Controlling the Period-Doubling Bifurcation of Logistic Model

Zhiqian Wang ∗, Jiashi Tang, Yingshe Luo 1 College of Civil Engineering and Mechanics, Central South University of Forestry and Technology, Changsha, Hunan 410004, PR China 2 College of Mechanical and Vehicle Engineering, Hunan University, Changsha, Hunan 410082, PR China 3 Hunan Province Key Laboratory of Engineering Rheology, Central South University of Forestry and Technology, Changsha, Hunan 410004, PR China (Received 21 December 2014, accepted 20 March 2015)


Introduction
With the development of the nonlinear science, chaos control and bifurcation control have been researched as a popular topic [1].As an engineering leading research field, bifurcation control has become more and more challenging [2].Bifurcation control means bifurcation characteristics of the system are changed with a designed controller [3,4].The typical applications of bifurcation controls include possible resolution for stall of compression system in jet engines, high incidence flight, voltage collapse in power systems, oscillatory behavior of tethered satellites, magnetic bearing systems, rotating chains, thermal convection loop, and cardiac alternans and heart rhythms [5].In general, the goal of bifurcation control is to avoid the appearance of the undesired bifurcation characteristics of the system so as to make the system to be monitored [6,7].
Bifurcation control has been designed for stationary, Hopf, period-doubling bifurcations and chaotic motions, etc [8,9].Many research works on chaos control and synchronization of two chaotic systems have born great fruits.However, there are fewer investigations on bifurcation control relative to chaos control [10].For Hopf bifurcation control, Abed and Fu designed a static state feedback controller by for discrete maps [11].Yagnoobi and Abed considered a nonlinear feedback (mainly cubic) controller for Hopf bifurcation of discrete maps.Recently, Chen et al [1,12].developed a dynamic state feedback control law incorporating a washout filter to control Hopf bifurcation in the Lorenz system.For period-doubling bifurcation control, few studies has been extended to this topic.Chen et al. developed the state feedback and parameter tuning control law to control period-doubling bifurcation in the Logistic system [14,15].The first and second bifurcation locations of Logistic mapping are precisely controlled [16].Some investigations proposed the state feedback control law to control the bifurcation of an acoustooptic bistable system [17].Chen et al. proposed a comprehensive survey of Logistic system.The investigation analysed the bifurcation characteristics of Logistic system and proposed some questions about bifurcation control of Logistic system [18,19].
For the controlled system, it should have the same structure of the original system and yet the controller is easy to be implemented.Thus, we can design a effective controller which satisfy the following conditions.Firstly, this competitive controller keeps the equilibria of a system unchanged and also preserves the dimension of the system.Secondly, the controller can delay the time of the first bifurcation of the period-doubling bifurcation, but it could not control the second bifurcation or others after the first bifurcation [20].Thus, the controller can only control one of the bifurcation locations [21].In this paper, we introduce the time-delay τ into the controlled system.Our goal is to design a controller with time-delay to satisfy the above conditions.

Parametric synchronization scheme
Consider a Logistic model described by where µ is a constant and satisfies µ > 0.
x n is the density of target insect in the given year.x n+1 is density of target insect in the next given year.
The equilibrium points can be obtained The Jacobi can be given Fig. 1 shows the bifurcation of the Logistic model.Consider the logistic model with designed controller described by Design the new feedback controller using time delay u as where τ is the time-delay.Assume Let τ = m△t.
IJNS homepage: http://www.nonlinearscience.org.uk/ The Jacobi can be given When the controlled system has 1 periodic orbit, the system has 1 periodic point.The Logistic model with designed controller described by where Because of x(t + (n + 1)∆t) = x(t + n∆t), the controller keeps the equilibria of the original system unchanged.The eigenvalue of the Jacobi should satisfy that Substitute the equilibrium points Eq. ( 2) into Eq.( 9) where k ∈ (−1, 0).Solve Eq. ( 11), we can obtain that where where b is the value which we want to define.Because of µ > µ cr , the controller u indeed expand the range of the first bifurcation.Fig. 2 presents the bifurcation of the controlled system.From Fig. 2, the controller u satisfies the first condition that keeps the equilibria of a system unchanged.Obviously, the controller delays the time of the first bifurcation from µ = 3.145 (the original system) to µ = 3.35.Moreover, the second and other bifurcation locations of controlled Logistic system are not controlled by the controller.We have achieved the goal with the controller.When the controlled system has 2 periodic orbit, the system has 2 periodic point.The logistic model with designed controller described by where The eigenvalue of the Jacobi should satisfy that Substitute the equilibrium points into Eq.( 16) IJNS email for contribution: editor@nonlinearscience.org.uk According to Eq. ( 17), we can obtain that where µ > 0. From Eq. ( 19), when k > 0, we can obtain From Eq. ( 20), we know that µ > µ cr .Thus, the controller delays the time of the second bifurcation.Fig. 3 is the bifurcation of the controlled system.From Fig. 3, the controller u keeps the equilibria of a system unchanged.The second bifurcation begins with µ = 3.145 in Fig. 3 which means the first bifurcation of the controlled system has not been controlled.We achieve the goal.Moreover, we delay the second bifurcation from µ = 3.55 (the original system) to µ = 3.58.We can modify the value of k to expand the range of µ.The second bifurcation can be delayed to µ = 3.615 with k = 0.1 (see Fig. 4).That is to say that we can control one of the bifurcation locations.In this case, others bifurcation locations remain unchanged.

Conclusions
In this paper, we introduce a competitive controller into the Logistic system.The new feedback controller is designed using time delay.It can keep the equilibria of a system unchanged and also preserve the dimension of the system.Secondly, the controller can delay the time of the first bifurcation of the period-doubling bifurcation, but it could not control the second bifurcation or others after the first bifurcation.The first and second bifurcation locations of Logistic mapping are precisely controlled by the designed controller.

Figure 1 :Figure 2 :
Figure 1: Bifurcation of the Logistic model

Figure 3 :Figure 4 :
Figure 3: The second bifurcation of the controlled system