Chaos control and Hopf bifurcation analysis of the Genesio system with distributed delays feedback
© Guan et al.; licensee Springer 2012
Received: 9 May 2012
Accepted: 6 September 2012
Published: 19 September 2012
In this paper, the Genesio system with distributed time delay feedback is investigated. Firstly, the stability of the equilibria of the system is investigated by analyzing the characteristic equation, and then the existence of Hopf bifurcations is verified by choosing the mean time delay as a bifurcation parameter. Subsequent to that, the direction and stability of the bifurcating periodic solutions are determined by using the normal form theory and the center manifold theorem. Finally, some numerical simulations are presented to verify the effectiveness of the theoretical results.
Chaos control has attracted considerable attention since the pioneering work of Ott and Grebogi . It is well known that in many practical applications, chaos is undesirable and needs to be controlled. Therefore, the investigation of controlling chaos is of great significance. Many schemes have been presented to implement chaos control, among which using time-delayed controlling forces proves to be a simple and viable method for a continuous dynamical system . It is noteworthy that time-delayed feedback controller can also be used to realize the control of a bifurcation, see [3–6] and references therein. It is known that if the steady state is stable or the bifurcating periodic solutions are orbitally asymptotically stable, then the chaotic system will not exhibit chaotic dynamical behaviors. As a consequence, bifurcation control in this sense may also help to control chaos.
In order to better model some complicated practical phenomena, recently, distributed time delay has been introduced into many modeling systems. There are extensive literature works dealing with such systems [7–11]. As the distributed time delay is incorporated in a system, some interesting dynamical behaviors occur near the equilibrium point. Inspired by these previous works, in this paper, we intend to introduce the distributed time delay as a feedback controller into the chaotic Genesio system with the aim to realize the control of chaos. The rest of this paper is organized as follows. In the next section, we present the mathematical models of the Genesio system with distributed time delay feedback and consider its local stability and Hopf bifurcation. In Section 3, the stability of the bifurcating periodic solutions and the direction of the Hopf bifurcation at the critical values of mean time delay are determined by using the normal form method and the center manifold reduction due to Hassard et al. . In Section 4, a numerical example is provided to verify the theoretical results. Finally, some concluding remarks are given in Section 5.
2 Stability analysis and Hopf bifurcation of the Genesio system with distributed delay feedback
where , , , .
It is easy to see that at , we always have , thus the equilibrium point is unstable. In what follows, we only analyze the equilibrium point . Straightforwardly, we have the following result.
thus the Hopf bifurcation occurs at as α passes through .
3 Direction and stability of bifurcating periodic solutions
and z, are the local coordinates of the center manifold in the directions of and respectively.
In order to guarantee the continuity of solutions, we further assume that is continuous at .
which determine the quantities of bifurcating periodic solutions on the center manifold at the critical value , i.e., determines the directions of the Hopf bifurcation: if (), then the Hopf bifurcation is supercritical (subcritical) and the bifurcating periodic solutions exist for ; determines the period of the bifurcating periodic solutions: the period increases (decreases) if (); determines the stability of the bifurcating periodic solutions: the bifurcating periodic solutions are stable (unstable) if ().
4 Numerical simulations
As compared with the former method, a chaotic model with distributed delay feedback is more general than that with discrete delay feedback [3–5], because the distributed delay becomes a discrete delay when the delay kernel is a delta function at a certain time. The distributed delay has found widespread applications in many fields such as neural network [8, 10], complicated real models , the modeling of aggregative processes involving the flow of entities with random transit times through a given process , and so on. Therefore, it is of considerable significance to propose distributed delays as control input to control the chaotic system.
From the numerical simulations, we see, as the distributed delay feedback is incorporated in the chaotic Genesio system, a rich spectrum of dynamical behaviors can occur by adjusting the mean time delay values. Chaotic behaviors vanish and the orbitally asymptotically stable Hopf bifurcation occurs as the mean time delay reaches a certain value. Also, we can determine the critical mean time delay value that the Hopf bifurcation occurs at, which is of great help when choosing appropriate parameter values to realize Hopf bifurcation control.
5 Concluding remarks
In this paper, the Genesio system with distributed time delay feedback has been studied. It has been demonstrated that the Hopf bifurcation occurs near the steady state as the average time delay crosses the critical value. The explicit formulae for determining the direction, stability and period of bifurcating periodic solutions have been presented by using the normal form theory and the center manifold theorem. A numerical example is provided to verify the theoretical results.
The authors would like to thank the anonymous referees for their valuable comments. This work was supported by the National Natural Science Foundation of China (No. 11171084, No. 61273077) and the Natural Science Foundation of Zhejiang Province (No. LY12A01001).
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