18].

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destabilizing process, in the sense that increasing the delay could cause the bio-economics to fluctuate. Formulas about the stability of bifurcat- ing periodic solution and the direction of **Hopf** **bifurcation** are exhibited by applying the normal form theory and the center manifold theorem. Numerical simulations are given to illustrate the results.

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Following the idea of Hassard [34], in this section, we investigate the direction and stabil- ity of the **Hopf** **bifurcation** at the critical value τ 2 ∗ by using the normal form theory and the center manifold theorem. Throughout this section, we assume that τ 1∗ < τ 1 ∗ , where τ 1∗ ∈ (0, τ 10 ). Let τ 2 = τ 2 ∗ + μ (μ ∈ R), u 1 = S(τ 2 t), u 2 = E 1 (τ 2 t), u 3 = E 2 (τ 2 t), and u 4 = I(τ 2 t).

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The paper is organized as follows. In Section , a producer-scrounger model with age- structure in scrounger is derived. In Section , the basic theory, including existence, uniqueness, positivity and boundedness of solutions for the model are discussed. The stability of equilibriums, uniform persistence, and **Hopf** **bifurcation** are investigated in Section . Some numerical simulations and concluding discussions are given in the last section.

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In this paper, the Turing–**Hopf** **bifurcation** of a ratio-dependent predator-prey model with diﬀusion and Neumann boundary condition is considered. Firstly, we present a kind of double parameters selection method, which can be used to analyze the Turing–**Hopf** **bifurcation** of a general reaction-diﬀusion equation under Neumann boundary condition. By analyzing the distribution of eigenvalues, the stable region, the unstable region (including Turing unstable region), and Turing–**Hopf** **bifurcation** point are derived in a double parameters plane. Secondly, by applying this method, the Turing–**Hopf** **bifurcation** of a ratio-dependent predator-prey model with diﬀusion is investigated. Finally, we compute normal forms near Turing–**Hopf** singularity and verify the theoretical analysis by numerical simulations.

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In this paper, local stability and **Hopf** **bifurcation** of a delayed heroin model with saturated treatment function are discussed. First of all, suﬃcient conditions for local stability and existence of **Hopf** **bifurcation** are obtained by regarding the time delay as a **bifurcation** parameter and analyzing the distribution of the roots of the associated characteristic equation. Directly afterward, properties of the **Hopf** **bifurcation**, such as the direction and stability, are investigated with the aid of the normal form theory and the manifold center theorem. Finally, numerical simulations are presented to justify the obtained theoretical results, and some suggestions are oﬀered for controlling heroin abuse in populations.

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By introducing a delayed fractional-order diﬀerential equation model, we deal with the dynamics of the stability and **Hopf** **bifurcation** of a paddy ecosystem with three main components: rice, weeds, and inorganic fertilizer. In the system, there exists an equilibrium for rice and weeds extinction and an equilibrium for rice extinction or weeds extinction. We obtain suﬃcient conditions for the stability and **Hopf** **bifurcation** by analyzing their characteristic equation. Some numerical simulations validate our theoretical results.

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to obtain more details of the **Hopf** **bifurcation** at (x ∗ , y ∗ ), we need to do a further analysis to system (1.2). Let x = x – x ∗ , y = y – y ∗ , we transform the equilibrium (x ∗ , y ∗ ) of system (1.2) to (0, 0) of a new system. For the sake of simplicity, we denote x, y by x, y, respectively. Thus, system (1.2) is transformed to

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Furthermore, **Hopf** **bifurcation** theory can be utilized as an important tool for the determination of the flutter and limit cycle vibrations of panels. In addition, **Hopf** **bifurcation** tool can be used to analyze the flutter speed of the system. Hence, with the use of thin panels in shuttles and large space stations, nonlinear dynamics, bifurcations, and the chaos of thin panels have become more and more important. In the past decade, researchers have made a number of studies into nonlinear oscillations, bifurcations, and the chaos of thin panels. Holmes [5] studied flow-induced oscillations and bifurcations of thin panels and gave a finite-dimensional analysis. Then based on the analysis in [5], Holmes and Marsden [7] considered an infinite-dimensional analysis for flow-induced oscillations and pitchfork and fold bifurcations of thin panels. Holmes [6] then simplified this problem to a two-degrees-of-freedom nonlinear system and used center manifolds and the theory of normal forms to study the degenerate bifurcations. Yang and Sethna [18] used an averaging method to study the local and global bifurcations in parametrically excited, nearly square plates. From the von Karman equation, they simplified this system to a parametrically excited two-degrees-of-freedom nonlinear oscillators and analyzed the global behaviour of the averaged equations. Based on the stud- ies in [18], Feng and Sethna [3] made use of the global perturbation method developed by Kovacic and Wiggins [8] to study further the global bifurcations and chaotic dynamics of a thin panel under parametric excitation, and ob- tained the conditions in which Silnikov-type homoclinic orbits and chaos can occur. Zhang et al. [19] investigated both the local and global bifurcations of a simply supported at the fore-edge, rectangular thin plate subjected to transversal and in-plane excitations simultaneously.

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We consider a time delay predator-prey model with Holling type-IV functional response and stage-structured for the prey. Our aim is to observe the dynamics of this model under the inﬂuence of gestation delay of the predator. We obtain suﬃcient conditions for the local stability of each of feasible equilibria of the system and the existence of a **Hopf** **bifurcation** at the coexistence equilibrium. By using the normal form theory and center manifold theory we also derive some explicit formulae determining the **bifurcation** direction and the stability of the bifurcated periodic solutions. Finally, numerical simulations are given to explain the theoretical results.

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The objective is apply **Hopf** **Bifurcation** Control to the system (3) in order to change the **Hopf** **bifurcation** from supercritical to subcritical, by using non-lineal control laws in state feedback. The used method for **Hopf** **bifurcation** analisys include the **Hopf** Theorem, the Central Manifold Theorem, Normal Forms, and a formule for the first Lyapunov coefficient given in [13].

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This work explored the center manifold reduction of a **Hopf**-**Hopf** **bifurcation** in a nonlinear differential delay equation. When analyzing a system of coupled oscillators that separately undergo **Hopf** **bifurcation**, there exists the possibility of the full system to undergo this codimension 2 **bifurcation**. In doing so, a wealth of sophisticated dynamics may arise that are not immediately anticipated, for instance the quasiperiodic motions. This work has served to rigorously show that a system inspired by the physical application of delay-coupled microbubble oscillators exhibits quasiperiodic motions because in part of the occurrence of a **Hopf**-**Hopf** **bifurcation**.

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In recent years we have witnessed an increasing interest in dynamical systems with time delays, especially in applied mathematics. Stability and direction of the **Hopf** **bifurcation** for the predator-prey system have been discussed by using normal form theory and center manifold theory [5,6,10,13,15,16,17]. Direction and stability of the equilibrium for a neural network model with two delays have been investigated [7,12]. **Bifurcation** analysis of the predtor-prey model has been detailed [8]. Direction and stability of the equilibrium involving various fields have been discussed [4,9,11,14]. We reported the shifting of algal dominated reef ecosystem due to the invasion of KA in Gulf of Mannar [1]. Subsequently, the dominance of KA over NA and corals in competing for space has also been reported. KA sexual reproduction by spores in the Gulf of Mannar Marine Biosphere Reserve (GoM) in future, when environmental conditions unanimously favor this alga has been deliberated [2]. To simulate the three way competition among corals, KA and NA, we proposed the following system of non-linear ODE’s [3].

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The rest of the paper is arranged as follows, the linear stability of the model and the local **Hopf** **bifurcation** are studied and the conditions for the stability and the existence of **Hopf** **bifurcation** at the equilibrium are derived in section 2. In section 3, according to the method of theory and applications of **Hopf** **bifurcation** by Hassard et al. [ 7 ] , the direction and stability of bifurcating periodic solutions are investigated. In section 4, the correctness of theoretical analysis are confirmed by some numerical simulation results. At last, some conclusions are obtained in section 5.

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In this paper we have investigated the stability nature of **Hopf** **bifurcation** in a two dimensional nonlinear differential equation, popularly known as the Brusselator model. The Brusselator model exhibits supercritical **Hopf** **bifurcation** for certain parameter values which marks the stability of limit cycles created in **Hopf** bifurcations. We have used the Center manifold theorem and the technique of Normal forms in our investigation.

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The paper is organized as follows. In section 2 we investigate the local stability of the equilibrium point associated to system (2). Choosing the delay as a **bifurcation** parameter some sufficient conditions for the existence of **Hopf** **bifurcation** are found. In section 3 there is the main aim of the paper, namely the direction, the stability and the period of a limit cycle solution. Section 4 gives some numerical simulations which show the existence and the nature of the periodic solutions. Finally, some conclusions are given.

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The structure of this paper is as follows. In the next section, we study the local stabil- ity of the viral equilibrium of system (2) and the existence of a local **Hopf** **bifurcation** of system (2). In Sect. 3, properties of the **Hopf** **bifurcation** are investigated. In Sect. 4, some numerical simulations are presented in order to verify our obtained theoretical results. Section 5 summarizes this work.

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The rest of this paper is organized as follows. In Sect. 2, distribution of equilibria of system (2) is presented. The problem of zero-**Hopf** **bifurcation** of system (2) is addressed in Sect. 3. In Sect. 4, the classical **Hopf** **bifurcation** is studied to illustrate the existence of periodic solution. Finally, we conclude this paper in Sect. 5.

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rium. In Section , we discuss the direction and stability of the **Hopf** **bifurcation** by using the normal form theory and center manifold theorem. In Section , we give an interest- ing numerical analysis to illustrate the main results. In the last section, we make a brief summary.

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then the positive equilibrium E is unstable, and system (4.1) undergoes a **Hopf** **bifurcation** at E, and a family of periodic solutions bifurcate from the positive equilibrium E. This property can be illustrated by in Figs. 1–3. Further, we can compute the values

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