Bifurcation Analysis and Chaos Control in Genesio System with Delayed Feedback

Abstract

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: 𝑧̇=𝑎𝑥+𝑥𝑦̇𝑏̇==𝑦+𝑧𝑦,,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. [3–5] 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:𝑧̇𝑦(𝑡̇()𝑡)==𝑎𝑥𝑧((𝑡𝑡))𝑥++̇(𝑡𝑀𝑏)𝑦=((𝑡𝑦)𝑦(+(𝑡𝑡))𝑐−,𝑧(𝑦𝑡)(𝑡+−𝑥𝜏₂)),(𝑡),(1.2) where 𝜏>0 and 𝑀∈𝑅.

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+(𝑀𝑐−𝑏)𝜆−𝑎+𝑀𝜆²−𝑀𝑐𝜆𝑒^−𝜆𝜏=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 𝑃𝜆,𝑒^−𝜆𝜏1,…,𝑒^{−𝜆𝜏𝑚}=𝜆^𝑛+𝑝¹⁽⁰⁾𝜆^𝑛−1+⋯+𝑝^(0)𝑛−1𝜆+𝑝^𝑛(0)++𝑝𝑝¹⁽¹⁾𝜆^𝑛−1+⋯+𝑝^(1)𝑛−1𝜆+𝑝^𝑛(1)𝑒^−𝜆𝜏1+⋯_1(𝑚)𝜆^𝑛−1+⋯+𝑝^{(𝑚)𝑛−1}𝜆+𝑝^𝑛(𝑚)𝑒^{−𝜆𝜏𝑚},(2.2) where 𝜏_𝑖≥0(𝑖=1,2,…,𝑚) and 𝑝^𝑗(𝑖)(𝑖=0,1,…,𝑚;𝑗=1,2,…,𝑛) are constants. As (𝜏₁,𝜏₂,…,𝜏_𝑚) vary, the sum of the order of the zeros of 𝑃(𝜆,𝑒^−𝜆𝜏1,…,𝑒^{−𝜆𝜏𝑚}) on the open right half plane can change only if a zero appears on or crosses the imaginary axis.

Denote 𝑝=𝑐²+2𝑏, 𝑞=𝑏²−2𝑀𝑐𝑏−2𝑀𝑎−2𝑎𝑐, 𝑟=𝑎², Δ=𝑝²−3𝑞=𝑐⁴+4𝑏𝑐²+6(𝑀𝑏+𝑎)𝑐+6𝑀𝑎+𝑏², ℎ(𝑣)=𝑣³+𝑝𝑣²+𝑞𝑣+𝑟, 𝑣=𝜔², 𝑣^∗1=(−𝑝+√Δ)/3, 𝑣^∗2=(−𝑝−√Δ)/3, 𝜏^𝑘(𝑗)=(1/𝜔_𝑘){cos⁻¹((𝑀𝜔^2𝑘+𝑀𝑐²−𝑏𝑐−𝑎)/𝑀(𝜔^2𝑘+𝑐²))+2𝑗𝜋}, 𝜏₀=min_{𝑘∈{1,2,3}}{𝜏^𝑘(0)}. Following the detailed analysis in [8], we have the following results.

Lemma 2.2. (i) If Δ≤0, 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, 𝑣^∗1>0, ℎ(𝑣^∗1)≤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 ℎ′(v_𝑘)≠0, then d(Re𝜆(𝜏^𝑗𝑘))/d𝜏≠0, and sign{d(Re𝜆(𝜏^𝑗𝑘))/d𝜏}=sign{ℎ^′(𝑣_𝑘)}.

Proof. Substituting 𝜆(𝜏) into (2.1) and taking the derivative with respect to 𝜏, we can easily calculate that dRe𝜆𝜏^𝑗𝑘d𝜏⁻¹=3𝑣^2𝑘+2𝑝𝑣_𝑘+𝑞𝑀²𝜔^2𝑘𝜔^2𝑘+𝑐²=ℎ^′𝑣_𝑘𝑀²𝜔^2𝑘𝜔^2𝑘+𝑐²,(2.3) thus the results hold.

Theorem 2.4. (i) If Δ≤0, then (2.1) has two roots with positive real parts for all 𝜏>0.
(ii) If Δ>0, 𝑣^∗1>0, ℎ(𝑣^∗1)≤0, then (2.1) has two roots with positive real parts for 0≤𝜏<𝜏₀.
(iii) If Δ>0, 𝑣^∗1>0, ℎ(𝑣^∗1)≤0 and ℎ′(𝑣_𝑘)≠0, then system (1.2) exhibits the Hopf bifurcation at the equilibrium 𝐸₀ for 𝜏=𝜏^𝑘(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=𝜏_𝑘+𝜇, 𝑥₁=𝑥−𝑥_∗, 𝑥₂=𝑦−𝑦_∗, 𝑥₃=𝑧−𝑧_∗, then system (1.2) can be replaced by the following system:𝑥̇(𝑡)=𝐿_𝜇𝑥_𝑡+𝑓𝜇,𝑥_𝑡,(3.1) where 𝑥(𝑡)=(𝑥₁(𝑡),𝑥₂(𝑡),𝑥₃(𝑡))^𝑇∈𝑅³, and for 𝜙=(𝜙₁,𝜙₂,𝜙₃)^𝑇∈𝐶, 𝐿_𝜇 and 𝑓 are, respectively, given as𝐿_𝜇(𝜙)=𝜏_𝑘+𝜇⎛⎜⎜⎜⎜⎝𝑎+002𝑥_∗𝑀101𝑏𝑐⎞⎟⎟⎟⎟⎠⎛⎜⎜⎜⎜⎝𝜙¹𝜙(0)₂𝜙(0)₃(0)⎞⎟⎟⎟⎟⎠+𝜏^𝑘+𝜇⎛⎜⎜⎜⎜⎝000−00𝑀000⎞⎟⎟⎟⎟⎠⎛⎜⎜⎜⎜⎝𝜙¹𝜙(−1)₂𝜙(−1)₃𝑓(𝜏,𝜙)=𝜏_𝑘(−1)⎞⎟⎟⎟⎟⎠,(3.2)+𝜇⎛⎜⎜⎜⎜⎝𝜙₂₃00(0)⎞⎟⎟⎟⎟⎠.(3.3) By the Riesz representation theorem, there exists a function 𝜂(𝜃,𝜇) of bounded variation for 𝜃∈[−1,0], such that𝐿_𝜇(𝜙)=⁰⁻¹d𝜂(𝜃,0)𝜙(𝜃),𝜙∈𝐶.(3.4) In fact, the above equation holds if we choose𝜂(𝜃,𝜇)=𝜏_𝑘+𝜇⎛⎜⎜⎜⎜⎝𝑎+002𝑥_∗𝑀101𝑏𝑐⎞⎟⎟⎟⎟⎠𝛿(𝜃)−𝜏^𝑘+𝜇⎛⎜⎜⎜⎜⎝000−00𝑀000⎞⎟⎟⎟⎟⎠𝛿(𝜃+1),(3.5) where 𝛿 is Durac function. For 𝜙∈𝐶¹([−𝜏,0],𝑅), let𝐴(𝜇)𝜙=⎧⎪⎨⎪⎩d𝜙(𝜃)d𝜃,−1≤𝜃<0,_0−𝜏𝑅(𝜇)𝜙=0𝐹,(𝜇d𝜂,𝜙(𝜃),,𝜇)𝜃−𝜙1=(𝜃≤0).,𝜃<𝜃0=,0,(3.6) Then (1.2) can be rewritten in the following form:𝑥̇(𝑡)=𝐴(𝜇)𝑥_𝑡+𝑅(𝜇)𝑥_𝑡.(3.7) For 𝜓∈𝐶[0,1], we consider the adjoint operator 𝐴^∗ of 𝐴 defined by𝐴^∗(𝜇)𝜓(𝑠)=⎧⎪⎨⎪⎩−d𝜓d𝑠(𝑠),0<𝑠≤1,_0−𝜏d𝜂^𝑇(𝑡,0)𝜓(−𝑡),𝑠=0.(3.8) For 𝜙∈𝐶[−1,0] and 𝜓∈𝐶[0,1], we define the bilinear inner product form as⟨𝜓,𝜙⟩=𝜓(0)𝜙(0)−^{0𝜃=−𝜏}^𝜃𝑠=0𝜓(𝑠−𝜃)d𝜂(𝜃)𝜙(𝑠)d𝑠.(3.9)

Suppose that 𝑞(𝜃)=(1,𝛼,𝛽)^𝑇𝑒^{i𝜃𝜔𝑘𝜏𝑘}(−1≤𝜃≤0) is the eigenvectors of 𝐴(0) with respect to i𝜔_𝑘𝜏_𝑘, then 𝐴(0)𝑞(𝜃)=i𝜔_𝑘𝜏_𝑘𝑞(𝜃). By the definition of 𝐴 and (3.2), (3.4), and (3.5) we have𝜏_𝑘⎛⎜⎜⎜⎜⎝i𝜔0_𝑘i𝜔_𝑘−10−𝑎−2𝑥_∗−𝑀+𝑀𝑒^{−i𝜔𝑘𝜏𝑘}−1−𝑏i𝜔_𝑘−𝑐⎞⎟⎟⎟⎟⎠𝑞(0)=⎛⎜⎜⎜⎜⎝000⎞⎟⎟⎟⎟⎠.(3.10) Hence𝑞(𝑠)=(1,𝛼,𝛽)^𝑇=1,i𝜔_𝑘,𝑎+2𝑥^∗+i𝑏𝜔_𝑘i𝜔_𝑘−𝑐^𝑇𝑒^{i𝜃𝜔𝑘𝜏𝑘}.(3.11) Similarly, let 𝑞^∗(𝑠)=𝐵(1,𝛼^∗,𝛽^∗)𝑒^{i𝑠𝜔𝑘𝜏𝑘}(0≤𝑠≤1) be the eigenvector of 𝐴^∗ with respect to −i𝜔_𝑘𝜏_𝑘, by the definition of 𝐴^∗ and (3.2), (3.4), and (3.5) we can obtain𝑞^∗(𝑠)=𝐵1,𝛼^∗,𝛽^∗𝑒^{i𝑠𝜔𝑘𝜏𝑘}=11+𝛼𝛼^∗+𝛽𝛽^∗−𝑀𝛼𝛼^∗𝜏_𝑘𝑒^{−i𝜔𝑘𝜏𝑘}1,i𝜔^𝑘i𝜔_𝑘+𝑐𝑎+2𝑥_∗,−i𝜔^𝑘𝑎+2𝑥_∗.(3.12) Furthermore, ⟨𝑞^∗,𝑞⟩=1, ⟨𝑞^∗,𝑞⟩=0.

Let 𝑧(𝑡)=⟨𝑞^∗,𝑥_𝑡⟩, where 𝑥_𝑡 is the solution of (3.7) when 𝜇=0. We denote 𝑤(𝑡,𝜃)=𝑢_𝑡(𝜃)−2Re{𝑧(𝑡)𝑞(𝜃)}, then𝑧̇(𝑡)=i𝜔_𝑘𝜏_𝑘𝑧(𝑡)+𝑞^∗(0)𝑓𝜇₀,𝑤𝑧,𝑧+2Re{𝑧𝑞(0)}=i𝜔_𝑘𝜏_𝑘𝑧(𝑡)+𝑞^∗(0)𝑓₀𝑧,𝑧.(3.13) We Rewrite (3.13) in the following form:𝑧̇(𝑡)=i𝜔_𝑘𝜏_𝑘𝑧(𝑡)+𝑔𝑧,𝑧,(3.14) where𝑔𝑧,𝑧=𝑔₂₀𝑧²2+𝑔¹¹𝑧𝑧+𝑔₀₂𝑧²2+𝑔²¹𝑧²𝑧2+⋯.(3.15) Noticing that𝑤𝑧,𝑧=𝑤₂₀𝑧²2+𝑤¹¹𝑧𝑧+𝑤₀₂𝑧²2+⋯,(3.16) we have𝑤̇=⎧⎪⎨⎪⎩𝐴𝑤−2Re𝐴𝑤−2Re𝑞^∗(0)𝑓₀𝑞(𝜃),−1≤𝜃<0,𝑞^∗(0)𝑓₀𝑞(𝜃)+𝑓₀,𝜃=0.(3.17) Define𝑤̇=𝐴𝑤+𝐻𝑧,𝑧,𝜃,(3.18) with𝐻𝑧,𝑧,𝜃=𝐻₂₀𝑧²2+𝐻¹¹𝑧𝑧+𝐻₀₂𝑧²2+⋯.(3.19) On the other hand,𝑤̇=𝑤_𝑧𝑧̇+𝑤_𝑧𝑧̇‌=𝐴𝑤+𝐻𝑧,𝑧,𝜃.(3.20) Expanding the above series and comparing the corresponding coefficients, we obtain𝐴−2𝐴i𝜔𝑤_𝑘𝜏_𝑘11𝑤₂₀(𝜃)=−𝐻₂₀(𝜃)(𝜃)⋮=−𝐻₁₁(𝜃)(3.21) While𝑔𝑧,𝑥_𝑡(𝜃)=𝑤𝑧,𝑧(𝜃)+𝑧𝑞(𝜃)+𝑧𝑞(𝜃),𝑧=𝑔₂₀𝑧²2+𝑔¹¹𝑧𝑧+𝑔₀₂𝑧²2+𝑔²¹𝑧²𝑧2+⋯=𝑞^∗(0)𝑓₀.(3.22) Let 𝑥_𝑡(𝜃)=(𝑥^𝑡(1)(𝜃),𝑥^𝑡(2)(𝜃),𝑥^𝑡(3)(𝜃)), then we have𝑥^𝑡(1)(0)=𝑧+𝑧+𝑤⁽¹⁾²⁰(0)𝑧²2+𝑤⁽¹⁾¹¹(0)𝑧𝑧+𝑤⁽¹⁾⁰²(0)𝑧²2+𝑂||𝑧,𝑥_𝑡(2)𝑧||³,(0)=𝛼𝑧+𝛼𝑧+𝑤⁽²⁾²⁰(0)𝑧²2+𝑤⁽²⁾¹¹(0)𝑧𝑧+𝑤⁽²⁾⁰²(0)𝑧²2+𝑂||𝑧,𝑥_𝑡(3)𝑧||³,(0)=𝛽𝑧+𝛽𝑧+𝑤⁽³⁾²⁰(0)𝑧²2+𝑤⁽³⁾¹¹(0)𝑧𝑧+𝑤⁽³⁾⁰²(0)𝑧²2+𝑂||𝑧,𝑧||³.(3.23) Therefore we have𝑔𝑧,𝑧=𝐵𝜏_𝑘1,𝛼^∗,𝛽^∗⎛⎜⎜⎜⎜⎝𝑥_𝑡(1)00(0)²=⎞⎟⎟⎟⎟⎠𝐵𝜏_𝑘𝛽^∗𝑧+𝑧+𝑤⁽¹⁾²⁰(0)𝑧²2+𝑤⁽¹⁾¹¹(0)𝑧𝑧+𝑤⁽¹⁾⁰²(0)𝑧²2+𝑂||𝑧,𝑧||³².(3.24) Comparing the corresponding coefficients, we have𝑔𝑔₂₀₂₁=𝑔₁₁=𝑔₀₂=2𝐵𝜏_𝑘𝛽^∗,=2𝐵𝜏_𝑘𝛽^∗𝑤⁽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) that𝐻𝑧,𝑧,𝜃=−2Re𝑞^∗(0)𝑓₀𝑞(𝜃)=−𝑔𝑞(𝜃)−𝑔𝑞(𝜃).(3.26) Substituting the above equation into (3.21) and comparing the corresponding coefficients yields𝐻𝐻₂₀(𝜃)=−𝑔₂₀𝑞(𝜃)−𝑔₀₂𝑞(𝜃),(3.27)₁₁(𝜃)=−𝑔₁₁𝑞(𝜃)−𝑔₁₁𝑞(𝜃).(3.28) By (3.21), (3.28), and the definition of 𝐴 we have𝑤̇₂₀(𝜃)=2i𝜔_𝑘𝜏_𝑘𝑤₂₀(𝜃)+𝑔₂₀𝑞(𝜃)+𝑔₀₂𝑞(𝜃).(3.29) Hence𝑤₂₀(𝜃)=i𝑔²⁰𝜔_𝑘𝜏_𝑘𝑞(0)𝑒^{i𝜔𝑘𝜏𝑘𝜃}+i𝑔₀₂3𝜔_𝑘𝜏_𝑘𝑞(0)𝑒^{−i𝜔𝑘𝜏𝑘𝜃}+𝐸₁𝑒^{2i𝜔𝑘𝜏𝑘𝜃}.(3.30) Similarly we have𝑤₁₁(𝜃)=−i𝑔¹¹𝜔_𝑘𝜏_𝑘𝑞(0)𝑒^{i𝜔𝑘𝜏𝑘𝜃}+i𝑔₁₁𝜔_𝑘𝜏_𝑘𝑞(0)𝑒^{−i𝜔𝑘𝜏𝑘𝜃}+𝐸₂.(3.31)

In what follows, we will seek appropriate 𝐸₁ and 𝐸₂ in (3.30) and (3.31). When 𝜃=0,𝐻𝑧,𝑧,𝜃=−2Re𝑞^∗(0)𝑓₀𝑞(𝜃)+𝑓₀=−𝑔𝑞(0)−𝑔𝑞(0)+𝑓₀(3.32) with𝑓₀=𝑓_0,𝑧2𝑧²2+𝑓^0,𝑧𝑧𝑧𝑧+𝑓_0,𝑧²𝑧²2+⋯.(3.33) Comparing the coefficients in (3.18) we have𝐻𝐻₂₀₁₁(0)=−𝑔₂₀𝑞(0)−𝑔₀₂𝑞(0)+𝑓_0,𝑧2,(0)=−𝑔₁₁𝑞(0)−𝑔₁₁𝑞(0)+𝑓_0,𝑧𝑧.(3.34) By (3.21) and the definition of 𝐴 we have⁰⁻¹d𝜂(𝜃)𝑤₂₀(𝜃)=2i𝜔_𝑘𝜏_𝑘𝑤₂₀(𝜃)+𝑔₂₀𝑞(0)+𝑔₀₂𝑞(0)+⎛⎜⎜⎜⎜⎝200𝜏_𝑘⎞⎟⎟⎟⎟⎠.(3.35)₀₋₁d𝜂(𝜃)𝑤₁₁(𝜃)=𝑔₁₁𝑞(0)+𝑔₁₁𝑞(0)+⎛⎜⎜⎜⎜⎝200𝜏_𝑘⎞⎟⎟⎟⎟⎠.(3.36) Substituting (3.30) into (3.36) and noticing that−ii𝜔𝜔_𝑘𝑘𝜏_𝑘𝑞(0)=⁰⁻¹𝑒^{i𝜔𝑘𝜏𝑘𝜃}d𝜂(𝜃)𝑞(0),𝜏_𝑘𝑞(0)=⁰⁻¹𝑒^{−i𝜔𝑘𝜏𝑘𝜃}d𝜂(𝜃)𝑞(0),(3.37) we have2i𝜔^𝑘𝜏_𝑘𝐼−⁰⁻¹𝑒^{2i𝜔𝑘𝜏𝑘𝜃}d𝜂(𝜃)𝐸₁=𝑓_0,𝑧2,(3.38) namely,⎛⎜⎜⎜⎜⎝2i0𝜔_𝑘2i𝜔_𝑘−10−𝑎−2𝑥_∗−𝑀+𝑀𝑒^{−2𝜔𝑘𝜏𝑘}−1−𝑏2i𝜔_𝑘−𝑐⎞⎟⎟⎟⎟⎠𝐸¹=⎛⎜⎜⎜⎜⎝002⎞⎟⎟⎟⎟⎠.(3.39) Thus𝐸¹⁽¹⁾=2𝑅,𝐸¹⁽²⁾=4i𝜔^𝑘𝑅,𝐸¹⁽³⁾=4i𝜔^𝑘2i𝜔_𝑘−𝑀+𝑀𝑒^{−2𝜔𝑘𝜏𝑘}𝑅,(3.40) where𝑅=|||||||||2i𝜔^𝑘02i𝜔_𝑘−10−𝑎−2𝑥_∗−𝑀+𝑀𝑒^{−2𝜔𝑘𝜏𝑘}−1−𝑏2i𝜔_𝑘−𝑐|||||||||.(3.41) Following the similar analysis, we also have⎛⎜⎜⎜⎜⎝−𝑎−002𝑥_∗−01−01−𝑏−𝑐⎞⎟⎟⎟⎟⎠𝐸²=⎛⎜⎜⎜⎜⎝002⎞⎟⎟⎟⎟⎠,(3.42) hence𝐸²⁽¹⁾=2𝑆,𝐸²⁽²⁾=0,𝐸²⁽³⁾=0,(3.43) where𝑆=|||||||||−𝑎−002𝑥_∗−01−01−𝑏−𝑐|||||||||.(3.44) Thus the following values can be computed:𝑐₁(0)=i2𝜔_𝑘𝜏_𝑘𝑔²⁰𝑔₁₁−2||𝑔₁₁||²−||𝑔⁰²||²𝜇₂3+𝑔2²¹,=−Re𝑐¹(0)𝜒₂Re𝜆^′𝜏_𝑘,=−Im𝑐¹(0)+𝜇₂Im𝜆^𝜏_𝑘𝜔_𝑘𝜏_𝑘𝛽₂,=2Re𝑐₁(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>𝜏_𝑘(𝜏<𝜏_𝑘); 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:𝑧̇(𝑡)=𝑦̇(−𝑡)6=𝑥(𝑧𝑡)(𝑡−)𝑥+2̇(.𝑡𝑀9)2=𝑦(𝑦(𝑦𝑡()(𝑡𝑡)−)−,1.𝑦2(𝑧𝑡(−𝑡)𝜏+))𝑥,₂(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−(𝑀−1.2)𝜆²+(−1.2𝑀+2.92)𝜆+6+𝑀𝜆²+1.2𝑀𝜆𝑒^−𝜆𝜏=0.(4.2) Hence we have 𝑝=4.4, 𝑞=−5.8736+4.992𝑀, 𝑟=36, Δ=36.9808−14.976𝑀, ℎ(𝑣)=𝑣³+𝑝𝑣²+𝑞𝑣+𝑟, 𝑣=𝜔², 𝑣^∗1=(−𝑝+√Δ)/3=(1/3)(4.4+√36.9808−14.976𝑀), 𝑣^∗2=(1/3)(4.4−√36.9808−14.976𝑀), 𝜏^𝑘(𝑗)=(1/𝜔_𝑘){cos⁻¹((𝑀𝜔^2𝑘+𝑀𝑐²−𝑏𝑐−𝑎)/(𝑀𝜔^2𝑘+𝑀𝑐²))+2𝑗𝜋}, 𝜏₀=min_{𝑘∈{1,2,3}}{𝜏^𝑘(0)}. By Theorem 2.4, when Δ=36.9808−14.976𝑀≤0, that is, 𝑀≥2.46934, (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 of𝑧̇(𝑡)=𝑦−̇(𝑡6)𝑥=(𝑡𝑧)(−𝑡𝑥)2̇−(.𝑡9)82=𝑦𝑦((𝑦𝑡𝑡))(𝑡+−),81𝑦.2(𝑧𝑡(−𝑡)𝜏+),𝑥₂(𝑡).(4.3) Thus we can compute Δ=156.789, 𝑣^∗1≐5.64051, ℎ(𝑣^∗1)≐−182.922, 𝑣₁≐4.30249, 𝑣₂≐0.873751, 𝜔₁≐2.07424, 𝜔₂≐0.934746, ℎ^(𝑣₁)≐−28.1373, ℎ^(𝑣₂)≐−51.2083, 𝜏¹⁽⁰⁾≐0.632012, 𝜏²⁽⁰⁾≐1.85965,𝜏₀≐0.632012. 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.

Acknowledgment

This work was supported by the Research Foundation of Hangzhou Dianzi University (KYS075609067).