# Chaos and Bifurcation.

## Top PDF Chaos and Bifurcation.: ### Chaos and bifurcation of discontinuous dynamical systems with piecewise constant arguments

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. ### Bifurcation and chaos control in a discrete time predator–prey model with nonlinear saturated incidence rate and parasite interaction

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. ### Chaos and bifurcation of the Logistic discontinuous dynamical systems with piecewise constant arguments

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. ### Bifurcation and Chaos in Delayed Cellular Neural Network Model

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. ### Bifurcation of Parameter Space and Chaos in Mira 2 Map

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 − ### Bifurcation, chaos analysis and control in a discrete time predator–prey system

–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 ### Hybrid Control of Bifurcation and Chaos in Dynamic Model of Compound TCP under RED

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. ### Fractional order Chua’s system: discretization, bifurcation and chaos

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. ### Bifurcation and Chaos of Gear Pair System Supported by Long Journal Bearings Based on Turbulent Flow Effect and Nonlinear Suspension Effect

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- ### Bifurcation and chaos of an SIS model with logistic term

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 . 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. ### Bifurcation Analysis and Chaos in Electronic Genetic Toggle Switch

map (5.1). Which shows the region of bistability . 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. ### Hopf bifurcation and chaos control for a Leslie–Gower type generalist predator model

of delayed feedback control in . 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. ### Chaos control and Hopf bifurcation analysis of the Genesio system with distributed delays feedback

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 ### Discretization of a fractional order ratio-dependent functional response predator-prey model, bifurcation and chaos

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. ### Bifurcation and Chaos in a Parasitoid Host Parasitoid Model

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  ), 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  . ### Bifurcation and chaos in a host parasitoid model with a lower bound for the host

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. ### Bifurcation and chaos of a particle motion system with holonomic constraint

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] ### Chaos Control of a Resource-Economic-Pollution Dynamic System

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. ### Bifurcation Analysis and Fractal Dimensions of a Non- Linear Map

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. 