Effective Control and Bifurcation Analysis in a Chaotic System with Distributed Delay Feedback

α ∈ (α 0 ,∞); Hopf bifurcation occurs when α crosses a critical value α 0 by choosing α as a bifurcation parameter. Meanwhile, the explicit algorithm for determining the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions is derived by normal form theorem and center manifold argument. Furthermore, regarding α as a bifurcation parameter, we explore variation tendency of the dynamics behavior of a chaotic system with the increase of the parameter value α.


Introduction
As one of the important discoveries in 21st century, chaos has been extensively investigated in many fields over the last several decades, which has been widely applied in secure communication, signal processing, radar, image processing, power system protection, flow dynamics, and so on.As is known chaos is undesirable and needs to be controlled in many practical applications.Therefore, the investigation of controlling chaos is of great significance.Many schemes have been presented to carry out chaos control [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17] of which using time-delayed controlling forces proves to be a simple and viable method for a continuous dynamical system.
In the controller, one can see that when || is large enough, chaos attractor disappears and the stable family of limit cycles appears; when || is small enough, a complete chaos attractor appears.
With purpose of reflecting and controlling the complex and unpredictable dynamical behavior of the model depending on the past information of the system, it is necessary to incorporate time delay into this system.The signal error of current state and past state of the continuous time system will be given distributed delay feedback to the system itself.
As compared with the former method, a chaotic model with distributed delay feedback is more general than that with discrete delay feedback [10][11][12][13], because the distributed delay becomes a discrete delay when the delay kernel is a delta function at a certain time.The distributed delay has found widespread applications in many fields such as neural network [14,16], complicated real models [15], and the modeling of aggregative processes involving the flow of entities with random transit times through a given process [17].Therefore, it is of considerable significance to propose distributed delays as control input to control the chaotic system.
Many studies have been made in [18,19] about the Lorenzlike system.In this paper, we present the Hopf bifurcation of a Lorenz-like system with the distributed delay.We not only display numerical simulation of Hopf bifurcation in delayed feedback Lorenz-like system, but also give the theoretical proof.The stability of the equilibrium point will vary complicatedly setting different values of the feedback intensity coefficient .Regarding the delay variable  as a branch of parameters, when  passes through a critical value, the stability of the equilibrium point will change from instability to stability, and then the chaos phenomenon of the system disappears.Finally, the stable periodic solution emerges.Furthermore, flat malicious point of the original system will not change and retain some features of the original system.Motivated by the above works, we, in this paper, add a time-delayed force  ∫ 0 −∞ [() − ( + )](−) to the third equation of system (1); then system (1) takes the form where , , , ℎ, ,  > 0, ∫ The rest of this paper is organized as follows.In Section 2, the stability and the existence of Hopf bifurcation are determined.In Section 3, based on the normal form method and the center manifold theorem presented in Hassard et al. [20], the direction, stability, and the period of the bifurcating period solutions are analyzed.In Section 4, numerical simulations are given to verify the theorem analysis.Finally, Section 5 concludes some discussions.

Bifurcation Analysis of Lorenz-Like System
In this section, we will study the stability of the equilibria and the existential conditions of local Hopf bifurcations.As the Lorena-like system (3) is symmetric about the -axis,  1 and  2 have the same stability.It is sufficient to discuss the stability of equilibrium  1 ( * ,  * ,  * ).

By the linear transform
The linearization of (5) near ( * ,  * ,  * ) is given by The Jacobian matrix of (6) at  1 ( * ,  * ,  * ) is written as whose characteristic equation is given by In this paper, we focus on considering the weak kernel case; that is, () =  − , where  > 0. As to the general gamma kernel case, we can make a similar analysis.We give the initial condition of system (6) as The characteristic equation (8) with the weak kernel case takes the form where In view of the well known Routh-Hurwitz criterion, we can conclude that all the roots of (10) have negative real parts if the following conclusions hold: Based on the analysis above, we can easily obtain the following result.
(52) It follows that where ) , Similarly, substituting (44) and (49) into (47), we have Consequently, we can determine  20 (0) and  11 (0).Thus, all   can be determined by [7].Following the basic idea of [7] and the method in [20], one can draw the conclusion about the bifurcation direction and the stability of the Hopf bifurcation, which are determined by the following parameters: which determines the quantities of bifurcating periodic solutions on the center manifold  0 ; namely, we have the following result.
Remark 5. Since the original system (1) is chaotic, there is no stabilized orbit.When we add distributed delayed feedback perturbations to the original system (1), then, under some suitable condition, stabilized orbits will occur.Thus, we can conclude that the stabilized orbits of the original system (1) are delay-induced.

Conclusion
In this paper, we investigate a Lorenz-like system within chaotic attractor responding to the local Hopf bifurcation and   the local stability of equilibrium  ± .Meanwhile, we study the direction of Hopf bifurcation and the stability of bifurcating periodic solutions by using the center manifold theorem and normal form method. Numerical simulation results confirm that the new feedback controller using distributed delay feedback is an effective method for chaos control.The results show the periodic solutions disappear and chaos attractor appears again with  increasing.Through further investigation, we expect that the chaos system and physicists can decide on which bifurcation arises in the chaos model by proper setting on the feedback parameters for building programs to suppress chaos.In addition, it will be pointed out that the real time evolutions of the chaos systems show marked discrete feature due to their small systems sizes.It is interesting for us to discuss the Lorenz-like system reaction dynamics of discrete Lorenz-like system systems.And it will be further investigated elsewhere in the future.