Bifurcation Analysis for a Kind of Nonlinear Finance System with Delayed Feedback and Its Application to Control of Chaos

Abstract

A kind of nonlinear finance system with time-delayed feedback is considered. Firstly, by employing the polynomial theorem to analyze the distribution of the roots to the associate characteristic equation, the conditions of ensuring the existence of Hopf bifurcation are given. Secondly, by using the normal form theory and center manifold argument, we derive the explicit formulas determining the stability, direction, and other properties of bifurcating periodic solutions. Finally, we give several numerical simulations, which indicate that when the delay passes through certain critical values, chaotic oscillation is converted into a stable steady state or a stable periodic orbit.

1. Introduction

Since the chaotic phenomenon in economics was first found in 1985, great impact has been imposed on the prominent western economics at present, because the chaotic phenomenon occurring in the economic system means that the macroeconomic operation has in itself the inherent indefiniteness. Although the government can adopt such macrocontrol measures as the financial policies or the monetary policies to interfere, the effectiveness of the interference is very limited. The instability and complexity make the precise economic prediction greatly limited, and the reasonable prediction behavior has become complicated as well. In the fields of finance, stocks, and social economics, because of the interaction between nonlinear factors, with all kinds of economic problems being more and more complicated and with the evolution process from low dimensions to high dimensions, the diversity and complexity have manifested themselves in the internal structure of the system and there exists extremely complicated phenomenon and external characteristics in such a kind of system. So it has become more and more important to study the control of the complicated continuous economic system and stabilize the instable periodic or stationary solutions, in order to make the precise economic prediction possible [1, 2].

Recent works [1, 2] have reported a dynamic model of finance, composed of three first-order differential equations. The model describes the time variations of three state variables: the interest rate 𝑥, the investment demand 𝑦, and the price index 𝑧. By choosing an appropriate coordinate system and setting appropriate dimensions for each state variable, [1, 2] offer the simplified finance system as (𝑡)𝑥(𝑡)(𝑡),(𝑡)(𝑡)(𝑡)𝑧(𝑡),𝑧̇(𝑡)(𝑡)𝑦(𝑡),𝑥̇=−𝑎+𝑦𝑦̇=−𝑦−𝑎𝑥=𝑏+𝑎𝑥 (1.1) which is chaotic when 𝑎=1.69,𝑏=4 (see Figure 1).

Figure 1 

Strange attractor of finance system (1.1).

Over the last years, [3, 4] studied impulsive control and state feedback control of the finance system (1.1). In this paper, we are interesting in delayed feedback control of the finance system (1.1). The effects of the time-delayed feedback on the finance system have long been investigated [5–8].

Recently, different techniques and methods have been proposed to achieve chaos control. The existing control methods can be classified, mainly, into two categories. The first one, developed by Ott et al. [9] is based on the invariant manifold structure of unstable orbits. It is theoretically well understood but difficult to apply to fast experimental systems. The second, proposed by Pyragas [10], uses time-delayed controlling forces. In contrast to the former one, it is simple and convenient method of controlling chaos in continuous dynamical system. Thus, we adopt the second one in the present paper.

For predigesting the investigation, here we only put time delay on investment demand 𝑦. By adding a time-delayed force 𝐾(𝑦(𝑡)−𝑦(𝑡−𝜏)) to the second equation of finance system (1.1), we obtain the following new system (𝑡)𝑥(𝑡)(𝑡),(𝑡)(𝑡)(𝑡)𝑧(𝑡)𝑦(𝑡)(),𝑧̇(𝑡)(𝑡)𝑦(𝑡).𝑥̇=−𝑎+𝑦𝑦̇=−𝑦−𝑎𝑥+𝐾−𝑦𝑡−𝜏=𝑏+𝑎𝑥 (1.2) Here we assume that (𝐶₁)𝑎,𝑏,𝜏∈(0,∞) and 𝐾∈𝑅. The time delay 𝜏 is taken as the bifurcation parameter and we show that when 𝜏 passes through some certain critical values, the equilibrium will lose its stability and hopf bifurcation will take place; by adjusting 𝐾 values, we achieve the purpose of chaos control. The research of this paper is a new investigation about the hopf bifurcation and chaos control on the finance system and has important theoretical and practical value.

2. Stability of Steady States and Bifurcations of Periodic Solutions

In this section, we investigate the effect of delay on the dynamic behavior of system (1.2). Obviously, when 𝜏=0, system (1.2) becomes the system (1.1). First, we introduce the following several lemmas in [1, 2] for T’s system(1.1).

We know that under the assumption (𝐶₁), the system (1.1) has two equilibrium points:𝑆₁=√𝑎𝑏𝑏√,−𝑎𝑏𝑏,1𝑎,𝑆₂=−√𝑎𝑏𝑏,√𝑎𝑏𝑏,1𝑎. (2.1)

The characteristic equation of the system (1.1) at 𝑆₁(𝑆₂) is 𝜆³+(1+𝑎)𝜆²+𝑎𝑏𝜆+2𝑎²𝑏=0. (2.2)

By analyzing the characteristic equation (2.2) and the Routh-Hurwitz criteria, we get the following.

Lemma 2.1. For 𝑎<1, the characteristic equation (2.2) has three eigenvalues with negative real parts, so two equilibrium points 𝑆₁,𝑆₂ of the system (1.1) are asymptotic stable.

Lemma 2.2. For 𝑎=1, the characteristic equation (2.2) has a pair of purely imaginary eigenvalues 𝜆_1,2=±𝑖𝜔₀(𝜔₀=√𝑏) and a negative real eigenvalue 𝜆₃=−2, and 𝜆̇(𝑎=1)=5𝑏2𝑏+8>0. (2.3) According to the hopf bifurcation theorem [11], a hopf bifurcation of the system (1.1) occurs at 𝑎=1.

Lemma 2.3. For 𝑎>1, the characteristic equation (2.2) has one negative real root and one pair of conjugate complex roots with positive real parts, so two equilibrium points 𝑆₁,𝑆₂ of the system (1.1) are unstable.

Clearly, the delayed feedback control system (1.2) has the same equilibria to the corresponding system (1.1). In this section, we analyze the effect of delay on the stability of these steady states. Due to the symmetry of 𝑆₁ and 𝑆₂, it is sufficient to analyze the stability of 𝑆₁. By the linear transform 𝑥₁(𝑡)(𝑡)−√=𝑥𝑎𝑏𝑏,𝑦₁(𝑡)(𝑡)+√=𝑦𝑎𝑏𝑏,𝑧₁(𝑡)(𝑡)−1=𝑧𝑎, (2.4) system (1.2) becomes 𝑥̇₁(𝑡)=−𝑎𝑥₁(𝑡)+𝑦₁(𝑡),𝑦₁̇(𝑡)=−𝑥₁(𝑡)−𝑦₁(𝑡)−√𝑎𝑏𝑧₁(𝑡)−𝑎𝑥₁(𝑡)𝑧₁(𝑡)+𝐾𝑦₁(𝑡)−𝑦₁(),𝑧̇𝑡−𝜏₁(𝑡)=−√𝑎𝑏𝑥₁(𝑡)+√𝑎𝑏𝑦₁(𝑡)+𝑎𝑥₁(𝑡)𝑦₁(𝑡). (2.5) It is easy to see that the origin 𝑆₀(0,0,0) is the equilibrium of system (2.5). The associated characteristic equation of system (2.5) at 𝑆₀(0,0,0) is det𝜆+𝑎𝑎01𝜆+1−𝐾+𝐾𝑒^−𝜆𝜏√√𝑎𝑏𝑎𝑏−√𝑎𝑏𝜆=0. (2.6) Expanding (2.6), we have 𝜆³+(1+𝑎−𝐾)𝜆²+(𝑎𝑏−𝑎𝐾)𝜆+2𝑎²𝑏+𝐾𝜆²+𝑎𝜆𝑒^−𝜆𝜏=0. (2.7) Thus, we need to study the distribution of the roots of the third-degree exponential polynomial equation: 𝜆³+𝑎₂𝜆²+𝑎₁𝜆+𝑎₀+𝑏₂𝜆²+𝑏₁𝜆+𝑏₀𝑒^−𝜆𝜏=0, (2.8) where 𝑎_𝑖,𝑏_𝑖∈𝑅(𝑖=0,1,2) and ∑²_𝑖=0𝑏²_𝑖≠0. We first introduce the following simple result which was proved by Ruan and Wei [12] using Rouche’s theorem.

Lemma 2.4. Consider the exponential polynomial 𝑃𝜆,𝑒^−𝜆𝜏1,…,𝑒^{−𝜆𝜏𝑚}=𝜆^𝑛+𝑝₁⁽⁰⁾𝜆^𝑛−1+⋯+𝑝⁽⁰⁾_𝑛−1𝜆+𝑝_𝑛⁽⁰⁾+𝑝₁⁽¹⁾𝜆^𝑛−1+⋯+𝑝⁽¹⁾_𝑛−1𝜆+𝑝_𝑛⁽¹⁾𝑒^−𝜆𝜏1+⋯+𝑝₁^(𝑚)𝜆^𝑛−1+⋯+𝑝^(𝑚)_𝑛−1𝜆+𝑝_𝑛^(𝑚)𝑒^{−𝜆𝜏𝑚}, (2.9) 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.

Obviously, 𝑖𝜔(𝜔>0) is a root of (2.8) if and only if 𝜔 satisfies −𝑖𝜔³−𝑎₂𝜔²+𝑎₁𝜔𝑖+𝑎₀+−𝑏₂𝜔²+𝑏₁𝜔𝑖+𝑏₀(cos𝜔𝜏−𝑖sin𝜔𝜏)=0. (2.10) Separating the real and imaginary parts, we have 𝑎₂𝜔²−𝑎₀=𝑏₀−𝑏₂𝜔²cos𝜔𝜏+𝑏₁𝜔sin𝜔𝜏,−𝜔³+𝑎₁𝜔=𝑏₀−𝑏₂𝜔²sin𝜔𝜏−𝑏₁𝜔cos𝜔𝜏, (2.11) which is equivalent to 𝜔⁶+𝑎²₂−𝑏²₂−2𝑎₁𝜔⁴+𝑎²₁−2𝑎₀𝑎₂−𝑏²₁+2𝑏₀𝑏₂𝜔²+𝑎²₀−𝑏²₀=0. (2.12) Let 𝑧=𝜔² and denote 𝑝=𝑎²₂−𝑏²₂−2𝑎₁,  𝑞=𝑎²₁−2𝑎₀𝑎₂−𝑏²₁+2𝑏₀𝑏₂, 𝑟=𝑎²₀−𝑏²₀, then (2.12) becomes 𝑧³+𝑝𝑧²+𝑞𝑧+𝑟=0. (2.13)

In the following, we need to seek conditions under which (2.12) has at least one positive root. Denoteℎ(𝑧)=𝑧³+𝑝𝑧²+𝑞𝑧+𝑟. (2.14)

Therefor, applying [13], we obtain the following lemma.

Lemma 2.5. For the polynomial equation (2.13), one has the following results.(i)If 𝑟<0, then (2.13) has at least one positive root.(ii)If 𝑟≥0 and Δ=𝑝²−3𝑞≤0, then (2.13) has no positive roots.(iii)If 𝑟≥0 and Δ=𝑝²−3𝑞>0, then (2.13) has positive roots if and only if 𝑧^∗₁=(1/3)(−𝑝+√Δ)>0 and ℎ(z^∗₁)≤0.

Suppose that (2.13) has positive roots. Without loss of generality, we assume that it has three positive roots, defined by 𝑧₁,𝑧₂, and 𝑧₃, respectively. Then (2.12) has three positive roots: 𝜔₁=√𝑧₁,𝜔₂=√𝑧₂,𝜔₃=√𝑧₃. (2.15) From (2.11), we have 𝑏cos𝜔𝜏=¹𝜔²𝜔²−𝑎₁−𝑎₂𝜔²−𝑎₀𝑏₂𝜔²−𝑏₀𝑏₂𝜔²−𝑏₀²+𝑏²₁𝜔². (2.16) Thus, if we denote 𝜏_𝑘^(𝑗)=1𝜔_𝑘cos⁻¹𝑏₁𝜔²_𝑘𝜔²_𝑘−𝑎₁−𝑎₂𝜔²_𝑘−𝑎₀𝑏₂𝜔²_𝑘−𝑏₀𝑏₂𝜔²_𝑘−𝑏₀²+𝑏²₁𝜔²_𝑘+2𝑗𝜋, (2.17) where 𝑘=1,2,3;𝑗=0,1,2,…, then ±𝑖𝜔_𝑘 is a pair of purely imaginary roots of (2.8) with 𝜏_𝑘^(𝑗). Define 𝜏₀=𝜏_𝑘⁽⁰⁾₀=min_𝑘∈1,2,3𝜏_𝑘⁽⁰⁾,𝜔₀=𝜔_𝑘0. (2.18) Note that when 𝜏=0, (2.8) becomes 𝜆³+𝑎₂+𝑏₂𝜆²+𝑎₁+𝑏₁𝜆+𝑎₀+𝑏₀=0. (2.19)

Therefor, applying Lemmas 2.4 and 2.5 to (2.8), we get the following lemma.

Lemma 2.6. For (2.8), one has(i)if 𝑟≥0 and Δ=𝑝²−3𝑞≤0, then all roots with positive real parts of (2.8) have the same sum to those of the polynomial equation (2.19) for all 𝜏≥0.(ii)if either 𝑟<0 or 𝑟≥0, Δ=𝑝²−3𝑞>0, 𝑧^∗₁=(1/3)(−𝑝+√Δ)>0 and ℎ(𝑧^∗₁)≤0, then all roots with positive real parts of (2.8) have the same sum to those of the polynomial equation (2.19) for 𝜏∈[0,𝜏₀).

Let 𝜆(𝜏)=𝛼(𝜏)+𝑖𝜔(𝜏) (2.20) be the root of (2.8) near 𝜏=𝜏_𝑘^(𝑗) satisfying 𝛼𝜏_𝑘^(𝑗)=0,𝜔𝜏_𝑘^(𝑗)=𝜔_𝑘. (2.21) Then by [13], we have the following transversality condition.

Lemma 2.7. Suppose that 𝑧_𝑘=𝜔²_𝑘 and ℎ^′(𝑧_𝑘)≠0. Then 𝑅𝜆𝜏_𝑘^(𝑗)𝑑𝜏≠0, (2.22) and 𝑅𝜆(𝜏_𝑘^(𝑗))/𝑑𝜏 and ℎ^′(𝑧_𝑘) have the same sign.

Now, we study the characteristic equation (2.7) of the system (2.5). Comparing (2.7) with (2.8), we know that 𝑎₂=1+𝑎−𝐾,𝑎₁=𝑎𝑏−𝑎𝐾,𝑎₀=2𝑎²𝑏,𝑏₂=𝐾,𝑏₁=𝑎𝐾,𝑏₀=0. (2.23) Thus, 𝑝=𝑎²₂−𝑏²₂−2𝑎₁=𝑎²+2𝑎+1−2𝑎𝑏−2𝐾,𝑞=𝑎²₁−2𝑎₀𝑎₂−𝑏²₁+2𝑏₀𝑏₂=𝑎²𝑏²+2𝑎²𝑏𝐾−4𝑎²𝑏−4𝑎³𝑏,𝑟=𝑎²₀−𝑏²₀=4𝑎⁴𝑏²>0, (2.24) and then we can compute Δ=𝑝²−3𝑞,ℎ(𝑧)=𝑧³+𝑝𝑧²+𝑞𝑧+𝑟,𝑧^∗₁=13−𝑝+√Δ. (2.25) When 𝜏=0, (2.7) becomes (2.2)𝜆³+(1+𝑎)𝜆²+𝑎𝑏𝜆+2𝑎²𝑏=0. (2.26)

Applying Lemmas 2.1, 2.2, 2.6, and 2.7 to (2.7), we have the following theorems.

Theorem 2.8. Let 𝜏_𝑘^(𝑗) and 𝜏₀ be defined by (2.17) and (2.18). Suppose that conditions (𝐶₁) and 𝑎<1 hold.(i)If Δ≤0, then (2.7) had all roots with negative real parts for all 𝜏≥0, and the equilibrium 𝑆₁ (or 𝑆₂) of the system (1.2) is stable.(ii)If Δ>0,𝑧^∗₁>0 and ℎ(𝑧^∗₁)≤0, (2.7) had all roots with negative real parts for 𝜏∈[0,𝜏₀), and the equilibrium 𝑆₁ (or 𝑆₂) of the system (1.2) is stable.(iii)If the conditions of (ii) are satisfied, and ℎ^′(𝑧_𝑘)≠0, then system (1.2) exhibits the Hopf bifurcation at the equilibrium 𝑆₁ (or 𝑆₂) for 𝜏=𝜏_𝑘^(𝑗).

Theorem 2.9. Let 𝜏_𝑘^(𝑗) and 𝜏₀ are defined by (2.17) and (2.18). Suppose that conditions (𝐶₁) and 𝑎>1 hold.(i)If Δ≤0, then (2.7) had two roots with positive real parts for all 𝜏≥0, and the equilibrium 𝑆₁ (or 𝑆₂) of the system (1.2) is unstable.(ii)If Δ>0,𝑧^∗₁>0 and ℎ(𝑧^∗₁)≤0, (2.7) has two roots with positive real parts for 𝜏∈[0,𝜏₀), and the equilibrium 𝑆₁ (or 𝑆₂) of the system (1.2) is unstable.(iii)If the conditions of (ii) are satisfied, and ℎ^′(𝑧_𝑘)≠0, then system (1.2) exhibits the Hopf bifurcation at the equilibrium 𝑆₁ (or 𝑆₂) for 𝜏=𝜏_𝑘^(𝑗).

3. Direction and Stability of the Hopf Bifurcation

In the Section 2, we obtained some conditions which guarantee that the system (1.2) undergoes the Hopf bifurcation at a sequence values of 𝜏. In this section, we shall study the direction and stability of the Hopf bifurcation. The method we used is based on the normal form theory and the center manifold theorem introduced by Hassard et al. [14]. Throughout this section, we always assume that system (1.2) undergoes Hopf bifurcations at the steady state (𝑥_∗,𝑦_∗,𝑧_∗) for 𝜏=𝜏_𝑘 and then ±𝑖𝜔_𝑘 is corresponding purely imaginary roots of the characteristic equation at the steady state (𝑥_∗,𝑦_∗,𝑧_∗).

Letting 𝑥₁=𝑥−𝑥_∗,𝑥₂=𝑦−𝑦_∗,𝑥₃=𝑧−𝑧_∗,𝑥_𝑖(𝑡)=𝑥_𝑖(𝜏𝑡),𝜏=𝜏_𝑘+𝜇 and dropping the bars for simplification of notations, system (1.2) is transformed into an FDE in 𝐶=𝐶([−1,0],𝑅³) as 𝑥̇(𝑡)=𝐿_𝜇𝑥_𝑡+𝑓𝜇,𝑥_𝑡, (3.1) where 𝑥(𝑡)=(𝑥₁(𝑡),𝑥₂(𝑡),𝑥₃(𝑡))^𝑇∈𝑅³, and 𝐿_𝜇∶𝐶→𝑅,𝑓∶𝑅×𝐶→𝑅 are given, respectively, by 𝐿_𝜇𝜙=𝜏_𝑘+𝜇−𝑎−𝑎0−𝑎𝑧_∗𝐾−1−𝑎𝑥_∗𝑎𝑦_∗𝑎𝑥_∗0𝜙₁(0)𝜙₂(0)𝜙₃(0)+𝜏_𝑘𝜙+𝜇0000−𝐾0000¹()𝜙−1₂()𝜙−1₃(),𝑓=𝜏−1𝜇,𝜙_𝑘0+𝜇−𝑎𝜙₁(0)𝜙₃(0)𝑎𝜙₁(0)𝜙₂(0). (3.2) By the Riesz representation theorem, there exists a function 𝜂(𝜃,𝜇) of bounded variation for 𝜃∈[−1,0], such that 𝐿_𝜇𝜙=⁰₋₁𝑑𝜂(𝜃,0)𝜙(𝜃) (3.3) for 𝜙∈𝐶[−1,0].

In fact, we can choose 𝜂𝜃,𝜇=𝜏_𝑘+𝜇−𝑎−𝑎0−𝑎𝑧_∗𝐾−1−𝑎𝑥_∗𝑎𝑦_∗𝑎𝑥_∗0𝛿(𝜃)−𝜏_𝑘+𝜇𝛿()0000−𝐾0000𝜃+1, (3.4) where 𝛿 is the Dirac delta function. For 𝜙∈𝐶¹([−1,0],(𝑅³)^∗), define 𝐴𝜇(𝜃)𝜙=𝑑𝜙[),𝑑𝜃,𝜃∈−1,0₀₋₁𝜙(𝑠)𝑅𝜇[),𝑓𝑑𝜂𝜇,𝑠,𝜃=0,𝜙=0,𝜃∈−1,0𝜇,𝜙,𝜃=0. (3.5) Then system (3.1) is equivalent to 𝑥̇_𝑡=𝐴𝜇𝑥_𝑡+𝑅𝜇𝑥_𝑡, (3.6) where 𝑥_𝑡(𝜃)=𝑥(𝑡+𝜃) for 𝜃∈[−1,0].

For 𝜓∈𝐶¹([0,1],𝑅³), define 𝐴^∗𝜓(𝑠)=−𝑑𝜓(𝑠)(],𝑑𝑠,𝑠∈0,1₀₋₁𝑑𝜂^𝑇(𝑡,0)𝜓(−𝑡),𝑠=0, (3.7) and a bilinear inner product 𝜓(𝑠),𝜙(𝜃)=𝜓(0)⋅𝜙(0)−⁰_𝜃=−1^𝜃_𝜉=0𝜓^𝑇(𝜉−𝜃)𝑑𝜂(𝜃)𝜙(𝜉)𝑑𝜉, (3.8) where 𝜂(𝜃)=𝜂(𝜃,0). Then 𝐴=𝐴(0) and 𝐴^∗=𝐴^∗(0) are adjoins operators.

By the discussion in Section 2, we know that ±𝑖𝜔_𝑘𝜏_𝑘 are eigenvalues of 𝐴, thus they are also eigenvalues of 𝐴^∗.

By direct computation, we obtain that 𝑞(𝜃)=𝑞₀𝑒^{𝑖𝜃𝜔𝑘𝜏𝑘}, with 𝑞(0)=1,𝛼,𝛽^𝑇=1,−𝑎+𝑖𝜔_𝑘𝑎,𝑎𝑦_∗−𝑥_∗−𝑖𝜔_𝑘𝑥_∗𝑖𝜔_𝑘^𝑇, (3.9) is the eigenvector of 𝐴 corresponding to 𝑖𝜔_𝑘𝜏_𝑘, and 𝑞^∗(𝑠)=𝐷𝑞^∗₀𝑒^{𝑖𝑠𝜔𝑘𝜏𝑘}, with 𝑞^∗₀=1,𝛼^∗,𝛽^∗𝑇=1,𝑖𝜔_𝑘𝑎−𝑖𝜔_𝑘𝑎²𝑥_∗𝑦_∗−𝑖𝜔_𝑘𝑎𝑧_∗,𝑎−𝑖𝜔_𝑘𝑎𝑥_∗𝑎²𝑥_∗𝑦_∗−𝑖𝜔_𝑘𝑎𝑧_∗^𝑇, (3.10) is the eigenvector of 𝐴^∗ corresponding to −𝑖𝜔_𝑘𝜏_𝑘, where 1𝐷=1+𝛼𝛼^∗+𝛽𝛽^∗−𝐾𝜏_𝑘𝛼𝛼^∗𝑒^{𝑖𝜔𝑘𝜏𝑘}. (3.11) Using the same notation as in [14], we compute the coordinates to describe the center manifold 𝐶₀ at 𝜇=0. Let 𝑥_𝑡 be the solution of (3.1) when 𝜇=0. Define 𝑧(𝑡)=𝑞^∗,𝑥_𝑡,𝑊(𝑡,𝜃)=𝑥_𝑡(𝜃)−2Re𝑧(𝑡)𝑞(𝜃). (3.12) On the center manifold 𝐶₀, we have𝑊(𝑡,𝜃)=𝑊(𝑧(𝑡),𝑧(𝑡),𝜃), (3.13)

where𝑊(𝑧,𝑧,𝜃)=𝑊₂₀(𝜃)𝑧²2+𝑊₁₁(𝜃)𝑧𝑧+𝑊₀₂(𝜃)𝑧²2+𝑊₃₀(𝜃)𝑧³6+⋯, (3.14)𝑧 and 𝑧 are local coordinates for center manifold 𝐶₀ in the direction of 𝑞^∗ and 𝑞^∗. Note that 𝑊 is real if 𝑥_𝑡 is real. We consider only real solutions. For the solution 𝑥_𝑡∈𝐶₀ of (3.1), since 𝜇=0, we have 𝑧̇(𝑡)=𝑖𝜏_𝑘𝜔_𝑘𝑧+𝑞^∗(𝜃)𝑓0,𝑊(𝑧,𝑧,𝜃)+2Re𝑧𝑞(𝜃)=𝑖𝜏_𝑘𝜔_𝑘𝑧+𝑞^∗(0)𝑓0,𝑊(𝑧,𝑧,0)+2Re𝑧𝑞(0)=^def𝑖𝜏_𝑘𝜔_𝑘𝑧+𝑞^∗(0)𝑓₀(𝑧,𝑧). (3.15) We rewrite this equation as 𝑧̇(𝑡)=𝑖𝜏_𝑘𝜔_𝑘𝑧(𝑡)+𝑔(𝑧,𝑧), (3.16) where 𝑔(𝑧,𝑧)=𝑞^∗(0)𝑓₀(𝑧,𝑧)=𝑔₂₀𝑧²2+𝑔₁₁𝑧𝑧+𝑔₀₂𝑧²2+𝑔₂₁𝑧²𝑧2+⋯. (3.17) Noticing 𝑥_𝑡(𝜃)=𝑥_1𝑡(𝜃),𝑥_2𝑡(𝜃),𝑥_3𝑡(𝜃)=𝑊(𝑡,𝜃)+𝑧𝑞(𝜃)+(𝜃),𝑞(𝜃)=𝑧𝑞1,𝛼,𝛽^𝑇𝑒^{𝑖𝜃𝜔𝑘𝜏𝑘}, (3.18) we have 𝑥_1𝑡(0)=𝑧+𝑧+𝑊⁽¹⁾₂₀(0)𝑧²2+𝑊⁽¹⁾₁₁(0)𝑧𝑧+𝑊⁽¹⁾₀₂(0)𝑧²2+𝑂|(𝑧,𝑧)|³,𝑥_2𝑡(0)=𝛼𝑧+𝛼𝑧+𝑊⁽²⁾₂₀(0)𝑧²2+𝑊⁽²⁾₁₁(0)𝑧𝑧+𝑊⁽²⁾₀₂(0)𝑧²2+𝑂|(𝑧,𝑧)|³,𝑥_3𝑡(0)=𝛽𝑧+𝛽𝑧+𝑊⁽³⁾₂₀(0)𝑧²2+𝑊⁽³⁾₁₁(0)𝑧𝑧+𝑊⁽³⁾₀₂(0)𝑧²2+𝑂|(𝑧,𝑧)|³. (3.19) Thus, form (3.17), we have𝑔(𝑧,𝑧)=𝑞^∗(0)𝑓₀(𝑧,𝑧)=𝐷𝜏_𝑘1,𝛼^∗,𝛽^∗0−𝑎𝑥_1𝑡(0)𝑥_3𝑡(0)𝑎𝑥_1𝑡(0)𝑥_2𝑡(0)=−𝑎𝐷𝜏_𝑘𝛼^∗𝑧+𝑧+𝑊⁽¹⁾₂₀(0)𝑧²2+𝑊⁽¹⁾₁₁(0)𝑧𝑧+𝑊⁽¹⁾₀₂(0)𝑧²2+𝑂|(𝑧,𝑧)|³𝛽𝑧+𝛽𝑧+𝑊⁽³⁾₂₀(0)𝑧²2+𝑊⁽³⁾₁₁(0)𝑧𝑧+𝑊⁽³⁾₀₂(0)𝑧²2+𝑂|(𝑧,𝑧)|³+𝑎𝐷𝜏_𝑘𝛽^∗𝑧+𝑧+𝑊⁽¹⁾₂₀(0)𝑧²2+𝑊⁽¹⁾₁₁(0)𝑧𝑧+𝑊⁽¹⁾₀₂(0)𝑧²2+𝑂|(𝑧,𝑧)|³𝛼𝑧+𝛼𝑧+𝑊⁽²⁾₂₀(0)𝑧²2+𝑊⁽²⁾₁₁(0)𝑧𝑧+𝑊⁽²⁾₀₂(0)𝑧²2+𝑂|(𝑧,z)|³. (3.20)

Comparing the coefficients of (3.17), we get 𝑔₂₀=−2𝑎𝐷𝜏_𝑘𝛽^∗𝛼−𝛼^∗𝛽,𝑔₁₁=2𝑎𝐷𝜏_𝑘𝛽^∗Re(𝛼)−𝛼^∗𝛽,𝑔Re₀₂=−2𝑎𝐷𝜏_𝑘𝛽^∗𝛼−𝛼^∗𝛽,𝑔₂₁=−𝑎𝐷𝜏_𝑘𝛼^∗2𝑊⁽³⁾₁₁(0)+𝑊⁽³⁾₂₀(0)+2𝛽𝑊⁽¹⁾₁₁(0)+𝛽𝑊⁽¹⁾₂₀(0)+𝑎𝐷𝜏_𝑘𝛽^∗2𝑊⁽²⁾₁₁(0)+𝑊⁽²⁾₂₀(0)+2𝛼𝑊⁽¹⁾₁₁(0)+𝛼𝑊⁽¹⁾₂₀(0). (3.21) Since there are 𝑊₂₀(𝜃) and 𝑊₁₁(𝜃) in 𝑔₂₁, we need to compute them.

From (3.6) and (3.12), we have 𝑊̇=𝑥̇_𝑡−𝑧𝑞̇−𝑧𝑞̇=𝐴𝑊−2Re𝑞^∗(0)𝑓₀𝑞(𝜃),𝜃∈[−1,0),𝐴𝑊−2Re𝑞^∗(0)𝑓₀𝑞(0)+𝑓₀,𝜃=0._def=𝐴𝑊+𝐻(𝑧,𝑧,𝜃), (3.22) where 𝐻(𝑧,𝑧,𝜃)=𝐻₂₀(𝜃)𝑧²2+𝐻₁₁(𝜃)𝑧𝑧+𝐻₀₂(𝜃)𝑧²2+⋯. (3.23) Expanding the above series and comparing the corresponding coefficients, we obtain 𝐴−2𝑖𝜏_𝑘𝜔_𝑘𝑊₂₀(𝜃)=−𝐻₂₀(𝜃),𝐴𝑊₁₁(𝜃)=−𝐻₁₁(𝜃),…. (3.24) From (3.22), we know that for 𝜃∈[−1,0),𝐻(𝑧,𝑧,𝜃)=−𝑞^∗(0)𝑓₀𝑞(𝜃)−𝑞^∗(0)𝑓₀𝑞(0)=−𝑔𝑞(𝜃)+𝑔𝑞(𝜃). (3.25) Comparing the coefficients with (3.23) gives that 𝐻₂₀(𝜃)=−𝑔₂₀𝑞(𝜃)−𝑔₀₂𝑞(𝜃)𝐻₁₁,(𝜃)=−𝑔₁₁𝑞(𝜃)−𝑔₁₁𝑞(𝜃). (3.26)(3.27) From (3.24), (3.26) and the definition of 𝐴, it follows that 𝑊̇₂₀(𝜃)=2𝑖𝜏_𝑘𝜔_𝑘𝑊₂₀(𝜃)+𝑔₂₀𝑞(𝜃)+𝑔₀₂𝑞(𝜃). (3.28) Notice that 𝑞(𝜃)=(1,𝛼,𝛽)^𝑇𝑒^{𝑖𝜃𝜔𝑘𝜏𝑘}, hence 𝑊₂₀(𝜃)=−𝑖𝑔₂₀𝜔_𝑘𝜏_𝑘𝑞(0)𝑒^{𝑖𝜃𝜔𝑘𝜏𝑘}+𝑖𝑔₀₂3𝜔_𝑘𝜏_𝑘𝑞(0)𝑒^{−𝑖𝜃𝜔𝑘𝜏𝑘}+𝐸₁𝑒^{2𝑖𝜃𝜔𝑘𝜏𝑘}, (3.29) where 𝐸₁=(𝐸₁⁽¹⁾,𝐸₁⁽²⁾,𝐸₁⁽³⁾)^𝑇∈𝑅³ is a constant vector.

Similarly, from (3.24) and (3.27), we can obtain 𝑊₁₁(𝜃)=−𝑖𝑔₁₁𝜔_𝑘𝜏_𝑘𝑞(0)𝑒^{𝑖𝜃𝜔𝑘𝜏𝑘}+𝑖𝑔₁₁𝜔_𝑘𝜏_𝑘𝑞(0)𝑒^{−𝑖𝜃𝜔𝑘𝜏𝑘}+𝐸₂, (3.30) where 𝐸₂=(𝐸₂⁽¹⁾,𝐸₂⁽²⁾,𝐸₂⁽³⁾)^𝑇∈𝑅³ is also a constant vector.

In what follows, we shall seek appropriate 𝐸₁ and𝐸₂. From the definition of 𝐴 and (3.24), we obtain 𝑑𝜂(⁰₋₁𝜃)𝑊₂₀(𝜃)=2𝑖𝜔_𝑘𝜏_𝑘𝑊₂₀(0)−𝐻₂₀(0),₀₋₁𝑑𝜂(𝜃)𝑊₁₁(𝜃)=−𝐻₁₁(0), (3.31)(3.32) where 𝜂(𝜃)=𝜂(𝜃,0). By (3.22), we have 𝐻₂₀(0)=−𝑔₂₀𝑞(0)−𝑔₀₂𝑞(0)+2𝜏_𝑘0𝛼𝐻−𝛽,₁₁(0)=−𝑔₁₁𝑞(0)−𝑔₁₁𝑞(0)+2𝜏_𝑘0𝛽(𝛼)−ReRe. (3.33)(3.34) Substituting (3.29) and (3.33) into (3.31), we obtain 2𝑖𝜔_𝑘𝜏_𝑘𝐼−⁰₋₁𝑒^{2𝑖𝜃𝜔𝑘𝜏𝑘}𝑑𝜂(𝜃)𝐸₁=2𝜏_𝑘0𝛼−𝛽, (3.35) which leads to 2𝑖𝜔_𝑘+𝑎𝑎0𝑎𝑧_∗2𝑖𝜔_𝑘−𝐾+1+𝐾𝑒^{−2𝜔𝑘𝜏𝑘}𝑎𝑥_∗−𝑎𝑦_∗−𝑎𝑥_∗2𝑖𝜔_𝑘𝐸₁0𝛼,𝐸=2−𝛽₁₍₁₎=2𝑎2𝑖𝜔_𝑘𝛽+𝑎𝛼𝑥_∗𝐴,𝐸₁⁽²⁾=−22𝑖𝜔_𝑘+𝑎2𝑖𝜔_𝑘𝛽+𝑎𝛼𝑥_∗𝐴,𝐸₁⁽³⁾=2𝐴𝐴₁, (3.36) where𝐴=det2𝑖𝜔_𝑘+𝑎𝑎0𝑎𝑧_∗2𝑖𝜔_𝑘−𝐾+1+𝐾𝑒^{−2𝜔𝑘𝜏𝑘}𝑎𝑥_∗−𝑎𝑦_∗−𝑎𝑥_∗2𝑖𝜔_𝑘,𝐴₁=det2𝑖𝜔_𝑘+𝑎𝑎0𝑎𝑧_∗2𝑖𝜔_𝑘−𝐾+1+𝐾𝑒^{−2𝜔𝑘𝜏𝑘}−𝛽−𝑎𝑦_∗−𝑎𝑥_∗𝛼. (3.37)

Similarly, substituting (3.30) and (3.34) into (3.32), we can get 𝑎𝑎0𝑎𝑧_∗1𝑎𝑥_∗−𝑎𝑦_∗−𝑎𝑥_∗0𝐸₂0𝛽(𝛼),𝐸=2−ReRe₂₍₁₎=2𝑎²𝑥_∗Re(𝛼)𝐵,𝐸₂⁽²⁾=−2𝑎²𝑥_∗Re(𝛼)𝐵,𝐸₂⁽³⁾=2𝐵𝐵₁, (3.38) where 𝐵=det𝑎𝑎0𝑎𝑧_∗1𝑎𝑥_∗−𝑎𝑦_∗−𝑎𝑥_∗0,𝐵₁=det𝑎𝑎0𝑎𝑧_∗1−Re𝛽−𝑎𝑦_∗−𝑎𝑥_∗Re(𝛼). (3.39) Thus, we can determine 𝑊₂₀(0) and 𝑊₁₁(0) from (3.29) and (3.30). Furthermore, we can determine 𝑔₂₁. Therefore, each 𝑔_𝑖𝑗 in (3.21) is determined by the parameters and delay in (3.1). Thus, we can compute the following values: 𝑐₁(0)=𝑖2𝜏_𝑘𝜔_𝑘𝑔₁₁𝑔₂₀−2𝑔₁₁²−𝑔₀₂²3+𝑔₂₁2,𝜇₂𝑐=−Re₁(0)Re𝜆^′𝜏_𝑘,𝛽₂=2Re𝑐₁(0),𝑇₂c=−Im₁(0)+𝜇₂Im𝜆^′𝜏_𝑘𝜏_𝑘𝜔_𝑘, (3.40) which determine the quantities of bifurcating periodic solutions in the center manifold at the critical value 𝜏_𝑘, that is, 𝜇₂ determines the directions of the Hopf bifurcation: if 𝜇₂>0(𝜇₂<0) then the Hopf bifurcation is supercritical (subcritical) and the bifurcating periodic solutions exist for 𝜏>𝜏_𝑘(𝜏<𝜏_𝑘); 𝛽₂ determines the stability of the bifurcating periodic solutions: the bifurcating periodic solutions are stable (unstable) if 𝛽₂<0(𝛽₂>0); and 𝑇₂ determines the period of the bifurcating periodic solutions: the period increases (decreases) if 𝑇₂>0(𝑇₂<0).

4. Application to Control Chaos

In the present section, we apply the results in the previous sections to system (1.2) for the purpose of control of chaos. From Section 2, we know that under certain conditions, a family of periodic solutions bifurcate from the steady states of system (1.2) at some critical values of 𝜏 and the stability of the steady state maybe change along with increase of 𝜏. If the bifurcating periodic solution is orbitally asymptotically stable or some steady state becomes local stable, then chaos may vanish. Following this ideal, we consider the following delayed feedback control system: (𝑡)𝑥(𝑡)(𝑡),(𝑡)(𝑡)(𝑡)𝑧(𝑡)𝑦(𝑡)(),𝑧̇(𝑡)(𝑡)𝑦(𝑡),𝑥̇=−1.69+𝑦𝑦̇=−𝑦−1.69𝑥+𝐾−𝑦𝑡−𝜏=4+1.69𝑥 (4.1) which has two steady states 𝑆₊=̇(0.65,−0.65,0.5917),𝑆₋=̇(−0.65,0.65,0.5917). Clearly, when 𝜏=0 or 𝐾=0, system (3.1) is chaotic (as depicted in Figure 1).

For the steady state 𝑆₊ or 𝑆₋, we have the corresponding characteristic equation of system (4.1) as follows: 𝜆³+(2.69−𝐾)𝜆²+(6.76−1.69𝐾)𝜆+22.8488+𝐾𝜆²+1.69𝜆𝑒^−𝜆𝜏=0. (4.2) Clearly, when 𝜏=0, (4.2) has a negative root and a pair of complex roots with positive real parts. Following Section 2, we can obtain 𝑝=−2𝐾−6.2839, 𝑞=22.8488𝐾−77.2289, 𝑟=522.0677>0, Δ=𝑝²−3𝑞>0, and 𝑧^∗₁=(1/3)(−𝑝+√Δ)>0 for all 𝐾∈𝑅. When 𝐾<−0.1907 or 𝐾>12.107, ℎ(𝑧^∗)<0. Thus, from Lemma 2.6 and Theorem 2.9, we know that (4.2) has roots with positive real parts. In particular, we have 𝐾=−1, that is, (𝑡)𝑥(𝑡)(𝑡),(𝑡)(𝑡)(𝑡)𝑧(𝑡)−𝑦(𝑡)(),𝑧̇(𝑡)(𝑡)𝑦(𝑡).𝑥̇=−1.69+𝑦𝑦̇=−𝑦−1.69𝑥−𝑦𝑡−𝜏=4+1.69𝑥 (4.3) In this case, we can compute 𝑧𝑝=̇−4.2839,𝑞=̇−100.077,𝑟=̇522.0677,Δ=̇318.5850,₁=̇8.9478,𝜔₁=2.9913,𝜏₁^(𝑗)=̇0.2205+2𝑗𝜋𝜔₁,ℎ^′𝑧₁𝑧=̇63.4487,₂=̇5.6545,𝜔₂=2.3779,𝜏₂^(𝑗)=̇0.5227+2𝑗𝜋𝜔₂,ℎ^′𝑧₂=̇−52.6043. (4.4)

Thus, from Lemma 2.7, we have Re𝜆(𝜏₁^(𝑗))/𝑑𝜏>0 and Re𝜆(𝜏₂^(𝑗))/𝑑𝜏<0. In addition, notice that 𝜏₁⁽⁰⁾=̇0.2205<𝜏₂⁽⁰⁾=̇0.5227<𝜏₁⁽¹⁾=̇2.3210<𝜏₂⁽¹⁾=̇3.1650. (4.5) Thus, from Theorem 2.8, we have the following conclusion about the stability of the steady states of system (4.3) and Hopf bifurcation.

5. Conclusion

Suppose that 𝜏_𝑘^(𝑗),𝑘=1,2;𝑗=0,1,2,… is defined by (4.4).(i)When 𝜏∈[0,𝜏₂⁽⁰⁾)∪(𝜏₁⁽¹⁾,∞), the steady states 𝑆₁ and 𝑆₂ of the system (4.1) are unstable (see Figures 2 and 3).(ii)When 𝜏∈(𝜏₂⁽⁰⁾,𝜏₁⁽¹⁾), the steady states 𝑆₁ and 𝑆₂ of the system (4.1) are asymptotically (see Figures 4 and 5).(iii)When 𝜏=𝜏_𝑘^(𝑗), system (4.1) undergoes a Hopf bifurcation at the steady states states 𝑆₁ and 𝑆₂.

Figure 2 

Chaos still exists for 𝐾=−1,𝜏=0.2.

Figure 3 

Chaos still exists for 𝐾=−1,𝜏=2.5.

Figure 4 

When 𝐾=−1,𝜏=0.8, chaos vanishes, and 𝑆1 becomes local stable. Here initial value is (0.5,−0.5,0.6).

Figure 5 

When 𝐾=−1,𝜏=0.8, chaos vanishes, and 𝑆2 becomes local stable. Here initial value is (−0.5,0.5,0.6).

The above simulations indicate that when the steady state is stable or the bifurcating periodic solutions are orbitally asymptotically stable, chaos vanishes (see Figures 4–6).

Figure 6 

When 𝐾=−1,𝜏=2, chaos vanishes, and 𝑆2 becomes a stable periodic solution. Here initial value is (5,−5,5).