# Double Delayed Feedback Control of a Nonlinear Finance System

Authors: Zhichao Jiang; Yanfen Guo; Tongqian Zhang
Journal: Discrete Dynamics in Nature and Society
 (2019)

---

## Abstract

In this paper, a class of chaotic finance system with double delayed feedback control is investigated. Firstly, the stability of equilibrium and the existence of periodic solutions are discussed when delays change and cross some threshold value. Then the properties of the branching periodic solutions are given by using center manifold theory. Further, we give an example and numerical simulation, which implies that chaotic behavior can be transformed into a stable equilibrium or a stable periodic solution. Also, we give the local sensitivity analysis of parameters on equilibrium.


---

## Body

## 1. Introduction

Chaos applied to various disciplines of natural science and social science is a complex dynamic phenomenon. Chaotic systems with nonlinearity have been extensively investigated by many communities [1–5], and so on. In 1985, chaos was first discovered in the economic field, which had great impact imposed on the prominent economies. In economic field, chaos means that the economic operation has its intrinsic uncertainty.Nonlinear methods are an important research method that has been widely used to explain complex economic phenomena [6–10]. In economic field, financial risks mean the possibilities of suffering losses caused by uncertainly endogenous factors in financial or investment activities, which displays irregularly fluctuations. The source of risks derives from the strange attractor, while the key of risk management is to control the chaotic attractor. In fact, one of the features of chaos in the economic system is financial crisis. In the passed few decades, a lot of ways to control or synchronize chaos have been proposed, such as OGY method [11], PC method [12], fuzzy control [13], impulsive control [14–18], linear feedback control [19–23], delayed feedback method [24–29], multiple delayed feedback control [30–35], and so on. Nowadays, the delayed dynamic systems occupy a central position in many fields, such as biology, transport control, and chemistry [36–39]. Since many economic processes have time-delay characteristics [40–44], they are unsuitable using ordinary differential equations (ODEs) to describe. Some authors concluded that the chaotic behavior of a microeconomic system could be stabilized to periodic orbits by using delayed feedback control, which seemed more applicable to experimental systems and avoided heavy data processing. In [6, 7], the authors proposed a financial system including four parts (production, money, stock, and labor force). Furthermore, they proposed a simplified model which is described using three variables: X represents the interest rate, Y represents the investment demand, and Z represents the price index. The three-dimensional model is given as follows:(1)X˙t=Zt+Yt-aXt,Y˙t=1-bYt-X2t,Z˙t=-Xt-cZt,where a>0 is the saving amount, b>0 is the cost per investment, and c>0 is the elasticity of demand of commercial markets. For system (1), the feedback control with one delay had been studied by many authors [45–47]. In 2008, Chen [48] firstly prevented the feedback control with three delays in system (1) as follows:(2)X˙t=Zt+Yt-aXt+k1Xt-Xt-τ1,Y˙t=1-bYt-X2t+k2Yt-Yt-τ2,Z˙t=-Xt-cZt+k3Zt-Zt-τ3,where ki represent the feedback strengths and τi represent delays, i=1,2,3. In [47], by the numerical simulations, Chen firstly gave the dynamic behavior of system (2) with only one ki≠0 and then gave the dynamic behavior when all ki≠0(i=1,2,3) and τ1=τ2=τ3=τ. After that, Woo-Sik Son et al. [49] gave the theory analysis for the above results. We find that system (2) with one delay has obtained complete results and the delayed feedback control method is valid. But system (2) with multiple different delays has not completely been investigated. Therefore, our objective is to study system (2) with two different delays. Next, we assume that k3=0 and k1,k2≠0, the other cases are similar to analyze, then system (2) becomes(3)X˙t=Zt+Yt-aXt+k1Xt-Xt-τ1,Y˙t=1-bYt-X2t+k2Yt-Yt-τ2,Z˙t=-Xt-cZt, and the delayed feedback control graph is shown in Figure 1, where x(t) is the state vector of the system.Figure 1
The delayed feedback control graph.We consider system (3) with the parameters a=0.9,b=0.2, and c=1.2 [50]. When τ1=τ2=0, system (3) has a chaotic attractor (see Figure 2).Figure 2
Chaos phenomenon exists for system (3) when τ1=τ2=0.We choose the initial conditions for system (3) as (4)Xθ=φ1θ,Yθ=φ2θ,Zθ=φ3θ,where φ(θ)=(φ1(θ),φ2(θ),φ3(θ))T∈C,θ∈[-τ,0],τ=max⁡{τ1,τ2} and C denotes the Banach space C([-τ,0],R3).This paper is arranged as follows. In Section2, the stability of equilibrium and existence of Hopf bifurcations are obtained by investigating the distribution of roots of characteristic equation. In Section 3, an algorithm is derived for deciding the properties of the branching periodic solutions by computing center manifold. In Section 4, some numerical simulations are given for verifying the theoretical analyses. In Section 5, the local sensitivity analyses of parameters on equilibrium are given. At last, we give a brief conclusion and discussion.


## 2. Stability of Equilibrium and Local Hopf Bifurcation

The existence and uniqueness of solutions and stability of equilibrium have always been an important issue for differential and difference systems [51–54]. Let the right sides of system (3) be zero; it can obtain the equilibrium as follows.Lemma 1.
(i) If the conditionc-b-abc≤0 holds, then system (3) has a unique boundary equilibrium E0(0,1/b,0).
(ii) If the conditionc-b-abc>0 holds, then system (3) has three equilibria: E0 and (5)E±∗±c-b-abcc,1+acc,∓c-3/2c-b-abc.Remark 2.
If the cost per investment is smaller than some value(c/(1+ac)), then E±∗ is feasible.In this paper, it always assumes thatc-b-abc>0 holds and only considers the stability of E+∗ and the other one can be analyzed in the same way.Letu1(t)=X(t)-x∗,u2(t)=Y(t)-y∗,u3(t)=Z(t)-z∗, where x∗=(c-b-abc)/c, y∗=(1+ac)/c, and z∗=-c-3/2(c-b-abc)/c, then system (3) becomes(6)u˙1t=k1+1cu1t+x∗u2t+u3t-k1u1t-τ1+u1tu2t,u˙2t=-2x∗u1t+k2-bu2t-k2u2t-τ2-u12t,u˙3t=-u1t-cu3t,whose characteristic equation is(7)λ3+a2λ2+a1λ+a0+k1e-λτ1λ2+b1λ+b0+k2e-λτ2λ2+c1λ+c0+k1k2e-λτ1+τ2λ+c=0,where(8)a0=ck1k2-b+2cx∗2,a1=k2-bk1+1c-c-ck1+2x∗2,a2=c-k1-1c-k2-b,b0=cb-k2,b1=c+b-k2,c0=-ck1,c1=c-k1-1c.Now we analyze the distribution of roots of (7) by using the method in [55, 56].Whenτ1=τ2=0, (7) becomes(9)λ3+a2+k1+k2λ2+a1+k1b1+k2c1+k1k2λ+a0+k1b0+k2c0+ck1k2=0.It has a0+k1b0+k2c0+ck1k2=2(c-b-abc)>0 if c-b-abc>0.LetH1c+b>1c,c-1cc-1c+b2+2cb-1cc-b-abc>0. Then a2+k1+k2=c-1/c+b>0 and (a2+k1+k2)(a1+k1b1+k2c1+k1k2)-(a0+k1b0+k2c0+ck1k2)=(c-1/c)2+b2(c-1/c)+(2/c)(b-1/c)(c-b-abc)>0 under H1. By using Routh-Hurwitz criterion, all roots of (9) have negatively real parts if H1 holds.Next, letτ2=0 and use τ1 as parameter, then (7) becomes(10)λ3+a2+k2λ2+a1+k2c1λ+a0+k2c0+k1e-λτ1λ2+b1+k2λ+b0+k2c=0.Letiω(ω>0) be the root of (10), then ω satisfies(11)-ω3+a1+k2c1ω=k1b0+k2c-ω2sin⁡ωτ1-k1b1+k2ωcos⁡ωτ1,a2+k2ω2-a0-k2c0=k1b0+k2c-ω2cos⁡ωτ1+k1b1+k2ωsin⁡ωτ1.Squaring and adding in the both sides of (11), it has(12)ω6+pω4+qω2+r=0,where (13)p=k2+a22-2a1+k2c1-k12,q=a1+k2c12-2a2+k2a0+k2c0+2k12b0+k2c2-k12b1+k22,r=2c-b-abcc-b-abc-k1bc.Letz=ω2, then (12) becomes(14)hz≔z3+pz2+qz+r=0.Hence h′(z)=3z2+2pz+q. Let Δ=p2-3q and z1=(-p+△)/3, from Ruan and Wei [55], it knows that (14) has at least one positive root if r<0; (14) has no positive roots if r≥0 and △<0; if r≥0, then (14) has positive roots iff h(z1)≤0. Hence, under the condition c-b-abc>0, it has the following lemma.Lemma 3.
(i) Equation (14) has at least one positive root if c-b-abc<k1bc.
(ii) Equation (14) has no positive root if c-b-abc≥k1bc and Δ≤0.
(iii) Ifc-b-abc≥k1bc and Δ>0, then (14) has one positive roots iff z1∗>0 and h(z1∗)≤0, where Δ=p2-3q,z1∗=(-p+Δ)/3.It assumes that (14) has three positive roots, denoted by zk(k=1,2,3). Then (12) has three positive roots ωk=zk. Substituting ωk into (11) yields (15)cos⁡ωτ1=b0+k2c-ω2a2+k2ω2-a0-k2c0+b1+k2ω2-a1-k2c1ω2k1b0+k2c-ω22+k1b1+k22ω2≔Q1ω,sin⁡ωτ1=ωb0+k2c-ω2-ω2+a1+k2c1+ωb1+k2a2+k2ω2-a0-k2c0k1b0+k2c-ω22+k1b1+k22ω2≔Q2ω,and furthermore, it has(16)
τ
1
k
j
=
1
ω
k
arccos
⁡
Q
1
ω
k
+
2
j
π
,
Q
2
ω
k
≥
0
,
1
ω
k
2
π
-
arccos
⁡
Q
1
ω
k
+
2
j
π
,
Q
2
ω
k
<
0
,
k
=
1,2
,
3
,
j
=
0,1
,
⋯
,where ωkτ1k(j)∈(0,2π] is determined by the sign of sin⁡ωkτ1k(j). Define (17)τ10=min1≤k≤3,j≥0τ1kj.Letλ(τ1)=σ(τ1)+iω(τ1) be the root of (10) satisfying σ(τ1k(j))=0,ω(τ1k(j))=ωk.Lemma 4.
It assumes thath′(zk)≠0. Then(18)signdReλτ1kjdτ1=signh′zk≠0.Proof.
Substitutingλ(τ1) into (10) and taking the derivative of both sides with respect to τ1, it has (19)dλdτ1-1=3λ2+2a2+k2λ+a1+k2c1eλτ1k1λλ2+b1+k2λ+b0+k2c+2λ+b1+k2λλ2+b1+k2λ+b0+k2c-τ1λ.Hence, using zk=ωk2 and (10), it has (20)dReλτ1kjdτ1-1=Re3λ2+2a2+k2λ+a1+k2c1eλτ1+k12λ+b1+k2k1λλ2+b1+k2λ+b0+k2c-τ1λτ1=τ1kj=Re-3λ2+2λa2+k2+a1+k2c1λ3+a2+b2λ2+a1+k2c1λ+a0+k2c0+2λ+b1+k2λ2+b1+k2λ+b0+k2cτ1=τ1kj=Re3ωk2-a1+k2c1-2iωka2+b2-a2+b2ωk2+a0+k2c0+i-ωk3+ωka1+k2c1+b1+k2+2iωk-ωk2+b0+k2c+iωkb1+k2=1Θ3ωk6+2pωk4+qωk2=zkΘh′zk,where Θ=k12[b12ωk4+(ωk2-b0)ωk2]. Since Θ>0 and zk>0, then it has (21)signdReλτ1kjdτ1=signh′zk.By Lemmas3 and 4 and applying the Hopf bifurcation theorem for FDE [57–61], it has the following theorem.Theorem 5.
Whenτ2=0, suppose that H1 is satisfied.
(i) Ifc-b-abc>k1bc and Δ≤0, then, for any τ1≥0, all roots of (10) has negatively real parts and E+∗ is locally asymptotically stable.
(ii) If eitherc-b-abc<k1bc or c-b-abc≥k1bc and Δ>0,z1∗>0,h(z1∗)≤0 hold, then h(z)=0 has at least a positive root zk, for τ1∈[0,τ10), all roots of (10) have negatively real parts, and E+∗ is locally asymptotically stable.
(iii) If conditions (ii) andh′(zk)≠0 hold, then system (3) undergoes Hopf bifurcations at E+∗ when τ1=τ1k(j), j=0,1,2,…,k=1,2,3.Remark 6.
Whenτ2=0 and H1 hold, Theorem 5 tells us that, through adjusting the cost per investment, the system will tend to E+∗ or vibrates around E+∗. Under this situation, the state of system goes from order to order, that is, the macroeconomic operation is definite.It has known thatH1 guarantees that all roots of (9) have negatively real parts. Now we assume that H1 is violated. For convenience, denote (22)A=a2+k1+k2=c-1c+b,B=2-3bc-2ab+bc,C=2c1-ab-2b,then (9) becomes(23)λ3+Aλ2+Bλ+C=0.Letλ=Λ-A/3, then (23) becomes(24)Λ3+p1Λ+q1=0,where p1=B-A2/3 and q1=2A3/27-AB/3+C.Define(25)Δ1=p133+q122,γ=-q12+Δ13,ρ=-q12-Δ13.From Cardano’s formula, it has the following theorem.Theorem 7.
IfΔ1>0, then (24) has a real root γ+ρ and a pair of complex roots -(γ+ρ)/2±i(3(γ-ρ)/2), that is to say, (23) has a real root γ+ρ-A/3, and a pair of complex roots -(γ+ρ/2+A/3)±i(3(γ-ρ)/2).
(ii) If△1<0, then (24) has three real roots and (23) has also three real roots.Furthermore, it assumes thatH2Δ1>0,γ+ρ-A3<0,γ+ρ2+A3<0,γ-ρ≠0.Theorem 8.
Whenτ2=0, suppose that the condition H2 is satisfied.
(i) Ifc-b-abc>k1bc and Δ≤0, then, for any τ1≥0, (10) has at least one root with positively real parts and E+∗ is unstable.
(ii) If eitherc-b-abc<k1bc or c-b-abc≥k1bc and Δ>0,z1∗>0,h(z1∗)≤0 hold, then h(z)=0 has at least one positive root zk, for τ1∈[0,τ10), (10) has at least one root with positively real parts, and E+∗ is unstable. In addition, if dReλ(τ10)/dτ1<0, then E+∗ is locally asymptotically stable when τ1∈(τ10,τ11), where τ11 is the second bifurcating value.
(iii) If conditions (ii) andh′(zk)≠0 hold, then system (3) undergoes Hopf bifurcations at E+∗ when τ1=τ1k(j), j=0,1,2,…, k=1,2,3.Remark 9.
Whenτ2=0 and H2 hold, Theorem 8 tells us that it can still adjust the cost per investment b, such that the system tends to E+∗ or vibrates around E+∗ under some conditions. Under this situation, the state of system goes from chaos to order; that is, the financial crisis may be eliminated.From the above discussions, it knows that system (3) possibly makes stability switches as τ1 varying when τ2=0. Define I is stable interval of τ1. Let τ1∈I and λ=iϖ(τ2)(ϖ>0) be the root of (7), then it has(26)ϖ6+a22-2a1+k12-k22ϖ4+a12-2a2a0+k12b12-2b0-k22c12+k12-2c0ϖ2+a02+k12b02-k22k12c2+c02+M1cos⁡ϖτ1+M2sin⁡ϖτ1=0,where (27)M1=2k1a2-b1ϖ4+2k1b1a1-a0+b0a2-k22c1-cϖ2+2k1a0b0-cc0k22,M2=-2k1ϖ5+a2b1-a1-b0-k22ϖ3+a1b0-b1a0+k22cc1+c0ϖ.We know that (26) has finite positive roots ϖi(i=1,2,…,k). For every fixed ϖi, there exists a sequence {τ2ij∣j=0,1,2,3,⋯} satisfying (26). Define τ20=τ2i00=min{τ2ij∣i=1,2,…,k;j=0,1,2,⋯}, ϖ0=ϖi0. When τ2=τ20, ±iϖ0 are a pair of roots of (7). H3dReλτ20dτ2≠0.Hence, by Hopf bifurcation theorem [57], it has the next result.Theorem 10.
Suppose that eitherH1 or H2 is satisfied and τ1∈I.
(i) If (26) has no positive roots, then for any τ2≥0, all roots of (7) have negatively real parts and E+∗ is locally asymptotically stable.
(ii) If (26) has positive roots, then all roots of (7) have negatively real parts when τ2∈[0,τ20) and E+∗ is locally asymptotically stable. In addition, if  H3 holds, then system (3) undergoes Hopf bifurcation at E+∗ when τ2=τ20.Remark 11.
Whenτ2>0 and H1 or H2 hold, Theorem 10 tells us that, through adjusting the parameters (a,b,c,τ2), the system will tend to E+∗ or vibrates around E+∗. Under this situation, the state of system goes from order to order or from chaos to order.


## 3. Property of Hopf Bifurcation

In the above section, the sufficient conditions that system (3) undergoes a Hopf bifurcation at E+∗ when τ2=τ20 have already been obtained. In this section, it assumes that Theorem 10 (ii) is satisfied and establishes the explicit formula for determining the properties of Hopf bifurcation at τ2=τ20 by using the method developed in [62].For convenience, it assumes thatτ1=τ1∗>τ20 and lets τ¯2=τ20+μ and drops the bar for simplifying. Then μ=0 is the Hopf bifurcation value. Since τ1∗>τ20, the phase space C=C([-τ1∗,0],R3), system (3) is transformed into the following FDE in C:(28)u˙t=Lμut+fμ,ut,where ut(θ)=u(t+θ)∈C and Lμ:C→R3,f:R×C→R3 are given, respectively, by (29)Lμφ=A1φ0+B1φ-τ1∗+B2φ-τ20,where (30)A1=k1+1cx∗1-2x∗k2-b0-10-c,B1=-k100000000,B2=0000-k20000,fμ,φ=φ10φ20-φ1200,where φ=(φ1,φ2,φ3)T.By Riesz representation theorem, there exists a bounded variation functionη(ν,μ) for ν∈[-τ1∗,0], such that (31)Lμφ=∫-τ1∗0dην,μφν. For φ∈C1([-τ1∗,0],R3), define (32)Aμφ=φ˙θ,θ∈-τ1∗,0,∫-τ1∗0dην,μφν,θ=0, and (33)Rμφ=0,θ∈-τ1∗,0,fμ,φ,θ=0.Forut=u(t+θ), it has dut/dθ=dut/dt and system (28) becomes(34)ut˙=Aμut+Rμut.Forψ∈C∗:=C([0,τ1∗],R3∗) and φ∈C([-τ1∗,0],R3), define (35)A∗ψs=-ψ˙ν,ν∈0,τ1∗,∫-τ1∗0dηTt,0ψ-t,ν=0, and the inner product (36)ψ,φ=ψ¯0φ0-∫-τ1∗0∫ξ=0θψ¯ξ-θdηθφξdξ, where η(θ)=η(θ,0). It knows that A∗ and A=A(0) are adjoint operators, then ±iϖ0 are eigenvalues of A(0) and A∗ when τ2=τ20.By computations, it can obtain thatq(θ)=(1,ϑ,ς)Teiϖ0θ is the eigenvector of A corresponding to the eigenvalue iϖ0, and q∗(s)=D¯(1,ϑ∗,ς∗)eiϖ0s is the eigenvector of A∗ corresponding to the eigenvalue -iϖ0. Therefore, it has that (37)q∗s,qθ=1,q∗s,q¯θ=0,where (38)ϑ=-2x∗b+k2e-iϖ0τ20-1+iϖ0,ς=-1c+iϖ0,ς∗=1c-iϖ0,ϑ∗=x∗b+k2e-iϖ0τ20-1-iϖ0,D=1+ϑ∗ϑ∗¯+ς∗ς∗¯-τ1∗k1e-iϖ0τ1∗-ϑϑ∗¯τ20k2e-iϖ0τ20-1.Letut be the solution of system (28) at μ=0. Define z(t)=〈q∗,ut〉, then(39)z˙t=q∗,u˙t=iω0zt+q¯∗0f^z,z¯,where (40)f^=f0,Wz,z¯+2Rezq,Wz,z¯=ut-2Rezq,Wz,z¯=W20z22+W11zz¯+W02z¯22+⋯.Rewrite (39) as(41)u˙t=iϖ0zt+gz,z¯,where(42)gz,z¯=g20z22+g11zz¯+g02z¯22+g21z2z¯2⋯.Substituting (3) and (39) into W˙=ut˙-z˙q-z¯˙q¯, it has (43)W˙=AW-2Req¯∗0f^qθ,θ∈-τ1∗,0AW-2Req¯∗0f^qθ+f^,θ=0=defAW+Hz,z¯,θ,where(44)Hz,z¯,θ=H20θz22+H11θzz¯+H02θz¯22+⋯.By comparing the coefficients, it can obtain(45)g20=2Dϑ-ϑ∗¯,g11=Dϑ+ϑ¯-2ϑ∗¯,g02=2Dϑ¯-ϑ∗¯,g21=2DW1120+12W2020+W201012ϑ¯-ϑ∗¯+W1110ϑ-2ϑ∗¯,where(46)W20θ=ig20ϖ0q0eiϖ0θ+ig¯023ϖ0q¯0e-iϖ0θ+E1e2iϖ0θ,W11θ=-ig11ϖ0q0eiϖ0θ+ig¯11ϖ0q¯0e-iϖ0θ+E2,E1=2iϖ0-k1+1c+k1e-2iϖ0τ1∗-x∗-12x∗2iϖ0+b+k2e-2iϖ0τ20-10102iϖ0+c-1×ϑ-10,E2=1cx∗1-2x∗-b0-10-c-1×-ϑ+ϑ¯20.SubstitutingE1 and E2 into W20(θ) and W11(θ), then g21 can be expressed. Thus we may compute the following important quantities: (47)C10=i2ϖ0g20g11-2g112-13g022+g212,κ2=-ReC10Re⁡λ′0,ι2=2ReC10,T2=-ImC10+κ2Im⁡λ′0ϖ0.Theorem 12.
Ifκ2>0(<0), Hopf bifurcation is supercritical (subcritical). If ι2<0(>0), periodic solution is stable (unstable). If T2>0(<0), the period of periodic solution is increase (decrease).


## 4. Numerical Simulations

In this section, we give an example:(48)X˙t=Z+Y-0.9X-Xt-Xt-τ1,Y˙t=1-0.2Y-X2-2Yt-Yt-τ2,Z˙t=-X-1.2Z.With these parameters, it can obtain E+∗(0.8083,1.7333,-0.6736) and H2 is satisfied. When τ2=0, by computation, it can obtain that (12) has two positive roots ω1≐1.9997, ω2≐0.9752. Substituting them into (16) gives, respectively, (49)τ11j=2.5811+3.1421j,τ12j=0.3795+6.4430j,j=0,1,2,⋯.Furthermore, Reλ(τ11(j))/dτ1>0, dReλ(τ12(j))/dτ1<0. By Theorem 8, E+∗ is asymptotically stable when τ1∈(0.3795,2.5911) and unstable for τ1∈[0,0.3795), which means that the stability switches occur. These results are illustrated in Figures 3–5.Figure 3
E + ∗ is unstable and chaos still exists with τ1=0.1∈[0,0.3795) and τ2=0.Figure 4
E + ∗ becomes stable and chaos disappears with τ1=0.8∈(2.2945,4.3572) and τ2=0.Figure 5
E + ∗ is unstable, and a stable periodic solution appears with τ1=2.77 and τ2=0.Letτ1∗=0.5∈(0.3795,2.5911); we obtain τ20≐5.7469. By Theorem 10, it knows that E+∗ is asymptotically stable for τ1∗=0.5 and τ2∈[0,5.7469). Furthermore, it has C1(0)=-0.8562-1.4758i,β2<0 and μ2>0. Therefore, at τ20≐5.7469, the periodic solution is orbitally asymptotically stable, and the Hopf bifurcation is forward (see Figures 6 and 7).Figure 6
E + ∗ is asymptotically stable with τ1=0.5 and τ2=1.Figure 7
E + ∗ is unstable, and a periodic solution appears with τ1=0.5 and τ2=5.8.Next, we investigate the effect of two different delays. Firstly, we chooseτ1=τ2=0.1 and find system (3) has chaos phenomenon (see Figure 8). When choosing τ1=0.1 and τ2=1.8, we find that chaos phenomenon disappears and the solutions of system (48) approach stable equilibrium (see Figure 9). When choosing τ1=0.1 and τ2=3.9, chaos phenomenon disappears and appears stable periodic solutions (see Figure 10). The above results show that double delayed feedback control is superior to delayed feedback control. Hence, we improve the results in [48].Figure 8
E + ∗ is unstable and chaos exists with τ1=0.1 and τ2=0.1.Figure 9
E + ∗ becomes stable and chaos disappears with τ1=0.1 and τ2=1.8.Figure 10
E + ∗ is unstable, and a periodic solution appears with τ1=0.1 and τ2=3.9.Using the same methods, it can also obtain the similar results in the following systems by choosing suitablek1,k2:(50)X˙t=Z+Y-aX+k1Xt-Xt-τ1,Y˙t=1-bY-X2,Z˙t=-X-cZ+k2Zt-Zt-τ2, and(51)X˙t=Z+Y-0.9X,Y˙t=1-0.2Y-X2+k1Yt-Yt-τ2,Z˙t=-X-1.2Z+k2Zt-Zt-τ1.In [48], Chen investigated the dynamics of system (2) with three delays by numerical simulations with k1=k2=k3=0.1 and τ1=τ2=τ3, and Chen found that the dynamics of this case have become more complex (the inverse period doubling, period doubling routes, and chaos). Next, we further consider system (2) with three delays by numerical simulations with a=0.9,b=0.2,c=1.2, k1=-1,k2=-2, and k3=-0.5. System (2) becomes(52)X˙t=Z+Y-0.9X-Xt-Xt-τ1,Y˙t=1-0.2Y-X2-2Yt-Yt-τ2,Z˙t=-X-1.2Z-0.5Zt-Zt-τ3.We find that when τ1=τ2=τ3=0.1, system (52) has chaos phenomenon (see Figure 11), while when τ1=0.1,τ2=2,τ3=0.2, chaos phenomenon disappears and the solutions of system (52) trend to stable equilibrium (see Figure 12). Using the same methods, we can also give the theory analysis of the above result for system (2) and the similar conclusions can be obtained.Figure 11
The equilibrium is unstable and chaos exists for system (52) with τ1=τ2=τ3=0.1.Figure 12
The equilibrium becomes stable and chaos disappears for system (52) with τ1=0.1,τ2=2, and τ3=0.2.Finally, it will investigate the effect ofb for system (1) for chaos to generate. Firstly, we fix a=0.9 and c>2.852; it can obtain the Hopf bifurcation curve in (c,b) plane (see Figure 13). When c>2.852 is chosen, it can obtain b value where Hopf bifurcation will occur. The conditions here are just sufficient for the existence of Hopf bifurcation about b parameter.Figure 13
The bifurcation curve in (c,b) plane for system (1) with a=0.9.Next, we fixa=0.9, c=1.2, choosing b=0.01,0.1,0.2, respectively. When b=0.01, system has a periodic solution. Increasing b, system will produce period doubling bifurcation and ultimately lead to chaos (see Figures 14 and 15). These show that the cost per investment b makes system change from order to chaos, which means the importance of the cost per investment to control chaos.Figure 14
For (a),b=0.01. For (b), b=0.1. For (c), b=0.2.
(a) (b) (c)Figure 15
For (a),b=0.01. For (b), b=0.1. For (c), b=0.2.
(a) (b) (c)


## 5. Local Sensitivity Analysis

Local sensitivity analysis index allows us to measure the relative change of a state variable as parameter changing. Next, we use the following definition of normalized forward sensitivity index to perform local sensitivity analysis and compute normalized sensitivity indices.Definition 13 (see [63]).
The normalized forward sensitivity index of a variable,u, that depends differentiably on a parameter, q, is defined as (53)Υqu=∂u∂q×qu.To perform local sensitivity analysis, we seta=0.9,b=0.2,c=1.2.Tables1–3 show the effect of parameters a,b,c on equilibrium E+∗(x∗,y∗,z∗).Table 1
Normalized sensitivity indexes and order of importance ofx∗ to the three parameters evaluated at the values a=0.9, b=0.2, and c=1.2.
Parameter Local sensitivity index Order of importance a -0.1378 2 b -0.2653 1 c +0.1276 3Table 2
Normalized sensitivity indexes and order of importance ofy∗ to the three parameters evaluated at the values a=0.9, b=0.2, and c=1.2.
Parameter Local sensitivity index Order of importance a +0.5192 1 b 0 3 c -0.4808 2Table 3
Normalized sensitivity indexes and order of importance ofz∗ to the three parameters evaluated at the values a=0.9, b=0.2, and c=1.2.
Parameter Local sensitivity index Order of importance a -0.1378 3 b -0.2653 2 c -0.8724 1Table1 shows that decreasing (respectively, increasing) the savings amount a by 1% will increase (respectively, decrease) the interest rate x∗ by 0.1378%. Decreasing (respectively, increasing) the cost per investment b by 1% will increase (respectively, decrease) the interest rate x∗ by 0.2653%. Increasing (respectively, decreasing) the elasticity of demand of commercial markets c by 1% will increase (respectively, decrease) the interest rate x∗ by 0.1276%. The conclusion is that the cost per investment is the most important factor to the interest rate.Table2 shows that increasing (decreasing) the savings amount a by 1% will increase (decrease) the investment demand y∗ by 0.5192%. Decreasing (respectively, increasing) the cost per investment c by 1% will increase (decrease) the investment demand y∗ by 0.4808%. The conclusion is that the savings amount is the most important factor to the investment demand.Table3 shows that decreasing (increasing) the savings amount a by 1% will increase (decrease) the price index z∗ by 0.1378%. Decreasing (increasing) the cost per investment b by 1% will increase (decrease) the price index z∗ by 0.2653%. Decreasing (increasing) the elasticity of demand of commercial markets c by 1% will increase (decrease) the price index z∗ by 0.8724%. The conclusion is that the elasticity of demand of commercial markets is the most important factor to the price index.


## 6. Conclusions and Discussions

Bifurcation in nonlinear finance system with one delay has been studied by many researchers. However, there are few papers to focus on nonlinear finance system with multiple delay feedback control. In this paper, we analyze a chaotic finance system using double delayed feedback control and find that the stability switches can occur whenτ1 varies in the case of τ2=0. The conclusion shows that if the saving amount, cost per investment, and the elasticity of demand are fixed, the feedback control used on the interest rate term can cause periodic fluctuations of the system when the feedback strength is fixed and chaotic phenomenon vanish. That is, it is effective in eliminating financial crisis using delayed feedback control in the interest rate term.Then fixτ1 in a stability interval, regarding τ2 as parameter; it can show that there exists the first critical value of τ2 at which the equilibrium loses its stability and the Hopf bifurcation occurs. These conclusions show that if the feedback control used on the interest rate term under some delay is invalid to remove chaos, then it may add the feedback control to the investment demand term at the same time, which can make chaos disappear and the system produces regular vibrations. The results tell us that the double delayed feedback control can be considered better method than single delayed feedback control for the control of chaotic attractor.Our results show that, for a class of chaotic finance system, the chaos oscillation can be controlled by delays. In addition, by choosing different delays and numerical simulations, we improve the results in [48] and show that the multiple delayed feedback control is more effective than one delayed feedback control.In addition, we also obtain that system can produce chaos by period doubling bifurcation when increasing the cost per investmentb, which means the importance of the cost per investment to control chaos. At last, local sensitivity analyses of parameters a,b,c on equilibrium are given. The conclusions are that the cost per investment is the most important factor to the interest rate; the savings amount is the most important factor to the investment demand; the elasticity of demand of commercial markets is the most important factor to the price index.



---

*Source: 7254121-2019-02-05.xml*