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ISRN Mathematical Physics
Volume 2012 (2012), Article ID 843962, 12 pages
Research Article

Bifurcation Analysis and Chaos Control in Genesio System with Delayed Feedback

School of Science, Hangzhou Dianzi University, Hangzhou 310018, China

Received 13 September 2011; Accepted 30 October 2011

Academic Editor: P. Minces

Copyright © 2012 Junbiao Guan. 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.


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.

1. Introduction

Since the pioneering work of Lorenz [1], 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 [2], is described by the following simple three-dimensional autonomous system with only one quadratic nonlinear term: ̇𝑥=𝑦,̇𝑦=𝑧,̇𝑧=𝑎𝑥+𝑏𝑦+𝑐𝑧+𝑥2,(1.1) where 𝑎, 𝑏, 𝑐 < 0 are parameters. System (1.1) exhibits chaotic behavior when 𝑎=6, 𝑏=2.92, 𝑐=1.2, as illustrated in Figure 1. In recent years, many researchers have studied this system from many different points of view; Park et al. [35] investigated synchronization of the Genesio chaotic system via backstepping approach, LMI optimization approach, and adaptive controller design. Wu et al. [6] investigated synchronization between Chen system and Genesio system. Chen and Han [7] investigated controlling and synchronization of Genesio chaotic system via nonlinear feedback control. Inspired by the control of chaos via time-delayed feedback force [8] and also following the idea of Pyragas [9], we consider the following Genesio system with delayed feedback control:̇𝑥(𝑡)=𝑦(𝑡),̇𝑦(𝑡)=𝑧(𝑡)+𝑀(𝑦(𝑡)𝑦(𝑡𝜏)),̇𝑧(𝑡)=𝑎𝑥(𝑡)+𝑏𝑦(𝑡)+𝑐𝑧(𝑡)+𝑥2(𝑡),(1.2) where 𝜏>0 and 𝑀𝑅.

Figure 1: Genesio system’s chaotic attractor.

2. Bifurcation Analysis of Genesio System with Delayed Feedback Force

It is easy to see that system (1.1) has two equilibria 𝐸0(0,0,0) and 𝐸1(𝑎,0,0), which are also the equilibria of system (1.2). The associated characteristic equation of system (1.2) at 𝐸0 appears as 𝜆3(𝑀+𝑐)𝜆2+(𝑀𝑐𝑏)𝜆𝑎+𝑀𝜆2𝑒𝑀𝑐𝜆𝜆𝜏=0.(2.1) As the analysis for 𝐸1 is similar, we here only analyze the characteristic equation at 𝐸0. First, we introduce the following result due to Ruan and Wei [10].

Lemma 2.1. Consider the exponential polynomial 𝑃𝜆,𝑒𝜆𝜏1,,𝑒𝜆𝜏𝑚=𝜆𝑛+𝑝1(0)𝜆𝑛1++𝑝(0)𝑛1𝜆+𝑝𝑛(0)+𝑝1(1)𝜆𝑛1++𝑝(1)𝑛1𝜆+𝑝𝑛(1)𝑒𝜆𝜏1+𝑝+1(𝑚)𝜆𝑛1++𝑝(𝑚)𝑛1𝜆+𝑝𝑛(𝑚)𝑒𝜆𝜏𝑚,(2.2) where 𝜏𝑖0(𝑖=1,2,,𝑚) and 𝑝𝑗(𝑖)(𝑖=0,1,,𝑚;𝑗=1,2,,𝑛) are constants. As (𝜏1,𝜏2,,𝜏𝑚) vary, the sum of the order of the zeros of 𝑃(𝜆,𝑒𝜆𝜏1,,𝑒𝜆𝜏𝑚) on the open right half plane can change only if a zero appears on or crosses the imaginary axis.

Denote 𝑝=𝑐2+2𝑏, 𝑞=𝑏22𝑀𝑐𝑏2𝑀𝑎2𝑎𝑐, 𝑟=𝑎2, Δ=𝑝23𝑞=𝑐4+4𝑏𝑐2+6(𝑀𝑏+𝑎)𝑐+6𝑀𝑎+𝑏2, (𝑣)=𝑣3+𝑝𝑣2+𝑞𝑣+𝑟, 𝑣=𝜔2, 𝑣1=(𝑝+Δ)/3, 𝑣2=(𝑝Δ)/3, 𝜏𝑘(𝑗)=(1/𝜔𝑘){cos1((𝑀𝜔2𝑘+𝑀𝑐2𝑏𝑐𝑎)/𝑀(𝜔2𝑘+𝑐2))+2𝑗𝜋}, 𝜏0=min𝑘{1,2,3}{𝜏𝑘(0)}. Following the detailed analysis in [8], we have the following results.

Lemma 2.2. (i) If Δ0, then all roots with positive real parts of (2.1) when 𝜏>0 has the same sum to those of (2.1) when 𝜏=0.
(ii) If Δ>0, 𝑣1>0, (𝑣1)0, then all roots with positive of (2.1) when 𝜏[0,𝜏0] has the same sum to those of (2.1) when 𝜏=0.

Lemma 2.3. Suppose that (v𝑘)0, then d(Re𝜆(𝜏𝑗𝑘))/d𝜏0, and sign{d(Re𝜆(𝜏𝑗𝑘))/d𝜏}=sign{(𝑣𝑘)}.

Proof. Substituting 𝜆(𝜏) into (2.1) and taking the derivative with respect to 𝜏, we can easily calculate that d𝜏Re𝜆𝑗𝑘d𝜏1=3𝑣2𝑘+2𝑝𝑣𝑘+𝑞𝑀2𝜔2𝑘𝜔2𝑘+𝑐2=𝑣𝑘𝑀2𝜔2𝑘𝜔2𝑘+𝑐2,(2.3) thus the results hold.

Theorem 2.4. (i) If Δ0, then (2.1) has two roots with positive real parts for all 𝜏>0.
(ii) If Δ>0, 𝑣1>0, (𝑣1)0, then (2.1) has two roots with positive real parts for 0𝜏<𝜏0.
(iii) If Δ>0, 𝑣1>0, (𝑣1)0 and (𝑣𝑘)0, then system (1.2) exhibits the Hopf bifurcation at the equilibrium 𝐸0 for 𝜏=𝜏𝑘(𝑗).

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 [11] to study some properties of bifurcated periodic solutions. Without loss of generality, let (𝑥,𝑦,𝑧) be the equilibrium point of system (1.2). For the sake of convenience, we rescale the time variable 𝑡=𝜏𝑡 and let 𝜏=𝜏𝑘+𝜇, 𝑥1=𝑥𝑥, 𝑥2=𝑦𝑦, 𝑥3=𝑧𝑧, then system (1.2) can be replaced by the following system:̇𝑥(𝑡)=𝐿𝜇𝑥𝑡+𝑓𝜇,𝑥𝑡,(3.1) where 𝑥(𝑡)=(𝑥1(𝑡),𝑥2(𝑡),𝑥3(𝑡))𝑇𝑅3, and for 𝜙=(𝜙1,𝜙2,𝜙3)𝑇𝐶, 𝐿𝜇 and 𝑓 are, respectively, given as𝐿𝜇𝜏(𝜙)=𝑘+𝜇0100𝑀1𝑎+2𝑥𝜙𝑏𝑐1𝜙(0)2(𝜙0)3+𝜏(0)𝑘𝜙+𝜇0000𝑀00001𝜙(1)2(𝜙1)3𝜏(1),(3.2)𝑓(𝜏,𝜙)=𝑘00𝜙+𝜇23(0).(3.3) By the Riesz representation theorem, there exists a function 𝜂(𝜃,𝜇) of bounded variation for 𝜃[1,0], such that𝐿𝜇(𝜙)=01d𝜂(𝜃,0)𝜙(𝜃),𝜙𝐶.(3.4) In fact, the above equation holds if we choose𝜏𝜂(𝜃,𝜇)=𝑘+𝜇0100𝑀1𝑎+2𝑥𝜏𝑏𝑐𝛿(𝜃)𝑘+𝜇0000𝑀0000𝛿(𝜃+1),(3.5) where 𝛿 is Durac function. For 𝜙𝐶1([𝜏,0],𝑅), let𝐴(𝜇)𝜙=d𝜙(𝜃)d𝜃,1𝜃<0,0𝜏d𝜂(𝜃,𝜇)𝜙(𝜃),𝜃=0,𝑅(𝜇)𝜙=0,1𝜃<0,𝐹(𝜇,𝜙),𝜃=0.(3.6) Then (1.2) can be rewritten in the following form:̇𝑥(𝑡)=𝐴(𝜇)𝑥𝑡+𝑅(𝜇)𝑥𝑡.(3.7) For 𝜓𝐶[0,1], we consider the adjoint operator 𝐴 of 𝐴 defined by𝐴(𝜇)𝜓(𝑠)=d𝜓(𝑠)d𝑠,0<𝑠1,0𝜏d𝜂𝑇(𝑡,0)𝜓(𝑡),𝑠=0.(3.8) For 𝜙𝐶[1,0] and 𝜓𝐶[0,1], we define the bilinear inner product form as𝜓,𝜙=𝜓(0)𝜙(0)0𝜃=𝜏𝜃𝑠=0𝜓(𝑠𝜃)d𝜂(𝜃)𝜙(𝑠)d𝑠.(3.9)

Suppose that 𝑞(𝜃)=(1,𝛼,𝛽)𝑇𝑒i𝜃𝜔𝑘𝜏𝑘(1𝜃0) is the eigenvectors of 𝐴(0) with respect to i𝜔𝑘𝜏𝑘, then 𝐴(0)𝑞(𝜃)=i𝜔𝑘𝜏𝑘𝑞(𝜃). By the definition of 𝐴 and (3.2), (3.4), and (3.5) we have𝜏𝑘i𝜔𝑘100i𝜔𝑘𝑀+𝑀𝑒i𝜔𝑘𝜏𝑘1𝑎2𝑥𝑏i𝜔𝑘000𝑐𝑞(0)=.(3.10) Hence𝑞(𝑠)=(1,𝛼,𝛽)𝑇=1,i𝜔𝑘,𝑎+2𝑥+i𝑏𝜔𝑘i𝜔𝑘𝑐𝑇𝑒i𝜃𝜔𝑘𝜏𝑘.(3.11) Similarly, let 𝑞(𝑠)=𝐵(1,𝛼,𝛽)𝑒i𝑠𝜔𝑘𝜏𝑘(0𝑠1) be the eigenvector of 𝐴 with respect to i𝜔𝑘𝜏𝑘, by the definition of 𝐴 and (3.2), (3.4), and (3.5) we can obtain𝑞(𝑠)=𝐵1,𝛼,𝛽𝑒i𝑠𝜔𝑘𝜏𝑘=11+𝛼𝛼+𝛽𝛽𝑀𝛼𝛼𝜏𝑘𝑒i𝜔𝑘𝜏𝑘1,i𝜔𝑘i𝜔𝑘+𝑐𝑎+2𝑥,i𝜔𝑘𝑎+2𝑥.(3.12) Furthermore, 𝑞,𝑞=1, 𝑞,𝑞=0.

Let 𝑧(𝑡)=𝑞,𝑥𝑡, where 𝑥𝑡 is the solution of (3.7) when 𝜇=0. We denote 𝑤(𝑡,𝜃)=𝑢𝑡(𝜃)2Re{𝑧(𝑡)𝑞(𝜃)}, theṅ𝑧(𝑡)=i𝜔𝑘𝜏𝑘𝑧(𝑡)+𝑞𝜇(0)𝑓0,𝑤𝑧,𝑧+2Re{𝑧𝑞(0)}=i𝜔𝑘𝜏𝑘𝑧(𝑡)+𝑞(0)𝑓0𝑧,𝑧.(3.13) We Rewrite (3.13) in the following form:̇𝑧(𝑡)=i𝜔𝑘𝜏𝑘𝑧(𝑡)+𝑔𝑧,𝑧,(3.14) where𝑔𝑧,𝑧=𝑔20𝑧22+𝑔11𝑧𝑧+𝑔02𝑧22+𝑔21𝑧2𝑧2+.(3.15) Noticing that𝑤𝑧,𝑧=𝑤20𝑧22+𝑤11𝑧𝑧+𝑤02𝑧22+,(3.16) we havė𝑤=𝐴𝑤2Re𝑞(0)𝑓0𝑞(𝜃),1𝜃<0,𝐴𝑤2Re𝑞(0)𝑓0𝑞(𝜃)+𝑓0,𝜃=0.(3.17) Definė𝑤=𝐴𝑤+𝐻𝑧,𝑧,𝜃,(3.18) with𝐻𝑧,𝑧,𝜃=𝐻20𝑧22+𝐻11𝑧𝑧+𝐻02𝑧22+.(3.19) On the other hand,̇𝑤=𝑤𝑧̇𝑧+𝑤𝑧̇𝑧=𝐴𝑤+𝐻𝑧,.𝑧,𝜃(3.20) Expanding the above series and comparing the corresponding coefficients, we obtain𝐴2i𝜔𝑘𝜏𝑘𝑤20(𝜃)=𝐻20(𝜃)𝐴𝑤11(𝜃)=𝐻11(𝜃)(3.21) While𝑥𝑡(𝜃)=𝑤𝑧,𝑧(𝜃)+𝑧𝑞(𝜃)+𝑧𝑞𝑔(𝜃),𝑧,𝑧=𝑔20𝑧22+𝑔11𝑧𝑧+𝑔02𝑧22+𝑔21𝑧2𝑧2+=𝑞(0)𝑓0.(3.22) Let 𝑥𝑡(𝜃)=(𝑥𝑡(1)(𝜃),𝑥𝑡(2)(𝜃),𝑥𝑡(3)(𝜃)), then we have𝑥𝑡(1)(0)=𝑧+𝑧+𝑤(1)20𝑧(0)22+𝑤(1)11(0)𝑧𝑧+𝑤(1)02(0)𝑧22||+𝑂𝑧,𝑧||3,𝑥𝑡(2)(0)=𝛼𝑧+𝛼𝑧+𝑤(2)20𝑧(0)22+𝑤(2)11(0)𝑧𝑧+𝑤(2)02(0)𝑧22||+𝑂𝑧,𝑧||3,𝑥𝑡(3)(0)=𝛽𝑧+𝛽𝑧+𝑤(3)20𝑧(0)22+𝑤(3)11(0)𝑧𝑧+𝑤(3)02(0)𝑧22||+𝑂𝑧,𝑧||3.(3.23) Therefore we have𝑔𝑧,𝑧=𝐵𝜏𝑘1,𝛼,𝛽00𝑥𝑡(1)(0)2=𝐵𝜏𝑘𝛽𝑧+𝑧+𝑤(1)20𝑧(0)22+𝑤(1)11(0)𝑧𝑧+𝑤(1)02(0)𝑧22||+𝑂𝑧,𝑧||32.(3.24) Comparing the corresponding coefficients, we have𝑔20=𝑔11=𝑔02=2𝐵𝜏𝑘𝛽,𝑔21=2𝐵𝜏𝑘𝛽𝑤(1)20(0)+2𝑤(1)11.(0)(3.25)

In what follows we will need to compute 𝑤11(𝜃) and 𝑤20(𝜃). Firstly we compute 𝑤11(𝜃), 𝑤20(𝜃) when 𝜃[1,0). It follows from (3.18) that𝐻𝑧,𝑧,𝜃=2Re𝑞(0)𝑓0𝑞(𝜃)=𝑔𝑞(𝜃)𝑔𝑞(𝜃).(3.26) Substituting the above equation into (3.21) and comparing the corresponding coefficients yields𝐻20(𝜃)=𝑔20𝑞(𝜃)𝑔02𝐻𝑞(𝜃),(3.27)11(𝜃)=𝑔11𝑞(𝜃)𝑔11𝑞(𝜃).(3.28) By (3.21), (3.28), and the definition of 𝐴 we havė𝑤20(𝜃)=2i𝜔𝑘𝜏𝑘𝑤20(𝜃)+𝑔20𝑞(𝜃)+𝑔02𝑞(𝜃).(3.29) Hence𝑤20(𝜃)=i𝑔20𝜔𝑘𝜏𝑘𝑞(0)𝑒i𝜔𝑘𝜏𝑘𝜃+i𝑔023𝜔𝑘𝜏𝑘𝑞(0)𝑒i𝜔𝑘𝜏𝑘𝜃+𝐸1𝑒2i𝜔𝑘𝜏𝑘𝜃.(3.30) Similarly we have𝑤11(𝜃)=i𝑔11𝜔𝑘𝜏𝑘𝑞(0)𝑒i𝜔𝑘𝜏𝑘𝜃+i𝑔11𝜔𝑘𝜏𝑘𝑞(0)𝑒i𝜔𝑘𝜏𝑘𝜃+𝐸2.(3.31)

In what follows, we will seek appropriate 𝐸1 and 𝐸2 in (3.30) and (3.31). When 𝜃=0,𝐻𝑧,𝑧,𝜃=2Re𝑞(0)𝑓0𝑞(𝜃)+𝑓0=𝑔𝑞(0)𝑔𝑞(0)+𝑓0(3.32) with𝑓0=𝑓0,𝑧2𝑧22+𝑓0,𝑧𝑧𝑧𝑧+𝑓0,𝑧2𝑧22+.(3.33) Comparing the coefficients in (3.18) we have𝐻20(0)=𝑔20𝑞(0)𝑔02𝑞(0)+𝑓0,𝑧2,𝐻11(0)=𝑔11𝑞(0)𝑔11𝑞(0)+𝑓0,𝑧𝑧.(3.34) By (3.21) and the definition of 𝐴 we have01d𝜂(𝜃)𝑤20(𝜃)=2i𝜔𝑘𝜏𝑘𝑤20(𝜃)+𝑔20𝑞(0)+𝑔0200𝑞(0)+2𝜏𝑘.(3.35)01d𝜂(𝜃)𝑤11(𝜃)=𝑔11𝑞(0)+𝑔1100𝑞(0)+2𝜏𝑘.(3.36) Substituting (3.30) into (3.36) and noticing thati𝜔𝑘𝜏𝑘𝑞(0)=01𝑒i𝜔𝑘𝜏𝑘𝜃d𝜂(𝜃)𝑞(0),i𝜔𝑘𝜏𝑘𝑞(0)=01𝑒i𝜔𝑘𝜏𝑘𝜃d𝜂(𝜃)𝑞(0),(3.37) we have2i𝜔𝑘𝜏𝑘𝐼01𝑒2i𝜔𝑘𝜏𝑘𝜃𝐸d𝜂(𝜃)1=𝑓0,𝑧2,(3.38) namely,2i𝜔𝑘1002i𝜔𝑘𝑀+𝑀𝑒2𝜔𝑘𝜏𝑘1𝑎2𝑥𝑏2i𝜔𝑘𝐸𝑐1=002.(3.39) Thus𝐸1(1)=2𝑅,𝐸1(2)=4i𝜔𝑘𝑅,𝐸1(3)=4i𝜔𝑘2i𝜔𝑘𝑀+𝑀𝑒2𝜔𝑘𝜏𝑘𝑅,(3.40) where|||||||||𝑅=2i𝜔𝑘1002i𝜔𝑘𝑀+𝑀𝑒2𝜔𝑘𝜏𝑘1𝑎2𝑥𝑏2i𝜔𝑘|||||||||𝑐.(3.41) Following the similar analysis, we also have010001𝑎2𝑥𝐸𝑏𝑐2=002,(3.42) hence𝐸2(1)=2𝑆,𝐸2(2)=0,𝐸2(3)=0,(3.43) where|||||||||𝑆=010001𝑎2𝑥|||||||||𝑏𝑐.(3.44) Thus the following values can be computed:𝑐1i(0)=2𝜔𝑘𝜏𝑘𝑔20𝑔11||𝑔211||2||𝑔02||23+𝑔212,𝜇2𝑐=Re1(0)𝜆Re𝜏𝑘,𝜒2𝑐=Im1(0)+𝜇2𝜆Im𝜏𝑘𝜔𝑘𝜏𝑘,𝛽2𝑐=2Re1.(0)(3.45)

It is well known in [11] that 𝜇2 determines the directions of the Hopf bifurcation: if 𝜇2>0(<0), then the Hopf bifurcation is supercritical(subcritical) and the bifurcated periodic solution exists if 𝜏>𝜏𝑘(𝜏<𝜏𝑘); 𝜒2 determines the period of the bifurcated periodic solution: if 𝜒2>0(<0), then the period increase(decrease); 𝛽2 determines the stability of the Hopf bifurcation: if 𝛽2<0(>0), 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:̇𝑥(𝑡)=𝑦(𝑡),̇𝑦(𝑡)=𝑧(𝑡)+𝑀(𝑦(𝑡)𝑦(𝑡𝜏)),̇𝑧(𝑡)=6𝑥(𝑡)2.92𝑦(𝑡)1.2𝑧(𝑡)+𝑥2(𝑡).(4.1) Obviously, system (4.1) has two equilibria 𝐸0(0,0,0) and 𝐸1(6,0,0). In what follows we analyze the case of 𝐸0 only, the analysis for 𝐸1 is similar. The corresponding characteristic equation of system (4.1) at 𝐸0 appears as𝜆3(𝑀1.2)𝜆2+(1.2𝑀+2.92)𝜆+6+𝑀𝜆2𝑒+1.2𝑀𝜆𝜆𝜏=0.(4.2) Hence we have 𝑝=4.4, 𝑞=5.8736+4.992𝑀, 𝑟=36, Δ=36.980814.976𝑀, (𝑣)=𝑣3+𝑝𝑣2+𝑞𝑣+𝑟, 𝑣=𝜔2, 𝑣1=(𝑝+Δ)/3=(1/3)(4.4+36.980814.976𝑀), 𝑣2=(1/3)(4.436.980814.976𝑀), 𝜏𝑘(𝑗)=(1/𝜔𝑘){cos1((𝑀𝜔2𝑘+𝑀𝑐2𝑏𝑐𝑎)/(𝑀𝜔2𝑘+𝑀𝑐2))+2𝑗𝜋}, 𝜏0=min𝑘{1,2,3}{𝜏𝑘(0)}. By Theorem 2.4, when Δ=36.980814.976𝑀0, that is, 𝑀2.46934, (4.2) has two roots with positive real parts for all 𝜏>0. In order to realize the control of chaos, we will consider 𝑀<2.46934. We take 𝑀=8 as a special case. In this case, system (4.1) takes the form oḟ𝑥(𝑡)=𝑦(𝑡),̇𝑦(𝑡)=𝑧(𝑡)8𝑦(𝑡)+8𝑦(𝑡𝜏),̇𝑧(𝑡)=6𝑥(𝑡)2.92𝑦(𝑡)1.2𝑧(𝑡)+𝑥2(𝑡).(4.3) Thus we can compute Δ=156.789, 𝑣15.64051, (𝑣1)182.922, 𝑣14.30249, 𝑣20.873751, 𝜔12.07424, 𝜔20.934746, (𝑣1)28.1373, (𝑣2)51.2083, 𝜏1(0)0.632012, 𝜏2(0)1.85965,𝜏00.632012. Therefore, using the results in the previous sections, we have the following conclusions: when the delay 𝜏=0.1<0.632012, the attractor still exists, see Figure 2; when the delay 𝜏=0.632, Hopf bifurcation occurs, see Figure 3. Moreover, 𝜇2>0, 𝛽2<0, the bifurcating periodic solutions are orbitally asymptotically stable; when the delay 𝜏=1.2[0.632012,1.85965], the steady state 𝑆0 is locally stable, see Figure 4; when the delay 𝜏=3.2>1.85965, the steady state 𝑆0 is unstable, see Figure 5. Numerical results indicate that as the delay sets in an interval, the chaotic behaviors really disappear. Therefore the parameter 𝜏 works well in control of chaos.

Figure 2: Phase diagram for system (4.3) with 𝜏=0.1 and initial value (0.1, 0.1, 0.2).
Figure 3: Phase diagram for system (4.3) with 𝜏=0.632 and initial value (0.1, 0.1, 0.2).
Figure 4: Phase diagram for system (4.3) with 𝜏=1.2 and initial value (0.1, 0.1, 0.2).
Figure 5: Phase diagram for system (4.3) with 𝜏=3.2 and initial value (0.1, 0.1, 0.2).

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).


  1. E. Lorenz, “Deterministic non-periodic flow,” Journal of the Atmospheric Sciences, vol. 20, pp. 130–141, 1963.
  2. R. Genesio and A. Tesi, “Harmonic balance methods for the analysis of chaotic dynamics in nonlinear systems,” Automatica, vol. 28, no. 3, pp. 531–548, 1992. View at Publisher · View at Google Scholar · View at Scopus
  3. J. H. Park, “Adaptive controller design for modified projective synchronization of Genesio-Tesi chaotic system with uncertain parameters,” Chaos, Solitons and Fractals, vol. 34, no. 4, pp. 1154–1159, 2007. View at Publisher · View at Google Scholar · View at Scopus
  4. J. H. Park, “Synchronization of Genesio chaotic system via backstepping approach,” Chaos, Solitons and Fractals, vol. 27, no. 5, pp. 1369–1375, 2006. View at Publisher · View at Google Scholar · View at Scopus
  5. J. H. Park, O. M. Kwon, and S. M. Lee, “LMI optimization approach to stabilization of Genesio-Tesi chaotic system via dynamic controller,” Applied Mathematics and Computation, vol. 196, no. 1, pp. 200–206, 2008. View at Publisher · View at Google Scholar · View at Scopus
  6. X. Wu, Z. H. Guan, Z. Wu, and T. Li, “Chaos synchronization between Chen system and Genesio system,” Physics Letters A, vol. 364, no. 6, pp. 484–487, 2007. View at Publisher · View at Google Scholar · View at Scopus
  7. M. Chen and Z. Han, “Controlling and synchronizing chaotic Genesio system via nonlinear feedback control,” Chaos, Solitons and Fractals, vol. 17, no. 4, pp. 709–716, 2003. View at Publisher · View at Google Scholar · View at Scopus
  8. Y. Song and J. Wei, “Bifurcation analysis for Chen's system with delayed feedback and its application to control of chaos,” Chaos, Solitons and Fractals, vol. 22, no. 1, pp. 75–91, 2004. View at Publisher · View at Google Scholar · View at Scopus
  9. K. Pyragas, “Continuous control of chaos by self-controlling feedback,” Physics Letters A, vol. 170, no. 6, pp. 421–428, 1992. View at Scopus
  10. 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 Scopus
  11. B. D. Hassard, N. D. Kazarinoff, and Y. H. Wan, Theory and Applications of Hopf Bifurcation, vol. 41, Cambridge University Press, Cambridge, UK, 1981.