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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 167065, 10 pages
Delayed Feedback Control and Bifurcation Analysis of an Autonomy System
1Institute of Information and System Computation Science, Beifang University of Nationalities, Yinchuan, Ningxia 750021, China
2Department of Mathematics and Information Science, Zhoukou Normal University, Zhoukou, Henan 466001, China
3Department of Applied Mathematics, Kunming University of Science and Technology, Kunming, Yunnan 650093, China
Received 28 February 2013; Accepted 26 March 2013
Academic Editor: Yisheng Song
Copyright © 2013 Zhen Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
An autonomy system with time-delayed feedback is studied by using the theory of functional differential equation and Hassard’s method; the conditions on which zero equilibrium exists and Hopf bifurcation occurs are given, the qualities of the Hopf bifurcation are also studied. Finally, several numerical simulations are given; which indicate that when the delay passes through certain critical values, chaotic oscillation is converted into a stable state or a stable periodic orbit.
Since the first chaotic attractor was found by Lorsenz in 1963, the theory of chaos has been developing rapidly. The topics of chaos and chaotic control are growing rapidly in many different fields such as biological systems and ecological and chemical systems [1–5]. The desirability of chaos depends on the particular application. Therefore, it is important that the chaotic response of a system can be controlled.
Many researchers have proposed chaos control and synchronization schemes in recent years [6–11]; among which, delayed feedback controller (DFC) is an effective method for chaos control; it has been receiving considerable attention recently [11–14]. The basic idea of DFC is to realize a continuous control for a dynamical system by applying a feedback signal which is proportional to the difference between the dynamical variable and its delayed value . Choosing the appropriate feedback strength and time delay can make the system stable. The delayed feedback control method does not require any computer analysis and can be simply implemented in various experiments. For example, Song and Wei in  investigated the chaos phenomena of Chen’s system using the method of delayed feedback control. Their results show that when taking some value, taking the delay as the bifurcation parameter when passes through a certain critical value, the stability of the equilibrium will be changed from unstable to stable, chaos vanishes and a periodic solution emerges.
Let us consider a system of nonlinear autonomous differential equations as follows: When , , the zero equilibrium of system (1) is unstable and system (1) is chaotic (see Figure 1). For control of chaos, we add a time-delayed force to the second equation of system (1), that is, the following delayed feedback control system: When or , we can see that system (2) is the same as system (1). In this paper, by taking as bifurcation parameter, we will show that when takes some value, with the increasing of , the stability of zero equilibrium of system (2) will change, and a family of periodic orbits bifurcates from zero equilibrium.
This paper is organized as follows. In Section 2, we first focus on the stability and Hopf bifurcation of the zero equilibrium of system (2). In Section 3, we derive the direction and stability of Hopf bifurcation by using normal form and central manifold theory. Finally in Section 4, an example is given for showing the effects of chaotic control.
2. Local Stability and Delay-Induced Hopf Bifurcations
In this section, by analyzing the characteristic equation of the linearized system of system (2) at the equilibriums, we investigate the stability of the equilibriums and the existence of local Hopf bifurcations occurring at the equilibriums.
System (2) has two equilibriums, and , where
The linearization of system (2) at is whose characteristic equation is where , , , , , , and . Thus, we need to study the distribution of the roots of the third-degree exponential polynomial equation. For this end, we first introduce the following simple result which was proved by Ruan and Wei  using Rouche’s theorem.
Lemma 1. Consider the exponential polynomial where and are constants. As vary, the sum of the order of the zeros of on the open right half-plane can change only if a zero appears on or crosses the imaginary axis.
Denote Then, we have the following lemma.
Lemma 2. For the polynomial equation (10), one has the following results.(i)If , then (10) has at least one positive root.(ii)If and , then (10) has no positive roots.(iii)If and , then (10) has positive roots if and only if and .
Suppose that (10) has positive roots, without loss of generality; we assume that it has three positive roots, defined by , , and , respectively. Then (9) has three positive roots Thus, we have If we denote then is a pair of purely imaginary roots of (9) with . Define Note that when , (5) becomes Therefore, applying Lemmas 1 and 2 to (5), we obtain the following lemma.
Lemma 3. For the third-degree transcendental equation (5), one has the following results.(i)If and , then all roots with positive real parts of (5) have the same sum to those of the polynomial equation (16) for all . (ii)If either or and , and , then all roots with positive real parts of (5) have the same sum to those of the polynomial equation (16) for .
Let be the root of (9) near satisfying Then it is easy to verify the following transversality condition.
Lemma 4. Suppose that and , where is defined by (15). Then , and and have the same sign.
Theorem 5. Let and be defined by (14) and (15), respectively. Then consider the following.(i)If and , then all roots with positive real parts of (5) have the same sum to those of the polynomial equation (16) for all .(ii)If either or and , , and , then has at least one positive root , and all roots with positive real parts of (5) have the same sum to those of the polynomial equation (16) for .(iii)If the conditions of (ii) are satisfied and , then system (2) exhibits Hopf bifurcation at the equilibrium for .
3. Stability and Direction of Bifurcating Periodic Orbits
In the previous section, we obtain the conditions under which family periodic solutions bifurcate from the equilibrium at the critical value of . As pointed by in Hassard et al. , it is interesting to determine the direction, stability, and period of these periodic solutions. Following the ideal of Hassard et al., we derive the explicit formulae for determining the properties of the Hopf bifurcation at the critical value of using the normal form and the center manifold theory. Throughout this section, we always assume that system (2) undergoes Hopf bifurcations at the equilibrium for , and then is corresponding to purely imaginary roots of the characteristic equation at the equilibrium .
Letting , system (2) is transformed into an FDE in as where and are given, respectively, by By the Riesz representation theorem, there exists a function of bounded variation for , such that In fact, we can choose where is Dirac-delta function.
For , define Then, when , system (18) is equivalent to where for . For , define and a bilinear inner product where . Then and are adjoint operators. By the discussion in Section 2, we know that are eigenvalues of . Thus, they are also eigenvalues of . We first need to compute the eigenvector of and corresponding to and , respectively. Suppose that is the eigenvector of corresponding to . Then . It follows from the definition of and (19), (21), and (22) that Thus, we can easily get Similarly, let be the eigenvector of corresponding to . By the definition of and (19), (21), and (22), we can compute In order to assure , we need to determine the value of . From (26), we have Thus, we can choose as In the following, we first compute the coordinates to describe the center manifold at . Let be the solution of (18) when . Define On the center manifold , we have where and are local coordinates for center manifold in the direction of and . Note that is real if is real. We consider only real solutions. For the solution of (18), since , we have where By (32), we have , and then It follows together with (20) that Comparing the coefficients with (35), we have In order to determine , in the sequel, we need to compute and . From (24) and (32), we have where Notice that near the origin on the center manifold , we have Thus, we have Since (39), for , we have Comparing the coefficients with (40) gives that From (43), (46), and the definition of , we can get Noticing that , we have where is a constant vector. In the same way, we can also obtain where is also a constant vector.
In what follows, we will seek appropriate and . From the definition of and (42), we obtain where . From (39) and (40), we have Substituting (46) and (50) into (48) and noticing that we obtain which leads to It follows that where Similarly, substituting (47) and (51) into (49), we can get Thus, we have where Thus, we can determine and . Furthermore, we can determine each . Therefore, each is determined by the parameters and delay in (2). Thus, we can compute the following values: which determine the quantities of bifurcating periodic solutions in the center manifold at the critical value ; that is, determines the directions of the Hopf bifurcation: if , then the Hopf bifurcation is supercritical (subcritical), and the bifurcation exists for ; determines the stability of the bifurcation periodic solutions: the bifurcating periodic solutions are stable (unstable) if ; and determines the period of the bifurcating periodic solutions: the period increases (decreases) if .
4. Numerical Simulations and Application to Control of Chaos
In this section, we apply the results we get in the previous sections to system (61) for the purpose of control of chaos. From Section 2, we know that under certain conditions, a family of periodic solutions bifurcates from the steady states of system (61) at some critical values of , and the stability of the steady state may be changed along with the increase of . If the bifurcating periodic solution is orbitally asymptotically stable or some steady state becomes local stable, then chaos may vanish. Following this idea, we consider the following delayed feedback control system for an autonomy system: which has two equilibriums, and . When or , is unstable and system (61) is chaotic (see Figure 1). The corresponding characteristic equation of system (61) at is When , (62) has one negative root and a pair of complex roots with positive parts. By the discussion of Section 2, we can get , and then we can compute , and it is easy to know that and when , always holds. Then, by Theorem 5, we have the following conclusion.
Conclusion 1. If or and , then there exist some , such that when , is always unstable.
For example, we fix . Through computing, we can get and . Furthermore, we can compute Thus, from Lemma 4, we have In addition, notice that Thus, from (64) and (65), we have the following conclusion about the stability of the zero equilibrium of system (61) and Hopf bifurcations.
Conclusion 2. Suppose that , is defined by (63). (i)When , the equilibrium is unstable (see Figures 3, 4, 6, 7, and 8). (ii)When , the equilibrium is asymptotically stable (see Figure 5).(iii)System (61) undergoes a Hopf bifurcation at the equilibrium when (see Figures 3, 4, and 6).
Thus, if we take , the stability of zero equilibrium of the system will change from unstable to stable. For this example, there are more complicated dynamical phenomena occurs. Bifurcation diagram is plotted with the parameter when (Figure 2). In Figure 2, we can see that chaos vanishes via a period-doubling bifurcation when varying from 0 to 0.3768. Figure 3 shows that, when , a period-2 orbits bifurcating from zero equilibrium. Figure 4 shows when , a period-1 periodical orbits bifurcating from zero equilibrium. In Figure 2, a stable window is obtained from to , which is demonstrated by Figure 5 with . When , Figure 2 shows that periodic orbits and strange attractors also occur, and system (61) through Hopf bifurcation route still goes to chaos, and it is demonstrated by Figures 6–8.
In this paper, an autonomy system with time-delayed feedback is studied by using the theory of functional differential equation and Hassard’s method. Illustrating with numerical simulations, we show that delayed feedback controller (DFC) is an effective method for chaos control.
In fact, system (2) has more complicated dynamical behaves. From Figure 2, we can see that chaos degeneration process, the orbits of system (2) continuously degenerate to a periodic solution region through reverse period-doubling bifurcation. With the increasing of , chaos occurs again through Hopf bifurcation route. However, the numerical simulations indicate that there were points of similarity between this route to chaos and quasiperiodicity route to chaos ; in our future research, we will consider this topic. And in this paper, we only give detailed analysis on the trivial equilibrium ; the detailed analysis of the interior equilibrium is the second interested topic in our future research.
The authors would like to thank the anonymous referee for the very helpful suggestions and comments which led to the improvement of the original paper. And this work is supported by 2013 Scientific Research Project of Beifang University of Nationalities (2013XYZ021), Institute of Information and System Computation Science of Beifang University (13xyb01), Science and Technology Department of Henan Province (122300410417), and Education Department of Henan Province (13A110108).
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