Bifurcation analysis of Rössler system with multiple delayed feedback ∗

In this paper, regarding the delay as parameter, we investigate the effect of delay on the dynamics of a Rossler system with multiple delayed feedback proposed by Ghosh and Chowdhury. At first we consider the stability of equilibrium and the existence of Hopf bifurcations. Then an explicit algorithm for determining the direction and the stability of the bifurcating periodic solutions is derived by using the normal form theory and center manifold argument. Finally, we give a numerical simulation example which indicates that chaotic oscillation is converted into a stable steady state or a stable periodic orbit when the delay passes through certain critical values.


Introduction
The study of chaotic systems has increasingly gained interest of many researchers since the pioneering work of Lorenz [8].For a quite long period of time, people thought that chaos was neither predictable nor controllable.Recently the trend of analyzing and understanding chaos has been extended to controlling and utilizing chaos.The main goal of chaos control was to eliminate chaotic behavior and to stabilize the chaotic system at one of the system's equilibrium points.More specially, when it is useful, we want to generate chaos intentionally.Until now, many advanced theories and methodologies have been developed for controlling chaos.Many scientists have more concerns with delayed control (see Guan, Chen and Peng [5], Shu et al. [16], Zhang and Su [21]).The existing control method can be classified, mainly, into two categories.The first one, the OGY method developed by Ott, Grebogi and Yorke [9] in 1990s of the last century has completely changed the chaos research topic.The second one, proposed by Pyragas [10,11], using time-delayed controlling forces.Compared with the first one, it is much simpler and more convenient on controlling chaos in continuous dynamics system.Here, we mainly study the Rössler system with delayed controlling method developed by Pyragas.Rössler system is described by the following three-dimensional smooth autonomous system (see Rössler [13]) ẋ(t) = −y(t) − z(t), ẏ(t) = x(t) + β 1 y(t), which is chaotic when β 1 = β 2 = 0.2, γ = 5.7.
Rössler system is a quite simple set of differential equations with chaos to simplify the Lorenz model of turbulence that contains just one (second order) nonlinearity in one variable.Due to its simplicity, the Rössler system has become a standard one to issue the effectiveness of the chaos control strategy.Recently, many literatures adopted controlling strategy for the Rössler system.In the last years there are many studies on Rössler system.For example, Pyragas [10] stabilized unstable periodic orbits of a Rössler system to a desired periodic orbit by self-controlling feedback.Tao et al. [18] used the speed feedback control such that the controlled Rössler system will gradually converge to unsteadily equilibrium point.Tian et al. [19] used a nonlinear open-plus-closed-loop (NOPCL) control to Rössler system.Yang et.al. [20] presented an impulsive control to achieve periodic motions for the Rössler system.Moreover there are extensive study, for example Amhed et al. [1], Chang et al. [2], Chen et al. [3], Rasussen et al. [12].Recently, Ghosh et al. [4] have proposed the multiple delayed system in the following form: EJQTDE, 2010 No. 63, p. 2 where α i , β i (i = 1, 2) and γ are all positive constants.They studied the system (2) in numerical simulations mainly.The purpose of the present paper is to investigate system (2) analytically and numerically.Our analytical results show that the stability changes as the delays vary.Meanwhile, our numerical simulations indicate that chaotic oscillation is converted into a stable steady state or a stable periodic orbit when the delay passes through certain critical values.This shows that the chaos property changes as the delay varies.
The rest of the paper is organized as follows.In Section 2, we study the distribution of the eigenvalues by using the result due to Ruan and Wei [14,15] on the analysis of distribution of the zeros of exponential polynomial.Hence the stability and existence of Hopf bifurcations are obtained.In Section 3, the direction and stability of the Hopf bifurcation are determined by using the center manifold and normal forms theory.Some numerical simulations are carried out for supporting the analysis results in Section 4. Conclusions and discussions are given in Section 5.

Analysis of stability and bifurcation
In this section, we shall study the stability of the interior equilibrium and the existence of local Hopf bifurcations.For convenience, denote (2) has no equilibrium, and if γ 2 A = 4β 1 β 2 , the system has only one equilibrium given by holds, then the system (2) has two equilibria (x 0 , y 0 , z 0 ) and (x 1 , y 1 , z 1 ), where EJQTDE, 2010 No. 63, p. 3 Thorough out this paper, we always assume that (H 0 ) is satisfied and only consider the dynamics of system (2) near the equilibrium (x 0 , y 0 , z 0 ).
Let u 1 = x − x 0 , u 2 = y − y 0 and u 3 = z − z 0 .Then system (2) can be written in the following form The characteristic equation of system (3) at the equilibrium (0, 0, 0) is where Now we employ the method due to Ruan and Wei [14,15] to investigate the distribution of roots of Eq.( 4).
By the Lemmas 2.1, 2.2 and applying the Hopf bifurcation theorem for functional differential equations (see Hale [6], Chapter 11, Theorem 1.1), we can conclude the existence of Hopf bifurcation as stated in the following theorem.
(iii) If all conditions as stated in (ii) and h ′ (z k ) = 0 hold, then system (2) undergoes a Hopf bifurcation at the equilibrium (x 0 , y 0 , z 0 ), when We have known that (H 1 ) ensures that all roots of Eq.( 5) have negative real parts.Now we consider (H 1 ) is not satisfied.For convenience, denote Then Eq.( 5) becomes Let λ = X − a/3.Then it reduces to where Then from Cardano's formula for the third degree algebra equation we have the followings.

Stability and direction of the Hopf bifurcation
In the previous section, we obtained conditions for Hopf bifurcation to occur when τ 2 = τ 0 2 .In this section we study the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions when τ 2 = τ 0 2 , using techniques from normal form and center manifold theory (see e.g.Hassard et al. [7]).
EJQTDE, 2010 No. 63, p. 11 By (16) we have Hence, we can choose so that < q * , q >= 1.Clearly, < q * , q >= 0. Following the algorithms in Hassard et al. [1981] to describe the center manifold C 0 at µ = 0. Let u t be the solution of Eq.( 15) when µ = 0. Define On the center manifold C 0 we have z and z are local coordinates for center manifold C 0 in the direction of q * and q * .Note that W is real if u t is real.We consider only real solution.
Notice that Similarly, we have Then g 21 can be expressed by the parameters.
Based on the above analysis, we can see that each g ij can be determined by the parameters.Thus we can compute the following quantities:

Conclusion
Bifurcation in Rössler system with single delay has been observed by many researchers.However, there are few papers on the bifurcation of Rössler system with multiple delays.
In this paper we have analyzed the Rössler system with multiple delays on two different conditions.We find out that there are stability switches for the interior EJQTDE, 2010 No. 63, p. 19 equilibrium when τ 1 varies in the case of τ 2 = 0. Then for τ 1 in a stability interval, regarding the delay τ 2 as parameter, we show that there exists a first critical value of τ 2 at which the interior equilibrium loses its stability and the Hopf bifurcation occurs.We also investigate the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions, by using the center manifold theory and normal form method.
Our theoretical results and numerical simulations show that, for a Rössler system with chaos phenomena, the chaos oscillation can be controlled by delays.For example, the multiple delayed Rössler system we studied possess chaos oscillation when τ 1 = τ 2 = 0.The chaos disappears when the delays increase, and the stability of the equilibrium is lost at same time, and the periodic solutions occur from Hopf bifurcation.As the delays increasing further, the numerical simulations show that the periodic solution disappears and the chaos oscillation appears again.