Delay Feedback Control of the Lorenz-Like System

We choose the delay as a variable parameter and investigate the Lorentz-like systemwith delayed feedback by usingHopf bifurcation theory and functional differential equations.The local stability of the positive equilibrium and the existence ofHopf bifurcations are obtained. After that the direction of Hopf bifurcation and stability of periodic solutions bifurcating from equilibrium is determined by using the normal form theory and center manifold theorem. In the end, some numerical simulations are employed to validate the theoretical analysis. The results show that the purpose of controlling chaos can be achieved by adjusting appropriate feedback effect strength and delay parameters.The applied delay feedback control method in this paper is general and can be applied to other nonlinear chaotic systems.


Introduction
Since the discovery of Lorenz chaotic attractor in 1963, chaos has been studied and developed by many scholars, for instance, Rössler system [1], Chua circuit [2], and Chen system [3] et al.However, due to the extremely sensitive characteristics of chaos to the system environment, it is sometimes necessary to suppress or stimulate chaos phenomena in real applications.Therefore, chaos control has attracted more and more attention, and the applications of dynamical systems and chaos involve mathematical biology, financial systems, chemistry, electronic circuits, secure communications, cryptography, and neuroscience research [4][5][6][7][8][9].
The main goal of chaos control is to eliminate chaotic behavior and stabilize the chaotic system.Especially, when it is useful, we want to generate chaos intentionally.So far, many advanced theories and methodologies have been developed in order to be better in controlling chaos.The existing control method can be classified, mainly, into two categories.The first one, the OGY method [10], which has completely changed the chaos research topic, is based on the invariant manifold structure of unstable orbits.The second one is DFC method, proposed by Pyragas [11,12], using timedelayed controlling forces.It provides an alternative effective method for feedback control of chaos.In sharp contrast to the formal one, the second is a simple and convenient method of controlling chaos in continuous dynamical system.In order to make further study of the control of chaos via time-delayed feedback, in this paper, we aim to investigate the dynamical behaviors of Lorenz-like system with time delayed.
The study of [15] shows that the chaotic behavior can be stabilized on various periodic orbits by use of Pyragas timedelayed feedback control.The results of the existence of Hopf bifurcation and effectiveness of delayed feedback have been given [16][17][18][19][20][21][22][23].Following the idea of Pyragas [12], we add a where , , , ℎ, ,  > 0. Regarding the time delay  as the bifurcation parameter, when  passes through some certain critical values, the equilibrium will lose its stability and Hopf bifurcation will take place.By tuning the feedback effect strength , we can implement the control of chaos phenomena of system.
The remainder of this paper is organized as follows.In Section 2, the local stability and the existence of Hopf bifurcation of the control system (2) are determined.In Section 3, some explicit formulas determining the direction and stability of periodic solutions bifurcating from Hopf bifurcations points are demonstrated by applying the normal form theory and the center manifold theorem [24,25].In Section 4, The numerical simulations are carried to demonstrate our theorem analysis.Concluding comments are given in Section 5.

Bifurcation Analysis of the Chaotic System
In this section, we will investigate the effect of delay on the dynamic behavior of system (2).More specifically, we study the local stability and the existential conditions of Hopf bifurcation.Obviously, when  = 0 system (2) becomes the Lorenz-like system (1).System (1) is linearized at the equilibrium  0 (0, 0, 0) to obtain the Jacobian matrix as follows: The associated characteristic equation of ( 3) is given by When  = 0, (4) has three roots Obviously, if  > 0,  > 0, the equilibrium  0 is stable.Therefore, when  = 0, system (2) undergoes a pitchfork bifurcation.Else, the equilibrium  0 is unstable.The Lorenz-like system ( 2) is symmetric about the -axis, so  + and  − have the same stability.It is sufficient to analyze the stability of  + .By linear transformation system (2) will be The Jacobian matrix of (7) at the any point ( 0 ,  0 ) is written as The characteristic equation of ( 7) is given by Thus, we will study the distribution of the roots of the third degree exponential polynomial equation where First of all, we will introduce the results of [26].

Direction and Stability of Hopf Bifurcation Period Solution
In previous section, we obtained the conditions when the Hopf bifurcation occurs.In this section, we shall study the direction and the stability of the bifurcations with normal form theory and central manifold theorem.During this section we always assume that system (2) undergoes Hopf bifurcation at the equilibrium  + for  =   .
(77) and, hence, where Consequently, we can determine  20 (0) and  11 (0) by ( 65) and (67).Furthermore, using them we can compute  21 .Therefore, all   are determined by the parameters and delay in (29).After that, we can easily compute the following values: Thus, we have the main results of this section.Theorem 6.For (80), we have the following: (1) The sign of  2 determines the direction of the Hopf bifurcation; if  2 > 0( 2 < 0), then the Hopf bifurcation is supercritical (subcritical) and the bifurcating period solutions exist for  >   ( <   ); (2) The sign of  2 determines the stability of the bifurcating period solutions; if  2 < 0( 2 > 0), then the bifurcation period solutions are orbitally stable (unstable); (3) The sign of  2 determines the period of the bifurcating periodic solutions; if  2 > 0( 2 < 0), then the period increases (decreases).

Application to Control of Chaos
In this section, we perform some numerical simulations to verify the results of theoretical analysis in the previous section.Now we consider the following system:  For the equilibrium point  0 (0, 0, 0), the characteristic equation of system (81) will be It is clear that (82) will be have a positive root and two negative roots.By Lemma 2 and Δ > 0 for all  ∈  it is clear that (82) has at least one root with positive real part for  ∈  which means that  0 (0, 0, 0) is always unstable.Furthermore, we conclude that the bifurcating periodic solution is also unstable in the phase space even though they are stable in the center manifold.

Conclusions
In this paper, we aim to investigate delay feedback control of a new butterfly-shaped Lorenz-like system.Taking the time delay as a bifurcation parameter, we explore the effect of time delay on its dynamics.The local stability and the existence of Hopf bifurcation of the chaotic system is investigated by analyzing the characteristic equations.After that, we also investigate the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions, which are determined by applying the normal form theory and the center manifold theorem.Our theoretical results and numerical simulations show that when  passes through some certain critical values, the stability of the system will change from unstable to stable, or the stable periodic orbit will be branched out from the equilibrium, which will lead to the chaos phenomenon in the system disappear, and achieve the purpose of controlling chaos.