In this paper we are concerned with the definition and some properties of the discontinuous dynamical systems generated by piecewise constant arguments. Then we study a discontinuous dynamical system of the Riccati type equation as an example. The local stability at the fixed points is studied. The **bifurcation** analysis and **chaos** are discussed. In addition, we compare our results with the discrete dynamical system of the Riccati type equation.

where x(t), y(t), and z(t) denote densities of susceptible Tilapia ﬁsh population, infected Tilapia ﬁsh population and predator population known as Pelican birds, respectively. Moreover, x and y are the state variables to represent the susceptible prey population den- sity and infected prey population density, respectively, at time t, and z represents predator population density at time t. Furthermore, it is supposed that the population for prey obeys the logistic growth function where s is taken as the growth rate and k represents the car- rying capacity. Moreover, α denotes the coeﬃcient for disease transmission, μ represents the death rate of the infected Tilapia ﬁsh population, δ is a constant for half-saturation rate, β is the capturing capacity of predators, γ is used for the conversion eﬃciency of predator, and η represents the food-dependent death rate related to the predator popu- lation. For more investigations and modiﬁcations related to system (1.1) we refer to [8– 12]. All these modiﬁcations and discussions are implemented to continuous counterparts of system (1.1). Hence it is worthwhile to discuss discrete counterparts of system (1.1) in which populations are treated as generations of non-overlapping type. Furthermore, there are a lot of mathematical investigations related to ﬂip **bifurcation**, Hopf **bifurcation** and controlling **chaos** in planar discrete-time models and a little work is performed for controlling **chaos** and **bifurcation** analysis for discrete-time systems of three or more di- mensions. Hence, it is worthwhile to discuss the **bifurcation** analysis and controlling **chaos** in the case of a three-dimensional discrete-time systems.

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In this paper we are concerned with the definition and some properties of the discontinuous dynamical systems generated by piecewise constant arguments. Then we study two discontinuous dynamical system of the Logistic equation as an example. The local stability at the fixed points is studied. The **bifurcation** analysis and **chaos** are discussed. In addition, we compare our results with the discrete dynamical systems of the Logistic equation.

This paper is structured as follows: In Section 2 we analyze a new family of simplified 4-CNNs, as shown in Figure 1. In Section 3, we investigate different dynamical behaviors including **chaos** for different range of weight parameters of the system. Using Matlab software we can observe limit cycle, torus and **chaos** in a new family of 4-CNNs for different range of adjustable parameters of main diagonal of the weight matrix. Numerical results are discussed showing changes of dynamics of the system from unstable to stable (**chaos** control) one through **bifurcation** due to variation in the individual weight parameter. In Section 4, a delayed Cellular Neural Networks (DCNNs) is considered where the chaotic dynamics is controlled [see Figure 2]. A sufficient condition for the Global asymptotic stability of the equilibrium point in the delayed model is found. Finally some concluding remarks have been drawn on the implication of our results in the context of related work mentioned above.

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For case (2) The **bifurcation** diagram of Mira 2 map (1) for A = 0.1 in ( , ) B x plane and the corresponding maximal Lyapunov exponents are given in Figure 4(a) and Figure 4(b), respectively. In Figure 4(a), Mira 2 map (1) un- dergoes a Naimark-Sacker **bifurcation** from period-1 orbit at B = 0.7 − . At B decreasing to B = 1.0022 − , quasi-period region suddenly disappears and six pieces of period-doubling to **chaos** occur. In the interval B ∈ − ( 1.705, 1.22) − , period-doubling, Naimark-Sacker **bifurcation** and quasi-period behaviors are immersed in **chaos** region. The phase portraits of Mira 2 map (1) are shown in Figures 4(c)-(g), respectively. In Figures 4(c)-(e), the size of chaotic attractors at B = 1.29 −

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–0.09270059512 + 0.1894262031i, and L = 0.1558240975 > 0. The numerical simulation result is shown in Fig. 4, which conﬁrms the existence of a repelling closed orbit in the phase space for s = 0.12. On the basis of the above analysis, we deduce that the hybrid control strategy can successfully delay the appearance of the Neimark–Sacker **bifurcation**. 4 Existence of **chaos** in the sense of Marotto and **chaos** control

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Abstract: Compound TCP(CTCP) is a practical congestion control algorithm for high-speed and long delay networks. In this paper we address the problem of **bifurcation** and **chaos** analysis and its control of a nonlinear congestion control model with CTCP connections and random early detection (RED) gateway. First, we briefly analyze the nonlinear dynamics of the CTCP network with respect to system parameters. Then a hybrid control strategy using both state feedback and parameter perturbation is employed to control the **bifurcation** and stabilize the chaotic orbits embedded in this discrete-time dynamic system of CTCP/RED. Theoretical analysis show that the controlled system can achieve a stable average queue size in an extended range of parameters and even completely eliminates the chaotic behavior in a particular range of parameters. Therefore it is possible to decrease the sensitivity of RED parameters. Finally, numerical simulations are provided to show the effectiveness of the theoretical results.

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A discretization process will be applied to obtain its discrete version. Fixed points and their asymptotic stability are investigated. Chaotic attractor, **bifurcation** and **chaos** for diﬀerent values of the fractional-order parameter are discussed. We show that the proposed discretization method is diﬀerent from other discretization methods, such as predictor-corrector and Euler methods, in the sense that our method is an approximation for the right-hand side of the system under study.

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In practical bearing systems, the rotational speed ratio s is commonly used as a control parameter. Accordingly, the dynamic behavior of the current gear-bearing system was examined using the dimensionless rotational speed ratio s as a **bifurcation** control parameter. Figure 3 pre- sents the **bifurcation** diagrams for the gear-bearing sys- tem displacement against the dimensionless rotational speed ratio, s and some dynamic trajectories and Poin- caré maps (e.g. s = 0.9, 1.0, 1.2, 1.6 and 2.0) are exem- plified to describe corresponding dynamic responses. The **bifurcation** diagrams show that the geometric center of gear and bearing perform synchronous 1T-periodic mo- tion at low values of the rotational speed ratio, i.e. s < 0.9 and then the chaotic motion can be found as the dimen- sionless rotational speed ratio is increased over s = 0.9 for bearing center. Nevertheless, gear center performs quasi-periodic motion at s = 0.9. We may also make sure of the chaotic dynamic responses occurring for bearing center at i = 0.9 from observing the dynamic orbit (dis- ordered trajectory) and Poincaré maps (irregularly-distri- buted points). At higher values of the dimensionless rota- tional speed ratio, i.e. s > 0.9, the dynamics of the centers of bearing 1 and bearing 2 behave as **chaos**, but gear center and pinion center are found to be quasi-periodic before the dimensionless rotational speed ratio s > 1.8. Then dynamic behavior of gear center and pinion center become chaotic while s ≥ 1.8. Therefore, we can find that the dynamic behaviors of geometric centers of bear- ing 1 (or bearing 2) and gear (or pinion) are not syn-

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analysis and numerical simulations have demonstrated that the model exhibits the variety of dynamical behaviors, which include the discrete epidemic model undergoes transcritical bifur- cation, flip **bifurcation**, Hopf **bifurcation** and **chaos**. The results show that there are different dynamical behaviors between discrete system and its corresponding continuous system and the results are different from [23]. Furthermore, **chaos** can cause the population to run a higher risk of extinction due to the unpredictably [24-25]. Thus, how to control **chaos** in the epidemic model is very important, which needs further consideration.

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map (5.1). Which shows the region of bistability [13]. In the fig. 3 by varying the parameter β we can see that the rate increase and the shift time is also increased. In the fig. 5 shows the **bifurcation** diagram clearly more complex dynamical than fig 4, where according to the **bifurcation** phenomena leads to **chaos**, this chaotic behavior brings interior crisis. So the selection of β and γ should be very wise.

of delayed feedback control in [16]. Their results show that, when the controlling param- eter K is some value, taking the delay as the **bifurcation** parameter, then passing through a certain critical value, the stability of the equilibrium will be changed from unstable to stable, **chaos** will vanish, and a periodic solution will emerge.

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In order to better model some complicated practical phenomena, recently, distributed time delay has been introduced into many modeling systems. There are extensive literature works dealing with such systems [–]. As the distributed time delay is incorporated in a system, some interesting dynamical behaviors occur near the equilibrium point. Inspired by these previous works, in this paper, we intend to introduce the distributed time delay as a feedback controller into the chaotic Genesio system with the aim to realize the control of **chaos**. The rest of this paper is organized as follows. In the next section, we present the mathematical models of the Genesio system with distributed time delay feedback and consider its local stability and Hopf **bifurcation**. In Section , the stability of the bifurcating periodic solutions and the direction of the Hopf **bifurcation** at the critical values of mean time delay are determined by using the normal form method and the center manifold

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The organization of the paper is as follows. In Section 2, after some fractional calculus preliminaries, we review some results for relations between properties of equilibria and possibility of existence of **chaos** in a fractional order system. In Section 3, we determine the equilibria of the model and then the discretization process of the system is given. In Section 4, we study the local stability of the equilibria, and we investigate the dynamics of the discretized model. Section 5 is devoted to some numerical simulations and **bifurcation** diagrams, to support the analytic results.

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In our paper, the parasitoid-host-parasitoid system (2) is investigated in fur- ther details. We mainly focus on its bifurcations and possible **chaos** qualitatively. Based on the center manifold theorem and **bifurcation** theory (see [12] [13]), we can obtain the detailed existence conditions of these bifurcations. Numerical si- mulations, including **bifurcation** diagrams, phase portraits, are used to verify theoretical analysis. The results obtained in the paper can be regarded as the beneficial supplement of the work in [10] [11].

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parametric condition system (1.2) admits a **bifurcation** in Section 3. Numerical simula- tions using MATLAB are applied in Section 4 to support the theoretical analyses and vi- sualize the newly observed complex dynamics of the system. These ﬁndings prove that there are possibilities for periodic and chaotic motions to exist in the parameter space. In addition, the phase diagrams of two control parameters are also presented. Finally, com- ments and conclusions are summarized.

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rich dynamic behaviors of the particle motion system, including the degenerate Hopf bifurcations at multiple.. equilibrium points, the chaotic behaviors of the particle motion1[r]

How to control **chaos** in the economic system has aroused the interest of researchers. We research the **chaos** control in a new Resource-Economic-Pollution system by time-delayed feedback control. By determining the appropriate range of time delay τ and feedback strength k, the chaotic phenomena of the system are controlled. We verify the linear stability and the existence of Hopf **bifurcation** of the system. Numerical simulations show that **chaos** control can eliminate the chaotic behavior of the system and stabilize the system at the equilibrium point. When the time lag term is in a certain interval, the chaotic phenomenon of the system will disappear and the system will be controlled in a stable state. In practice, due to capacity and financial constraints, the firm or the government often restrains output through many methods to confine the range of fluctuations in these variables. This shows that the government or corporate decision makers have often used this approach consciously or unconsciously to promote steady economic growth.

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It is a visual summery of the succession of period doubling produced as the parameter increases. Initially the map has one stable fixed point up to certain value of the parameter “a”. The **bifurcation** diagram nicely shows the forking of the periods of stable orbits from 1 to 2, then 2 to 4 etc. The interesting thing about the diagram is that as the periods go to infinity, still the parameter remains finite. For further investigation numerical procedure is adopted to get the **bifurcation** point, which may help to confirm **chaos**. From the diagram it has been clear that the map follows period doubling route to **chaos** as “a” is varied.

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Nowadays the dc to dc converters are superior converters with renewable energy source. To get the pollution free environment, these converters plays a major role in DC micro grid applications and Electric Vehicle applications. The nonlinear analysis like **bifurcation** and **chaos** in dc to dc converter are used to study the complex behavior under load and line variation. This complex behavior commonly observed in the basic power electronic converters to find the stability region. By varying the different **bifurcation** parameters like load resistance, reference current and voltage etc., the converter loses its stability [1]-[4].

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