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

Authors: Rongyan Zhang
Journal: Journal of Applied Mathematics
 (2012)

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## 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.


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## Body

## 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
(1.1)ẋ(t)=-a(x(t)+y(t)),ẏ(t)=-y(t)-ax(t)z(t),ż(t)=b+ax(t)y(t),
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 demandy. 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
(1.2)ẋ(t)=-a(x(t)+y(t)),ẏ(t)=-y(t)-ax(t)z(t)+K(y(t)-y(t-τ)),ż(t)=b+ax(t)y(t).
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:(2.1)S1=(abb,-abb,1a),S2=(-abb,abb,1a).The characteristic equation of the system (1.1) at S1(S2) is
(2.2)λ3+(1+a)λ2+abλ+2a2b=0.By analyzing the characteristic equation (2.2) and the Routh-Hurwitz criteria, we get the following.Lemma 2.1.
Fora<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.
Fora=1, the characteristic equation (2.2) has a pair of purely imaginary eigenvalues λ1,2=±iω0(ω0=b) and a negative real eigenvalue λ3=-2, and
(2.3)λ̇(a=1)=5b2b+8>0.
According to the hopf bifurcation theorem [11], a hopf bifurcation of the system (1.1) occurs at a=1.Lemma 2.3.
Fora>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
(2.4)x1(t)=x(t)-abb,y1(t)=y(t)+abb,z1(t)=z(t)-1a,
system (1.2) becomes
(2.5)ẋ1(t)=-a(x1(t)+y1(t)),ẏ1(t)=-x1(t)-y1(t)-abz1(t)-ax1(t)z1(t)+K(y1(t)-y1(t-τ)),ż1(t)=-abx1(t)+aby1(t)+ax1(t)y1(t).
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
(2.6)det⁡⁡(λ+aa01λ+1-K+Ke-λτabab-abλ)=0.
Expanding (2.6), we have
(2.7)λ3+(1+a-K)λ2+(ab-aK)λ+2a2b+K(λ2+aλ)e-λτ=0.
Thus, we need to study the distribution of the roots of the third-degree exponential polynomial equation:
(2.8)λ3+a2λ2+a1λ+a0+(b2λ2+b1λ+b0)e-λτ=0,
where ai,bi∈R(i=0,1,2) and ∑i=02bi2≠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(2.9)P(λ,e-λτ1,…,e-λτm)=λn+p1(0)λn-1+⋯+pn-1(0)λ+pn(0)+[p1(1)λn-1+⋯+pn-1(1)λ+pn(1)]e-λτ1+⋯+[p1(m)λn-1+⋯+pn-1(m)λ+pn(m)]e-λτm,
where τi≥0(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,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
(2.10)-iω3-a2ω2+a1ωi+a0+(-b2ω2+b1ωi+b0)(cos⁡ωτ-isin⁡ωτ)=0.
Separating the real and imaginary parts, we have
(2.11)a2ω2-a0=(b0-b2ω2)cos⁡ωτ+b1ωsin⁡ωτ,-ω3+a1ω=(b0-b2ω2)sin⁡ωτ-b1ωcos⁡ωτ,
which is equivalent to
(2.12)ω6+(a22-b22-2a1)ω4+(a12-2a0a2-b12+2b0b2)ω2+a02-b02=0.
Let z=ω2 and denote p=a22-b22-2a1,  q=a12-2a0a2-b12+2b0b2, r=a02-b02, then (2.12) becomes
(2.13)z3+pz2+qz+r=0.In the following, we need to seek conditions under which (2.12) has at least one positive root. Denote(2.14)h(z)=z3+pz2+qz+r.Therefor, applying [13], we obtain the following lemma.Lemma 2.5.
For the polynomial equation (2.13), one has the following results.(i)
Ifr<0, then (2.13) has at least one positive root.(ii)
Ifr≥0 and Δ=p2-3q≤0, then (2.13) has no positive roots.(iii)
Ifr≥0 and Δ=p2-3q>0, then (2.13) has positive roots if and only if z1*=(1/3)(-p+Δ)>0 and h(z1*)≤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:
(2.15)ω1=z1,ω2=z2,ω3=z3.
From (2.11), we have
(2.16)cos⁡ωτ=b1ω2(ω2-a1)-(a2ω2-a0)(b2ω2-b0)(b2ω2-b0)2+b12ω2.
Thus, if we denote
(2.17)τk(j)=1ωk[cos⁡-1(b1ωk2(ωk2-a1)-(a2ωk2-a0)(b2ωk2-b0)(b2ωk2-b0)2+b12ωk2)+2jπ],
where k=1,2,3;j=0,1,2,…, then ±iωk is a pair of purely imaginary roots of (2.8) with τk(j). Define
(2.18)τ0=τk0(0)=min⁡k∈1,2,3τk(0),ω0=ωk0.
Note that when τ=0, (2.8) becomes
(2.19)λ3+(a2+b2)λ2+(a1+b1)λ+a0+b0=0.Therefor, applying Lemmas2.4 and 2.5 to (2.8), we get the following lemma.Lemma 2.6.
For (2.8), one has(i)
ifr≥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 eitherr<0 or r≥0, Δ=p2-3q>0, z1*=(1/3)(-p+Δ)>0 and h(z1*)≤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(2.20)λ(τ)=α(τ)+iω(τ)
be the root of (2.8) near τ=τk(j) satisfying
(2.21)α(τk(j))=0,ω(τk(j))=ωk.
Then by [13], we have the following transversality condition.Lemma 2.7.
Suppose thatzk=ωk2 and h′(zk)≠0. Then
(2.22)Rλ(τk(j))dτ≠0,
and Rλ(τk(j))/dτ and h′(zk) 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
(2.23)a2=1+a-K,a1=ab-aK,a0=2a2b,b2=K,b1=aK,b0=0.
Thus,
(2.24)p=a22-b22-2a1=a2+2a+1-2ab-2K,q=a12-2a0a2-b12+2b0b2=a2b2+2a2bK-4a2b-4a3b,r=a02-b02=4a4b2>0,
and then we can compute
(2.25)Δ=p2-3q,h(z)=z3+pz2+qz+r,z1*=13(-p+Δ).
When τ=0, (2.7) becomes (2.2)(2.26)λ3+(1+a)λ2+abλ+2a2b=0.Applying Lemmas2.1, 2.2, 2.6, and 2.7 to (2.7), we have the following theorems.Theorem 2.8.
Letτk(j) and τ0 be defined by (2.17) and (2.18). Suppose that conditions (C1) and a<1 hold.(i)
IfΔ≤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)
IfΔ>0,z1*>0 and h(z1*)≤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, andh′(zk)≠0, then system (1.2) exhibits the Hopf bifurcation at the equilibrium S1 (or S2) for τ=τk(j).Theorem 2.9.
Letτk(j) and τ0 are defined by (2.17) and (2.18). Suppose that conditions (C1) and a>1 hold.(i)
IfΔ≤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)
IfΔ>0,z1*>0 and h(z1*)≤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, andh′(zk)≠0, then system (1.2) exhibits the Hopf bifurcation at the equilibrium S1 (or S2) for τ=τk(j).


## 3. Direction and Stability of the Hopf Bifurcation

In the Section2, 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=τk and then ±iωk is corresponding purely imaginary roots of the characteristic equation at the steady state (x*,y*,z*).Lettingx1=x-x*,x2=y-y*,x3=z-z*,x¯i(t)=xi(τt),τ=τk+μ and dropping the bars for simplification of notations, system (1.2) is transformed into an FDE in C=C([-1,0],R3) as
(3.1)ẋ(t)=Lμ(xt)+f(μ,xt),
where x(t)=(x1(t),x2(t),x3(t))T∈R3, and Lμ:C→R,f:R×C→R are given, respectively, by
(3.2)Lμ(ϕ)=(τk+μ)(-a-a0-az*K-1-ax*ay*ax*0)(ϕ1(0)ϕ2(0)ϕ3(0))+(τk+μ)(0000-K0000)(ϕ1(-1)ϕ2(-1)ϕ3(-1)),f(μ,ϕ)=(τk+μ)(0-aϕ1(0)ϕ3(0)aϕ1(0)ϕ2(0)).
By the Riesz representation theorem, there exists a function η(θ,μ) of bounded variation for θ∈[-1,0], such that
(3.3)Lμϕ=∫-10dη(θ,0)ϕ(θ)
for ϕ∈C[-1,0].In fact, we can choose(3.4)η(θ,μ)=(τk+μ)(-a-a0-az*K-1-ax*ay*ax*0)δ(θ)-(τk+μ)(0000-K0000)δ(θ+1),
where δ is the Dirac delta function. For ϕ∈C1([-1,0],(R3)*), define
(3.5)A(μ)ϕ={dϕ(θ)dθ,θ∈[-1,0),∫-10dη(μ,s)ϕ(s),θ=0,R(μ)ϕ={0,θ∈[-1,0),f(μ,ϕ),θ=0.
Then system (3.1) is equivalent to
(3.6)ẋt=A(μ)xt+R(μ)xt,
where xt(θ)=x(t+θ) for θ∈[-1,0].Forψ∈C1([0,1],R3), define
(3.7)A*ψ(s)={-dψ(s)ds,s∈(0,1],∫-10dηT(t,0)ψ(-t),s=0,
and a bilinear inner product
(3.8)〈ψ(s),ϕ(θ)〉=ψ¯(0)⋅ϕ(0)-∫θ=-10∫ξ=0θψ¯T(ξ-θ)dη(θ)ϕ(ξ)dξ,
where η(θ)=η(θ,0). Then A=A(0) and A*=A*(0) are adjoins operators.By the discussion in Section2, we know that ±iωkτk are eigenvalues of A, thus they are also eigenvalues of A*.By direct computation, we obtain thatq(θ)=q0eiθωkτk, with
(3.9)q(0)=(1,α,β)T=(1,-a+iωka,a(y*-x*)-iωkx*iωk)T,
is the eigenvector of A corresponding to iωkτk, and q*(s)=Dq0*eisωkτk, with
(3.10)q0*=(1,α*,β*)T=(1,iωk(a-iωk)a2x*y*-iωkaz*,(a-iωk)ax*a2x*y*-iωkaz*)T,
is the eigenvector of A* corresponding to -iωkτk, where
(3.11)D=11+α¯α*+β¯β*-Kτkα¯α*eiωkτk.
Using the same notation as in [14], we compute the coordinates to describe the center manifold C0 at μ=0. Let xt be the solution of (3.1) when μ=0. Define
(3.12)z(t)=〈q*,xt〉,W(t,θ)=xt(θ)-2Re{z(t)q(θ)}.
On the center manifold C0, we have(3.13)W(t,θ)=W(z(t),z¯(t),θ),where(3.14)W(z,z¯,θ)=W20(θ)z22+W11(θ)zz¯+W02(θ)z¯22+W30(θ)z36+⋯,z and z¯ are local coordinates for center manifold C0 in the direction of q* and q¯*. Note that W is real if xt is real. We consider only real solutions. For the solution xt∈C0 of (3.1), since μ=0, we have
(3.15)ż(t)=iτkωkz+q¯*(θ)f(0,W(z,z¯,θ)+2Re{zq(θ)})=iτkωkz+q¯*(0)f(0,W(z,z¯,0)+2Re{zq(0)})=defiτkωkz+q¯*(0)f0(z,z¯).
We rewrite this equation as
(3.16)ż(t)=iτkωkz(t)+g(z,z¯),
where
(3.17)g(z,z¯)=q¯*(0)f0(z,z¯)=g20z22+g11zz¯+g02z¯22+g21z2z¯2+⋯.
Noticing
(3.18)xt(θ)=(x1t(θ),x2t(θ),x3t(θ))=W(t,θ)+zq(θ)+zq¯(θ),q(θ)=(1,α,β)Teiθωkτk,
we have
(3.19)x1t(0)=z+z¯+W20(1)(0)z22+W11(1)(0)zz¯+W02(1)(0)z¯22+O(|(z,z¯)|3),x2t(0)=αz+αz¯+W20(2)(0)z22+W11(2)(0)zz¯+W02(2)(0)z¯22+O(|(z,z¯)|3),x3t(0)=βz+βz¯+W20(3)(0)z22+W11(3)(0)zz¯+W02(3)(0)z¯22+O(|(z,z¯)|3).
Thus, form (3.17), we have(3.20)g(z,z¯)=q¯*(0)f0(z,z¯)=D¯τk(1,α*¯,β*¯)(0-ax1t(0)x3t(0)ax1t(0)x2t(0))=-aD¯τkα*¯[z+z¯+W20(1)(0)z22+W11(1)(0)zz¯+W02(1)(0)z¯22+O(|(z,z¯)|3)][βz+βz¯+W20(3)(0)z22+W11(3)(0)zz¯+W02(3)(0)z¯22+O(|(z,z¯)|3)]+aD¯τkβ*¯[z+z¯+W20(1)(0)z22+W11(1)(0)zz¯+W02(1)(0)z¯22+O(|(z,z¯)|3)][αz+αz¯+W20(2)(0)z22+W11(2)(0)zz¯+W02(2)(0)z¯22+O(|(z,z¯)|3)].Comparing the coefficients of (3.17), we get
(3.21)g20=-2aD¯τk(β*¯α-α*¯β),g11=2aD¯τk(β*¯Re(α)-α*¯Re(β)),g02=-2aD¯τk(β*¯α¯-α*¯β¯),g21=-aD¯τkα*¯[2W11(3)(0)+W20(3)(0)+2βW11(1)(0)+β¯W20(1)(0)]+aD¯τkβ*¯[2W11(2)(0)+W20(2)(0)+2αW11(1)(0)+α¯W20(1)(0)].
Since there are W20(θ) and W11(θ) in g21, we need to compute them.From (3.6) and (3.12), we have
(3.22)Ẇ=ẋt-żq-z¯q¯̇={AW-2Re{q¯*(0)f0q(θ)},θ∈[-1,0),AW-2Re{q¯*(0)f0q(0)}+f0,θ=0.=defAW+H(z,z¯,θ),
where
(3.23)H(z,z¯,θ)=H20(θ)z22+H11(θ)zz¯+H02(θ)z¯22+⋯.
Expanding the above series and comparing the corresponding coefficients, we obtain
(3.24)(A-2iτkωk)W20(θ)=-H20(θ),AW11(θ)=-H11(θ),….
From (3.22), we know that for θ∈[-1,0),(3.25)H(z,z¯,θ)=-q¯*(0)f0q(θ)-q*(0)f¯0q¯(0)=-gq(θ)+gq¯(θ).
Comparing the coefficients with (3.23) gives that
(3.26)H20(θ)=-g20q(θ)-g¯02q¯(θ),(3.27)H11(θ)=-g11q(θ)-g¯11q¯(θ).
From (3.24), (3.26) and the definition of A, it follows that
(3.28)Ẇ20(θ)=2iτkωkW20(θ)+g20q(θ)+g¯02q¯(θ).
Notice that q(θ)=(1,α,β)Teiθωkτk, hence
(3.29)W20(θ)=-ig20ωkτkq(0)eiθωkτk+ig¯023ωkτkq¯(0)e-iθωkτk+E1e2iθωkτk,
where E1=(E1(1),E1(2),E1(3))T∈R3 is a constant vector.Similarly, from (3.24) and (3.27), we can obtain
(3.30)W11(θ)=-ig11ωkτkq(0)eiθωkτk+ig¯11ωkτkq¯(0)e-iθωkτk+E2,
where E2=(E2(1),E2(2),E2(3))T∈R3 is also a constant vector.In what follows, we shall seek appropriateE1 andE2. From the definition of A and (3.24), we obtain
(3.31)∫-10dη(θ)W20(θ)=2iωkτkW20(0)-H20(0),(3.32)∫-10dη(θ)W11(θ)=-H11(0),
where η(θ)=η(θ,0). By (3.22), we have
(3.33)H20(0)=-g20q(0)-g¯02q¯(0)+2τk(0-βα),(3.34)H11(0)=-g11q(0)-g¯11q¯(0)+2τk(0-Re(β)Re(α)).
Substituting (3.29) and (3.33) into (3.31), we obtain
(3.35)(2iωkτkI-∫-10e2iθωkτkdη(θ))E1=2τk(0-βα),
which leads to
(3.36)(2iωk+aa0az*2iωk-K+1+Ke-2ωkτkax*-ay*-ax*2iωk)E1=2(0-βα),E1(1)=2a(2iωkβ+aαx*)A,E1(2)=-2(2iωk+a)(2iωkβ+aαx*)A,E1(3)=2AA1,
where(3.37)A=det⁡(2iωk+aa0az*2iωk-K+1+Ke-2ωkτkax*-ay*-ax*2iωk),A1=det⁡(2iωk+aa0az*2iωk-K+1+Ke-2ωkτk-β-ay*-ax*α).Similarly, substituting (3.30) and (3.34) into (3.32), we can get
(3.38)(aa0az*1ax*-ay*-ax*0)E2=2(0-Re(β)Re(α)),E2(1)=2a2x*Re(α)B,E2(2)=-2a2x*Re(α)B,E2(3)=2BB1,
where
(3.39)B=det⁡⁡(aa0az*1ax*-ay*-ax*0),B1=det⁡⁡(aa0az*1-Re(β)-ay*-ax*Re(α)).
Thus, we can determine W20(0) and W11(0) from (3.29) and (3.30). Furthermore, we can determine g21. Therefore, each gij in (3.21) is determined by the parameters and delay in (3.1). Thus, we can compute the following values:
(3.40)c1(0)=i2τkωk(g11g20-2|g11|2-|g02|23)+g212,μ2=-Re(c1(0))Re(λ′(τk)),β2=2Re(c1(0)),T2=-Im⁡(c1(0))+μ2Im⁡(λ′(τk))τkωk,
which determine the quantities of bifurcating periodic solutions in the center manifold at the critical value τk, 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>τk(τ<τk); 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:
(4.1)ẋ(t)=-1.69(x(t)+y(t)),ẏ(t)=-y(t)-1.69x(t)z(t)+K(y(t)-y(t-τ)),ż(t)=4+1.69x(t)y(t),
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 stateS+ or S-, we have the corresponding characteristic equation of system (4.1) as follows:
(4.2)λ3+(2.69-K)λ2+(6.76-1.69K)λ+22.8488+K(λ2+1.69λ)e-λτ=0.
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 z1*=(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,
(4.3)ẋ(t)=-1.69(x(t)+y(t)),ẏ(t)=-y(t)-1.69x(t)z(t)-(y(t)-y(t-τ)),ż(t)=4+1.69x(t)y(t).
In this case, we can compute
(4.4)p=̇-4.2839,q=̇-100.077,r=̇522.0677,Δ=̇318.5850,z1=̇8.9478,ω1=2.9913,τ1(j)=̇0.2205+2jπω1,h′(z1)=̇63.4487,z2=̇5.6545,ω2=2.3779,τ2(j)=̇0.5227+2jπω2,h′(z2)=̇-52.6043.Thus, from Lemma2.7, we have Reλ(τ1(j))/dτ>0 and Reλ(τ2(j))/dτ<0. In addition, notice that
(4.5)τ1(0)=̇0.2205<τ2(0)=̇0.5227<τ1(1)=̇2.3210<τ2(1)=̇3.1650.
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 thatτk(j),k=1,2;j=0,1,2,… is defined by (4.4).(i)
Whenτ∈[0,τ2(0))∪(τ1(1),∞), the steady states S1 and S2 of the system (4.1) are unstable (see Figures 2 and 3).(ii)
Whenτ∈(τ2(0),τ1(1)), the steady states S1 and S2 of the system (4.1) are asymptotically (see Figures 4 and 5).(iii)
Whenτ=τk(j), system (4.1) undergoes a Hopf bifurcation at the steady states states S1 and S2.Figure 2
Chaos still exists forK=-1,τ=0.2.Figure 3
Chaos still exists forK=-1,τ=2.5.Figure 4
WhenK=-1,τ=0.8, chaos vanishes, and S1 becomes local stable. Here initial value is (0.5,-0.5,0.6).Figure 5
WhenK=-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 Figures4–6).Figure 6
WhenK=-1,τ=2, chaos vanishes, and S2 becomes a stable periodic solution. Here initial value is (5,-5,5).



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*Source: 316390-2012-04-05.xml*