Abstract

This paper mainly investigates the dynamical behaviors of a chaotic system without ilnikov orbits by the normal form theory. Both the stability of the equilibria and the existence of local Hopf bifurcation are proved in view of analyzing the associated characteristic equation. Meanwhile, the direction and the period of bifurcating periodic solutions are determined. Regarding the delay as a parameter, we discuss the effect of time delay on the dynamics of chaotic system with delayed feedback control. Finally, numerical simulations indicate that chaotic oscillation is converted into a steady state when the delay passes through a certain critical value.

1. Introduction

In the past decades, both the chaotic synchronization and chaos control have excited amusements in many areas of science and technology due to their potential application. As is known some chaos is desirable and needs not to be controlled, but others are undesirable and need to be controlled. Therefore, the investigation of chaotic control is of great significance. Moreover, many schemes have been presented to implement chaotic control [1–18], of which the time-delayed feedback controlling forces is proved to be a simple and viable method for a continuous dynamical system. Fortunately, the time-delayed feedback controller can also be used to realize the control of a bifurcation. As is well discussed the chaotic dynamical behaviors of the chaotic system will be varied if the steady state is stable or the bifurcating periodic solutions are orbitally asymptotically stable.

Some authors study the chaotic systems with ilnikov orbit, such as Elhadj and Sprott [9]. Some others investigate chaotic systems without ilnikov orbit, like Wang and Chen [19–21], and so on. Recently, a chaotic system of non-ilnikov type has been introduced in [19, 22] as follows:where , , and . The bifurcations and period orbits of (1) will appear taking as the bifurcation parameter values and fixing , . When especially, there appears a chaotic attractor (see Figure 1). The system has two equilibria: , .

(a) The trajectories

(b) Phase graph

(a) The trajectories
(b) Phase graph

Figure 1 

The trajectories and phase graphs of system (2) with .

Inspired by the above, in this paper, we focus on the control of chaotic attractor of chaotic system (1). In order to better reflect the dynamical behaviors of system (1) depending on the past system information, it is reasonable to incorporate time delay into this system. The signal error of the current state and the past state of the continuous time system will be fed back to the system itself. Following the idea of Pyragas [13], we add a time-delayed force to the second equation of system (1) and then system (1) takes the form as follows:where is a real constant.

The structure 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. [23], the formulae of determining the direction, the stability, and the period of the bifurcating period solutions are presented. To verify the theoretic analysis, numerical simulations are given in Section 4. Finally, a summary of the main conclusions is provided in Section 5.

2. Bifurcation Analysis of the Chaotic System without ilnikov Orbits

In this section, we analyze the influence of time delay on the stability of the steady state by taking as bifurcation parameter and fixing the value of . The stability of system (2) will be changed and a family of periodic orbits will bifurcate from the equilibria with the increasing of .

By the linear transformationsystem (2) becomesThen, the Jacobian matrix of (4) at any point can be written as The characteristic equation of (4) is given bywhere , , , , , and . Thus, we will study the distribution of the roots of the third-degree exponential polynomial equation of (6), obviously, if is a root of (6) and satisfiesSeparating the real and imaginary parts of (7), we getwhich is equivalent toLet and denote , , and ; (10) becomesFrom (11), we havewhere .

Then, .

Denote and ; the equation has two real roots as follows:

Notice that and . We introduce the following results proved by Lü and Chen [24].

Lemma 1. For the polynomial equation (11), we have the following results:(1)If , then (11) does not have positive real root.(2)If , then (11) has positive real roots if and only if and .

Without loss of generality, we give the following assumption:If we make an assumption that (14) is true, then (11) has two positive roots and . Supposing , we have . Substituting into (8), we find thatwhere and ; then is a pair of pure imaginary roots of (7) with . Define

Lemma 2. If (14) holds, when , then (11) has a pair of pure imaginary roots and all the other roots of (11) have nonzero real parts.

Lemma 3. If (14) holds, then we have the following transversality condition:(i) and .(ii) and have the same sign.

Proof. Substituting into (6) and taking the derivative with respect to , we obtainFrom (9), we find thatwhere . Since , we conclude that and have the same sign. This completes the proof.

Theorem 4. Suppose (14) and Routh-Hurwitz criteria for (7) are satisfied. Moreover, if , there exists such that .(i)For , the equilibriums of system (2) are unstable.(ii)For the equilibriums are asymptotically stable.(iii)When and , system (2) undergoes a Hopf bifurcation at the equilibriums .

3. Direction and Stability of Hopf Bifurcation Period Solution

In the previous section, we obtained the conditions of Hopf bifurcation occurring. In this section, the direction and the stability of the bifurcations are analyzed using the central manifold theorem. Assuming that system (2) always undergoes Hopf bifurcation at the equilibrium for and , then, letting , , , , and and dropping the bars for simplifications of notations, the nonlinear system (2) can be transformed into an FDE in aswhere and and are given, respectively, byIf we choosewhere is a Dirac delta function and , then there is

For , defineFor convenience, we can write system (9) into an operator equationwhere

For , defineand a bilinear inner productwhere . Obviously, and are adjoint operators. In line with the discussion in Section 2, we know that are eigenvalues of . Thus they are eigenvalues of . Next, we need to calculate the eigenvectors of and corresponding to and , respectively. Let be the eigenvalue of ; that is, ; we haveThen it is easy to obtainSimilarly, supposing , from the definition of , we haveSo we havewhere is a constant such that . By (26), we getTherefore, we can choose asBelow we apply the idea in Hassard et al. [23] to compute the coordinates describing the center manifold coat . Let be the solution of (2) when . DefineOn the center manifold , we getwhereand and are local coordinates for center manifold in the direction of and . Noting that is also real if of (2), so we deal with real solutions only. For solution , since , we haveThat is,whereThen we havewhereNoticing and , we haveIn view of (38), (39), and (41), we getwherewhereSince in are unknown, we still need to compute them. By (24) and (36), we driveWe can rewritewhere