Bifurcation Analysis and Chaos Control in Genesio System with Delayed Feedback
We investigate the local Hopf bifurcation in Genesio system with delayed feedback control. We choose the delay as the parameter, and the occurrence of local Hopf bifurcations are verified. By using the normal form theory and the center manifold theorem, we obtain the explicit formulae for determining the stability and direction of bifurcated periodic solutions. Numerical simulations indicate that delayed feedback control plays an effective role in control of chaos.
Since the pioneering work of Lorenz , much attention has been paid to the study of chaos. Many famous chaotic systems, such as Chen system, Chua circuit, Rössler system, have been extensively studied over the past decades. It is well known that chaos in many cases produce bad effects and therefore, in recent years, controlling chaos is always a hot topic. There are many methods in controlling chaos, among which using time-delayed controlling forces serves as a good and simple one.
In order to gain further insights on the control of chaos via time-delayed feedback, in this paper, we aim to investigate the dynamical behaviors of Genesio system with time-delayed controlling forces. Genesio system, which was proposed by Genesio and Tesi , is described by the following simple three-dimensional autonomous system with only one quadratic nonlinear term: where , , < 0 are parameters. System (1.1) exhibits chaotic behavior when , , , as illustrated in Figure 1. In recent years, many researchers have studied this system from many different points of view; Park et al. [3–5] investigated synchronization of the Genesio chaotic system via backstepping approach, LMI optimization approach, and adaptive controller design. Wu et al.  investigated synchronization between Chen system and Genesio system. Chen and Han  investigated controlling and synchronization of Genesio chaotic system via nonlinear feedback control. Inspired by the control of chaos via time-delayed feedback force  and also following the idea of Pyragas , we consider the following Genesio system with delayed feedback control: where and .
2. Bifurcation Analysis of Genesio System with Delayed Feedback Force
It is easy to see that system (1.1) has two equilibria and , which are also the equilibria of system (1.2). The associated characteristic equation of system (1.2) at appears as As the analysis for is similar, we here only analyze the characteristic equation at . First, we introduce the following result due to Ruan and Wei .
Lemma 2.1. 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 , , , , , , , , , . Following the detailed analysis in , we have the following results.
(i) If , then all roots with positive real parts of (2.1) when has the same sum to those of (2.1) when .
(ii) If , , , then all roots with positive of (2.1) when has the same sum to those of (2.1) when .
Lemma 2.3. Suppose that , then , and .
Proof. Substituting into (2.1) and taking the derivative with respect to , we can easily calculate that thus the results hold.
(i) If , then (2.1) has two roots with positive real parts for all .
(ii) If , , , then (2.1) has two roots with positive real parts for .
(iii) If , , and , then system (1.2) exhibits the Hopf bifurcation at the equilibrium for .
3. Some Properties of the Hopf Bifurcation
In this section, we apply the normal form method and the center manifold theorem developed by Hassard et al. in  to study some properties of bifurcated periodic solutions. Without loss of generality, let be the equilibrium point of system (1.2). For the sake of convenience, we rescale the time variable and let , , , , then system (1.2) can be replaced by the following system: where , and for , and are, respectively, given as By the Riesz representation theorem, there exists a function of bounded variation for , such that In fact, the above equation holds if we choose where is Durac function. For , let Then (1.2) can be rewritten in the following form: For , we consider the adjoint operator of defined by For and , we define the bilinear inner product form as
Suppose that is the eigenvectors of with respect to , then . By the definition of and (3.2), (3.4), and (3.5) we have Hence Similarly, let be the eigenvector of with respect to , by the definition of and (3.2), (3.4), and (3.5) we can obtain Furthermore, , .
Let , where is the solution of (3.7) when . We denote , then We Rewrite (3.13) in the following form: where Noticing that we have Define with On the other hand, Expanding the above series and comparing the corresponding coefficients, we obtain While Let , then we have Therefore we have Comparing the corresponding coefficients, we have
In what follows we will need to compute and . Firstly we compute , when . It follows from (3.18) that Substituting the above equation into (3.21) and comparing the corresponding coefficients yields By (3.21), (3.28), and the definition of we have Hence Similarly we have
In what follows, we will seek appropriate and in (3.30) and (3.31). When , with Comparing the coefficients in (3.18) we have By (3.21) and the definition of we have Substituting (3.30) into (3.36) and noticing that we have namely, Thus where Following the similar analysis, we also have hence where Thus the following values can be computed:
It is well known in  that determines the directions of the Hopf bifurcation: if , then the Hopf bifurcation is supercritical(subcritical) and the bifurcated periodic solution exists if ; determines the period of the bifurcated periodic solution: if , then the period increase(decrease); determines the stability of the Hopf bifurcation: if , then the bifurcated periodic solution is stable(unstable).
4. Numerical Simulations
In this section, we apply the analysis results in the previous sections to Genesio chaotic system with the aim to realize the control of chaos. We consider the following system: Obviously, system (4.1) has two equilibria and . In what follows we analyze the case of only, the analysis for is similar. The corresponding characteristic equation of system (4.1) at appears as Hence we have , , , , , , , , , . By Theorem 2.4, when , that is, , (4.2) has two roots with positive real parts for all . In order to realize the control of chaos, we will consider . We take as a special case. In this case, system (4.1) takes the form of Thus we can compute , , , , , , , , , , ,. Therefore, using the results in the previous sections, we have the following conclusions: when the delay , the attractor still exists, see Figure 2; when the delay , Hopf bifurcation occurs, see Figure 3. Moreover, , , the bifurcating periodic solutions are orbitally asymptotically stable; when the delay , the steady state is locally stable, see Figure 4; when the delay , the steady state is unstable, see Figure 5. Numerical results indicate that as the delay sets in an interval, the chaotic behaviors really disappear. Therefore the parameter works well in control of chaos.
5. Concluding Remarks
In this paper we have introduced time-delayed feedback as a simple and powerful controlling force to realize control of chaos of Genesio system. Regarding the delay as the parameter, we have investigated the dynamics of Genesio system with delayed feedback. To show the effectiveness of the theoretical analysis, numerical simulations have been presented. Numerical results indicate that the delay works well in control of chaos.
This work was supported by the Research Foundation of Hangzhou Dianzi University (KYS075609067).
E. Lorenz, “Deterministic non-periodic flow,” Journal of the Atmospheric Sciences, vol. 20, pp. 130–141, 1963.View at: Google Scholar
K. Pyragas, “Continuous control of chaos by self-controlling feedback,” Physics Letters A, vol. 170, no. 6, pp. 421–428, 1992.View at: Google Scholar
S. Ruan and J. Wei, “On the zeros of a third degree exponential polynomial with applications to a delayed model for the control of testosterone secretion,” IMA Journal of Mathemathics Applied in Medicine and Biology, vol. 18, no. 1, pp. 41–52, 2001.View at: Google Scholar
B. D. Hassard, N. D. Kazarinoff, and Y. H. Wan, Theory and Applications of Hopf Bifurcation, vol. 41, Cambridge University Press, Cambridge, UK, 1981.