# Feedback Control of a Chaotic Finance System with Two Delays

Authors: Zhichao Jiang; Tongqian Zhang
Journal: Complexity
 (2020)

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## Abstract

In this research, we use the double-delayed feedback control (DDFC) method in order to control chaos in a finance system. Taking delays as parameters, the dynamic behavior of the system is investigated. Firstly, we study the local stability of equilibrium and the existence of local Hopf bifurcations. It can find that the delays can make chaos disappear and generate a stable equilibrium or periodic solution, which means the effectiveness of DDFC method. By using the normal form theory and center manifold argument, one derives the explicit algorithm for determining the properties of bifurcation. In addition, we also apply some mathematical methods (stability crossing curves) to show the stability changes of the financial system in two parameters’τ1,τ2 plane. Finally, we give some numerical simulations by Matlab Microsoft to show the validity of theoretical analyses.


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## Body

## 1. Introduction

In the past few decades, many scholars produced the increasing interest in nonlinear dynamic economic methods [1–11]. In the fields of finance, because of the influence of nonlinear factors, all sorts of economy problems become more and more complicated. The misalignment of certain parameters in the economic system can lead to runaway markets and possibly even a financial crisis [12–15]. Therefore, it is more and more important to study the internal structure characteristics of a complex financial system and uncover its causes, so as to predict and control the system.A lot of work has been carried out in modeling nonlinear economic dynamics, such as Goodwin’s model, van der Pol model, IS-LM model, and nonlinear finance system [14, 16–25]. However, it is well known that even a simple nonlinear system can exhibit chaotic behavior. Chaos is the inherent randomness of deterministic systems. Since the first discovery of chaos in economics from 1985, a great impact has been produced on the study of western economics because chaos in the economic system means the inherent uncertainty in macroeconomic operation. Over the past two decades, many efforts had been made to control chaos, such as stability and chaos synchronization, at unstable fixed points. In recent years, many methods had been put forward to control and synchronize chaos, such as OGY method [26], PC method [27], fuzzy control [28], impulsive control method [29, 30], stochastic control [31–33], linear feedback control [34], delay feedback approach [35–44], and multiple delay feedback control (MDFC) [45]. Delayed feedback control (DFC) was first proposed by Pyragas [46] in order to stabilize unstable periodic orbits (UPO). Then, the DFC method was extended to the multidelay [47]. One of the main characteristics of the DFC method is that it does not need the knowledge of the internal dynamics of the system beyond the period nor does it require a preliminary understanding of the required UPO. At the same time of UPO control, using the DFC method to realize USS stability had become an area of concern and had been applied to some real systems. It is very successful in stabilizing UPO for the DFC method, but the control of USS is less efficient. In [45], authors put forward the MDFC method and conducted numerical simulations, which showed that the MDFC method preceded the DFC method in USS stability.In [16], authors put forward a financial system describing the temporal changes using three variables: xt denotes the interest rate, yt expresses the investment demand, and zt represents the price index:(1)x˙t=y−ax+z,y˙t=1−by−x2,z˙t=−x−cz,where the parameters a,b,andc represent the saving amount, the investment cost, and the elasticity of market demand, respectively, and a,b,andc are positive constants. From [48], it is known that, under the parameter values a=0.9,b=0.2, and c=1.2, system (1) exists a strange attractor, as shown in Figure 1.Figure 1
There exists chaotic attractor for system (1). (a) Time series of the solutions of system (1). (b) Three-dimensional phase diagram of system (1).
(a)(b)In this paper, our object is to control the strange attractor by using the DDFC method and study the following system:(2)x˙t=y−ax+z+k1xt−xt−τ1+k2xt−xt−τ2,y˙t=1−by−x2,z˙t=−x−cz,where k1∈ℝ and k2∈ℝ are the feedback strengths and τ1 and τ2 are nonnegative delays.The initial conditions of system (2) are given as(3)x♭=φ1♭,y♭=φ2♭,z♭=φ3♭,♭∈−τ,0,where φ=φ1,φ2,φ3T∈C=C−τ,0,ℝ3 and τ=maxτ1,τ2.The purpose of this paper is to analyze and numerically study system (2). Our results show that the stability of system varies with delays. When the delay passes a certain critical value, the chaotic oscillation disappears and can be transformed into stable equilibrium or periodic orbit, which indicates that the chaotic property changes with the changes of delays.This article is organized as follows. In Section2, by studying the distribution of eigenvalues of exponential polynomials and using the results in [49, 50], the local stability and existence of local Hopf bifurcation are obtained. In Section 3, the properties of Hopf bifurcation are given by using central manifold theory and normal form method. In Section 4, using the crossing curve methods, it can obtain the stable changes of equilibrium in τ1,τ2 plane to overcome the problem that no information is given on the plane τ1,τ2 that comes into being stable or unstable equilibrium in Section 2. To support the analysis results, some numerical simulations are carried out in Section 5. Finally, some conclusions and discussions are given.


## 2. Stability of Equilibrium and Hopf Bifurcation

Firstly, it gives the existence of equilibria.Lemma 1.
(i)
Ifc1−ab−b≤0 holds, then system (2) has only a boundary equilibrium E00,1/b,0(ii)
Ifc1−ab−b>0 holds, then system (2) has two interior equilibria E±∗±κ,1+ac/c,∓κc−3/2 besides E0, where κ=1−ab−b/c1/2.
In the following text, it always assumes thatc1−ab−b>0 is satisfied and only considers the stability of E+∗ and the other equilibria can be analyzed similarly.
Letu1=x−κ,u2=y−1+ac/c, and u3=z+κc−3/2, then system (2) becomes(4)u˙1t=1c+k1+k2u1t+κu2t+u3t−k1u1t−τ1−k2u1t−τ2+u1tu2t,u˙2t=−2κu1t−bu2t−u12t,u˙3t=−u1t−cu3t,whose characteristic equation is(5)∇λ,τ1,τ2=λ3+a2λ2+a1λ+a0+k1e−λτ1λ2+b1λ+b0+k2e−λτ2λ2+b1λ+b0=0,where(6)a0=c2κ2−bk1+k2,a1=2κ2+bc−bc−b+ck1+k2,a2=b+c−1c+k1+k2,b0=bc,b1=b+c.
Now, we use the method in [49, 50] to study the root distribution of (5). When τ1=τ2=0, (5) becomes(7)∇λ,0,0=λ3+k1+k2+a2λ2+k1b1+k2b1+a1λ+k1b0+k2b0+a0=0.
By Routh–Hurwitz criterion, all roots of (7) have negative real parts if and only if(8)H1a2+k1+k2>0,a0+k1b0+k2b0>0,a2+k1+k2a1+k1b1+k2b1>a0+k1b0+k2b0,holds.


### 2.1. The Caseτ1>0 and τ2=0

In this part, letτ2=0, and choose τ1 as the parameter to study the distribution of the root of (5). Let iω be the root of (5), then ω must satisfy the following equations:(9)−ω3+a1+k2b1ω=k1b0−ω2sinωτ1−b1ωcosωτ1,a2ω2−a0−k2b0−ω2=k1b0−ω2cosωτ1+b1ωsinωτ1.Adding the squares of both sides of (9), it yields to(10)ω6+pω4+qω2+r=0,where(11)p=k2+a22−2a1+k2b1−k12,q=a1+k2b12−2a2+k2a0+b0k2+2b0k1−k12b12,r=a0+k2b02−k12b02.Furthermore, from (9), it can be obtained that(12)cosωτ1=Qb0−ω2−pb1k1b0−ω22+k1b12ω2≔S1,sinωτ1=pb0−ω2+Qb1ωk1b0−ω22+k1b12ω2≔S2,where p=−ω3+a1+k2b1ω and Q=a2ω2−a0−k2b0−ω2.Letz=ω2, then (10) becomes(13)hz≔z3+pz2+qz+r=0.Applying the results in [49], the following conclusions hold.Lemma 2.
(i)
Ifr≥0 and △≤0 hold, then (13) has no positive root(ii)
Ifr<0 holds, then (13) has at least a positive root(iii)
Ifr≥0 and △>0 hold, then (13) has a positive roots iff z1∗>0 and hz1∗≤0, where z1∗=−p+△/3 and △=p2−3q
Without loss of generality, it supposes that (13) has three positive roots, denoted by z1,z2,andz3, respectively. Then, (10) has three positive roots ωk=zkk=1,2,3. Substituting ωk into (9) gives(14)τ1kj=1ωkarccosS1+2jπ,S2≥0,1ωk−arccosS1+2j+1π,S2<0,where k=1,2,3,j=0,1,….
Defineτ10=mink=1,2,3τ1k0. Let λτ1=γτ1+iωτ1 be the root of (5) with τ2=0 satisfying γτ1kj=0 and ωτ1kj=ωk.Lemma 3.
Suppose thath′zk≠0, then dγτ1kj/dτ1≠0 and Signdγτ1kj/dτ1=Signh′zk.Proof.
Letτ2=0, and differentiate both sides of (5) about τ1, and it has(15)dλdτ1−1=3λ2+2b1+k2λ+a1+k2b1eλτ1k1λλ2+b1λ+b0+2λ+b1λλ2+b1λ+b0−τ1λ.
Hence,(16)dγτ1kjdτ1−1=Re3λ2+2b1+k2λ+a1+k2b1eλτ1+k12λ+b1k1λλ2+b1λ+b0−τ1λτ1=τ1kj=1Γ3ωk6+2pωk4+qωk2=zkΓh′zk,where Γ=k12b12ωk4+ωk2−b0ωk2. Since Γ>0 and zk>0, then we have(17)Signdγτ1kjdτ1=Signh′zk.
By3 and applying the Hopf bifurcation theorem in [51], for system (2) it has the following theorem.Theorem 1.
It assumes thatH1 holds:(i)
Ifr>0 and △≤0 hold, then, for all τ1≥0, E+∗ is locally asymptotically stable (LAS)(ii)
If eitherr<0 or r≥0 and △>0, z1∗>0, hz1∗≤0 hold, then for τ1∈0,τ10, E+∗ is LAS(iii)
If all conditions in (ii) andh′zk≠0 hold, then system (2) undergoes Hopf bifurcations at E+∗ when τ1=τ1kj, j=0,1,2,…,k=1,2,3
We know that the condition (H1) guarantees that all roots of (7) have negative real parts. If (H1) is violated, we define(18)A=a2+k1+k2,B=a1+k1+k2b1,C=a0+k1+k2b0.
Letλ=Λ−A/3, then (7) becomes(19)Λ3+p1Λ+q1=0,where p1=B−A2/3 and q1=2A3/27−AB/3+C.
Define(20)Δ1=p133+q122,α=3−q12+Δ1,β=3−q12−Δ1.
Then, from Cardano’s formula, it has the following Theorem.Theorem 2.
(i)
If△1<0, then (19) has three real roots(ii)
IfΔ1>0, then (19) has a real root α+β−A/3 and a pair of conjugate complex roots −α+β/2+A/3±i3α−β/2
Furthermore, we assume that(21)H2Δ1>0,α+β2+A3<0,α+β−A3<0,α−β≠0.Theorem 3.
It assumes thatH2 holds. For system (2), it has the following results.(i)
Ifr>0 and Δ≤0, then, for all τ1≥0, E+∗ of system (2) is unstable.(ii)
If eitherr<0 or r≥0 and Δ>0, z1∗>0, hz1∗≤0 hold, then for τ1∈0,τ10, E+∗ of system (2) is unstable. In addition, if dReλτ10/dτ1<0, then E+∗ is LAS when τ1∈τ10,τ11, where τ11 is the second critical value.(iii)
If all conditions in (ii) andh′zk≠0 hold, then system (2) undergoes Hopf bifurcations at E+∗ when τ1=τ1kj, j=0,1,2,…,k=1,2,3.
From the abovementioned discussion, one can know that the stable switch may exist asτ1 varies for system (2) with τ2=0. Define I as stable interval of τ1.


### 2.2. The Caseτ1∈I and τ2>0

In this part, letτ1∈I, τ2>0, and λ=iϖϖ=ϖτ2>0 be the root of (5), and it has(22)−ϖ3+a1ϖ−k1b0−ϖ2sinϖτ1+k1b1ϖcosϖτ1=k2b0−ϖ2sinϖτ2−b1ϖcosϖτ2,a2ϖ2−a0−k1b0−ϖ2cosϖτ1−k1b1ϖsinϖτ1=k2b0−ϖ2cosϖτ2+b1ϖsinϖτ2,which yields to(23)cosϖτ2=Q1b0−ϖ2−p1b1k2b0−ϖ22+k2b12ϖ2≔T1,sinϖτ2=p1b0−ϖ2+Q1b1ϖk2b0−ϖ22+k2b12ϖ2≔T2,where(24)p1=−ϖ3+a1ϖ−k1b0−ϖ2sinϖτ1+k1b1ϖcosϖτ1,and(25)Q1=a2ϖ2−a0−k1b0−ϖ2cosϖτ1−k1b1ϖsinϖτ1.Hence, we have(26)gϖ≔ϖ6+b12−2a1−k22+k12ϖ4+a12−2a0b1+2b0−b12k22−k12ϖ2+a02−b02k22−k12+2k1a0−b1ϖ2b0−ϖ2+k1b1ϖa1ϖ−ϖ3cosϖτ1+2k1b1ϖa0−b1ϖ2−k1a1ϖ−ϖ3b0−ϖ2sinϖτ1=0,with g0=a02−b02k22−k12+2k1a0−b1ϖ2b0−ϖ2+k1b1ϖa1ϖ−ϖ3 and g+∞=+∞.It can be easily known that equation (26) has at most N positive roots, denoted by ϖkk=1,2,…,N. From (22), we have(27)τ2ki=1ϖkarccosT1+2iπ,T2≥0,1ϖk−arccosT1+2i+1π,T2<0,i=0,1,2,….Denote(28)τ20=τ2k00=mink∈1,2,…,Nτ2k0,ϖ0=ϖk0.Letλτ2=ατ2+iϖτ2 be the root of (5) satisfying ατ2kj=0 and ϖτ2kj=ϖk. By computation, we obtain(29)α′τ20=℧RS−TU−2k22ϖ02+2b0k22ϖ02−b12k22ϖ02+RDϖ0−ETϖ0sinϖ0τ20−QD+PEϖ0cosϖ0τ1+τ20+−EQ+PDϖ0sinϖ0τ1+τ20−ERϖ0+DTϖ0cosϖ0τ20+SP+UQcosϖ0τ1+QS−PUsinϖ0τ1−1,where(30)℧=b12k22ϖ04+b02−ϖ022k22ϖ02,P=−k1−b1+b0τ1−τ1ϖ02,Q=−k1−2ϖ0+b1τ1ϖ0,R=−3ϖ02+a1,D=−k1b0−ϖ0,E=−k1b1ϖ0,S=a1ϖ02−ϖ04,T=2b1ϖ0,U=−b1ϖ03+a0ϖ0.To sum up, we have the following theorem.Theorem 4.
Suppose that eitherH1 or H2 is satisfied, and τ1∈I for system (2).(i)
If (26) has no positive roots, then for all τ2≥0, E+∗ is LAS.(ii)
If (26) has positive roots, then E+∗ of is LAS when τ2∈0,τ20. In addition, if α′τ20≠0, then system (2) undergoes Hopf bifurcation at E+∗ when τ2=τ20.Remark 1.
Obviously, there exists a Hopf bifurcation atτ20 when τ1 is fixed in the stable interval I. However, if we choose τ1 in the unstable interval, then there may be no τ2∗ such that when system (2) is unstable in τ2∈0,τ2∗, it is stable in τ2>τ2∗. The result will be discussed in the latter section by using the stability crossing curve method in [52].Remark 2.
For someτ1 and τ2, if (5) has two pairs of purely imaginary roots ±iw1 and ±iw2, all the other roots have negatively real parts. Let w1:w2=l1:l2; then, system (2) undergoes a double Hopf bifurcation (DHB) with the ratio l1:l2. If l1,l2∈ℤ+, then it is called a resonant DHB; otherwise, it is called a nonresonant DHB. Since in system (2) there are several parameters besides τ1 and τ2, the co-dimension 2 bifurcation may occur. An interesting study can be found in [53].


## 2.1. The Caseτ1>0 and τ2=0

In this part, letτ2=0, and choose τ1 as the parameter to study the distribution of the root of (5). Let iω be the root of (5), then ω must satisfy the following equations:(9)−ω3+a1+k2b1ω=k1b0−ω2sinωτ1−b1ωcosωτ1,a2ω2−a0−k2b0−ω2=k1b0−ω2cosωτ1+b1ωsinωτ1.Adding the squares of both sides of (9), it yields to(10)ω6+pω4+qω2+r=0,where(11)p=k2+a22−2a1+k2b1−k12,q=a1+k2b12−2a2+k2a0+b0k2+2b0k1−k12b12,r=a0+k2b02−k12b02.Furthermore, from (9), it can be obtained that(12)cosωτ1=Qb0−ω2−pb1k1b0−ω22+k1b12ω2≔S1,sinωτ1=pb0−ω2+Qb1ωk1b0−ω22+k1b12ω2≔S2,where p=−ω3+a1+k2b1ω and Q=a2ω2−a0−k2b0−ω2.Letz=ω2, then (10) becomes(13)hz≔z3+pz2+qz+r=0.Applying the results in [49], the following conclusions hold.Lemma 2.
(i)
Ifr≥0 and △≤0 hold, then (13) has no positive root(ii)
Ifr<0 holds, then (13) has at least a positive root(iii)
Ifr≥0 and △>0 hold, then (13) has a positive roots iff z1∗>0 and hz1∗≤0, where z1∗=−p+△/3 and △=p2−3q
Without loss of generality, it supposes that (13) has three positive roots, denoted by z1,z2,andz3, respectively. Then, (10) has three positive roots ωk=zkk=1,2,3. Substituting ωk into (9) gives(14)τ1kj=1ωkarccosS1+2jπ,S2≥0,1ωk−arccosS1+2j+1π,S2<0,where k=1,2,3,j=0,1,….
Defineτ10=mink=1,2,3τ1k0. Let λτ1=γτ1+iωτ1 be the root of (5) with τ2=0 satisfying γτ1kj=0 and ωτ1kj=ωk.Lemma 3.
Suppose thath′zk≠0, then dγτ1kj/dτ1≠0 and Signdγτ1kj/dτ1=Signh′zk.Proof.
Letτ2=0, and differentiate both sides of (5) about τ1, and it has(15)dλdτ1−1=3λ2+2b1+k2λ+a1+k2b1eλτ1k1λλ2+b1λ+b0+2λ+b1λλ2+b1λ+b0−τ1λ.
Hence,(16)dγτ1kjdτ1−1=Re3λ2+2b1+k2λ+a1+k2b1eλτ1+k12λ+b1k1λλ2+b1λ+b0−τ1λτ1=τ1kj=1Γ3ωk6+2pωk4+qωk2=zkΓh′zk,where Γ=k12b12ωk4+ωk2−b0ωk2. Since Γ>0 and zk>0, then we have(17)Signdγτ1kjdτ1=Signh′zk.
By3 and applying the Hopf bifurcation theorem in [51], for system (2) it has the following theorem.Theorem 1.
It assumes thatH1 holds:(i)
Ifr>0 and △≤0 hold, then, for all τ1≥0, E+∗ is locally asymptotically stable (LAS)(ii)
If eitherr<0 or r≥0 and △>0, z1∗>0, hz1∗≤0 hold, then for τ1∈0,τ10, E+∗ is LAS(iii)
If all conditions in (ii) andh′zk≠0 hold, then system (2) undergoes Hopf bifurcations at E+∗ when τ1=τ1kj, j=0,1,2,…,k=1,2,3
We know that the condition (H1) guarantees that all roots of (7) have negative real parts. If (H1) is violated, we define(18)A=a2+k1+k2,B=a1+k1+k2b1,C=a0+k1+k2b0.
Letλ=Λ−A/3, then (7) becomes(19)Λ3+p1Λ+q1=0,where p1=B−A2/3 and q1=2A3/27−AB/3+C.
Define(20)Δ1=p133+q122,α=3−q12+Δ1,β=3−q12−Δ1.
Then, from Cardano’s formula, it has the following Theorem.Theorem 2.
(i)
If△1<0, then (19) has three real roots(ii)
IfΔ1>0, then (19) has a real root α+β−A/3 and a pair of conjugate complex roots −α+β/2+A/3±i3α−β/2
Furthermore, we assume that(21)H2Δ1>0,α+β2+A3<0,α+β−A3<0,α−β≠0.Theorem 3.
It assumes thatH2 holds. For system (2), it has the following results.(i)
Ifr>0 and Δ≤0, then, for all τ1≥0, E+∗ of system (2) is unstable.(ii)
If eitherr<0 or r≥0 and Δ>0, z1∗>0, hz1∗≤0 hold, then for τ1∈0,τ10, E+∗ of system (2) is unstable. In addition, if dReλτ10/dτ1<0, then E+∗ is LAS when τ1∈τ10,τ11, where τ11 is the second critical value.(iii)
If all conditions in (ii) andh′zk≠0 hold, then system (2) undergoes Hopf bifurcations at E+∗ when τ1=τ1kj, j=0,1,2,…,k=1,2,3.
From the abovementioned discussion, one can know that the stable switch may exist asτ1 varies for system (2) with τ2=0. Define I as stable interval of τ1.


## 2.2. The Caseτ1∈I and τ2>0

In this part, letτ1∈I, τ2>0, and λ=iϖϖ=ϖτ2>0 be the root of (5), and it has(22)−ϖ3+a1ϖ−k1b0−ϖ2sinϖτ1+k1b1ϖcosϖτ1=k2b0−ϖ2sinϖτ2−b1ϖcosϖτ2,a2ϖ2−a0−k1b0−ϖ2cosϖτ1−k1b1ϖsinϖτ1=k2b0−ϖ2cosϖτ2+b1ϖsinϖτ2,which yields to(23)cosϖτ2=Q1b0−ϖ2−p1b1k2b0−ϖ22+k2b12ϖ2≔T1,sinϖτ2=p1b0−ϖ2+Q1b1ϖk2b0−ϖ22+k2b12ϖ2≔T2,where(24)p1=−ϖ3+a1ϖ−k1b0−ϖ2sinϖτ1+k1b1ϖcosϖτ1,and(25)Q1=a2ϖ2−a0−k1b0−ϖ2cosϖτ1−k1b1ϖsinϖτ1.Hence, we have(26)gϖ≔ϖ6+b12−2a1−k22+k12ϖ4+a12−2a0b1+2b0−b12k22−k12ϖ2+a02−b02k22−k12+2k1a0−b1ϖ2b0−ϖ2+k1b1ϖa1ϖ−ϖ3cosϖτ1+2k1b1ϖa0−b1ϖ2−k1a1ϖ−ϖ3b0−ϖ2sinϖτ1=0,with g0=a02−b02k22−k12+2k1a0−b1ϖ2b0−ϖ2+k1b1ϖa1ϖ−ϖ3 and g+∞=+∞.It can be easily known that equation (26) has at most N positive roots, denoted by ϖkk=1,2,…,N. From (22), we have(27)τ2ki=1ϖkarccosT1+2iπ,T2≥0,1ϖk−arccosT1+2i+1π,T2<0,i=0,1,2,….Denote(28)τ20=τ2k00=mink∈1,2,…,Nτ2k0,ϖ0=ϖk0.Letλτ2=ατ2+iϖτ2 be the root of (5) satisfying ατ2kj=0 and ϖτ2kj=ϖk. By computation, we obtain(29)α′τ20=℧RS−TU−2k22ϖ02+2b0k22ϖ02−b12k22ϖ02+RDϖ0−ETϖ0sinϖ0τ20−QD+PEϖ0cosϖ0τ1+τ20+−EQ+PDϖ0sinϖ0τ1+τ20−ERϖ0+DTϖ0cosϖ0τ20+SP+UQcosϖ0τ1+QS−PUsinϖ0τ1−1,where(30)℧=b12k22ϖ04+b02−ϖ022k22ϖ02,P=−k1−b1+b0τ1−τ1ϖ02,Q=−k1−2ϖ0+b1τ1ϖ0,R=−3ϖ02+a1,D=−k1b0−ϖ0,E=−k1b1ϖ0,S=a1ϖ02−ϖ04,T=2b1ϖ0,U=−b1ϖ03+a0ϖ0.To sum up, we have the following theorem.Theorem 4.
Suppose that eitherH1 or H2 is satisfied, and τ1∈I for system (2).(i)
If (26) has no positive roots, then for all τ2≥0, E+∗ is LAS.(ii)
If (26) has positive roots, then E+∗ of is LAS when τ2∈0,τ20. In addition, if α′τ20≠0, then system (2) undergoes Hopf bifurcation at E+∗ when τ2=τ20.Remark 1.
Obviously, there exists a Hopf bifurcation atτ20 when τ1 is fixed in the stable interval I. However, if we choose τ1 in the unstable interval, then there may be no τ2∗ such that when system (2) is unstable in τ2∈0,τ2∗, it is stable in τ2>τ2∗. The result will be discussed in the latter section by using the stability crossing curve method in [52].Remark 2.
For someτ1 and τ2, if (5) has two pairs of purely imaginary roots ±iw1 and ±iw2, all the other roots have negatively real parts. Let w1:w2=l1:l2; then, system (2) undergoes a double Hopf bifurcation (DHB) with the ratio l1:l2. If l1,l2∈ℤ+, then it is called a resonant DHB; otherwise, it is called a nonresonant DHB. Since in system (2) there are several parameters besides τ1 and τ2, the co-dimension 2 bifurcation may occur. An interesting study can be found in [53].


## 3. Property of Hopf Bifurcation atE+∗

In Section3, we have obtained some sufficient conditions to guarantee that the Hopf bifurcation occurs in system (2) at E+∗ when τ2=τ20. In this section, we assume that Theorem 4 (ii) is satisfied to establish the explicit formula for the property of Hopf bifurcation at τ2=τ20 using the method proposed by Hassard et al. [54].For convenience, we assumeτ1>τ2 and the phase space C=C−τ1,0,ℝ3. Let τ¯2=τ20+ϑ, ϑ∈ℝ and dropping “–”. Then, system (2) occurs Hopf bifurcation at ϑ=0. System (2) can be transformed into the following system:(31)U˙t=LϑUt+fϑ,Ut,where Utθ=Ut+θ∈C, and Lϑ:C⟶ℝ3,f:ℝ×C⟶ℝ3 are given, respectively, by(32)Lϑφ=A1φ0+B1φ−τ1+B2φ−τ20,where(33)A1=k1+k2+1cκ1−2κ−b0−10−c,B1=−k100000000,B2=−k200000000,fϑ,φ=φ10φ20−φ1200,where φ=φ1,φ2,φ3T∈C.By the Riesz representation theorem, forΘ∈−τ1,0, there exists a bounded variation function ζΘ,ϑ such that(34)Lϑφ=∫−τ10dζΘ,ϑφΘ.In fact, one may choose(35)ζΘ,ϑ=0,Θ=−τ1,B1,Θ∈−τ1,−τ20,B1+B2,Θ∈−τ20,0,A1+B1+B2,Θ=0.Forφ∈C1−τ1,0,R3, define(36)Aϑφ=∫−τ10dζs,ϑφs,Θ=0,φ˙Θ,Θ∈−τ1,0,ℝϑφ=fϑ,φ,Θ=0,0,Θ∈−τ1,0.ForUt=Ut+θ∈C1, it has dUt/dΘ=dUt/dt. Then, system (31) can be rewritten as(37)U˙t=AϑUt+ℝϑUt,where Utθ=Ut+θ.Forα1∈C−τ1,0,ℝ3 and ψ,α2∈C10,τ1,ℝ3∗, define(38)A∗ψs=∫−τ10dζTt,0ψ−t,s=0,−ψ˙s,s∈0,τ1,and the inner product(39)α1,α2=α¯10α20−∫−τ10∫η=0Θα¯1η−ΘdζΘα2ηdη,where ζΘ=ζΘ,0. By direct computations, we obtain that qΘ=1,ν,ςTeiϖ0Θ is an eigenvector of A corresponding to the eigenvalue iϖ0, and q∗η=D¯1,ν∗,ς∗eiϖ0η is an eigenvector of A∗ corresponding to the eigenvalue −iϖ0. Furthermore, it has that(40)q∗η,qΘ=1,q∗η,q¯Θ=0,where(41)ν=−2κb+iϖ0,ς=−1c+iϖ0,ν∗=κb−iϖ0,ς∗=1c−iϖ0,D=1+ν∗ν¯∗+ς∗ς∗¯−τ1k1e−iϖ0τ1−τ20k2e−iϖ0τ20−1.LetUt be the solution of system (31) when ϑ=0. Define Zt=q∗,Ut; then,(42)Z˙t=iϖ0Zt+q¯∗0f^Z,Z¯,where(43)f^=f0,WZ,Z¯+2ReZq,WZ,Z¯=Ut−2ReZq,WZ,Z¯=W20Z22+W11ZZ¯+W02Z¯22+⋯.Rewriting (42) as(44)U˙t=iϖ0Zt+gZ,Z¯,where(45)gZ,Z¯=g20Z22+g11ZZ¯+g02Z¯22+g21Z2Z¯2+⋯.Furthermore,(46)W˙=AW−2Req¯∗0f^qΘ,Θ∈−τ,0,AW−2Req¯∗0f^qΘ+f^,Θ=0,=defAW+ℍZ,Z¯,Θ,where(47)ℍZ,Z¯,Θ=ℍ20ΘZ22+ℍ11ΘZZ¯+ℍ02ΘZ¯22+⋯.Notice that(48)U1t=Z+Z¯+W2010Z22+W1110ZZ¯+⋯,U2t=νZ+ν¯Z¯+W2020Z22+W1120ZZ¯+⋯.Hence, we can obtain the following important quantities:(49)g20=2Dν−ν¯∗,g11=Dν+ν¯−2ν¯∗,g02=2Dν¯−ν¯∗,g21=2DW1120+12W2020+W201012ν¯−ν¯∗+W1110ν−2ν¯∗,where(50)W20Θ=ig20ϖ0q0eiϖ0Θ+ig¯023ϖ0q¯0e−iϖ0Θ+E1e2iϖ0Θ,W11Θ=−ig11ϖ0q0eiϖ0Θ+ig¯11ϖ0q¯0e−iϖ0Θ+E2,E1=G−κ−12κ2iϖ0+b0102iϖ0+c−1×ν−10,E2=1cκ1−2κ−b0−10−c−1×−ν+ν¯20,where G=2iϖ0−k1+k2+1/c+k1e−2iϖ0τ1+k2e−2iϖτ20.SubstitutingE1 and E2 into W20Θ and W11Θ, respectively, furthermore, g21 can be computed. Thus, it can obtain the following quantities:(51)C10=i2ϖ0g20g11−2g112−g0223+g212,ℰ2=−ReC10Reλ′τ20,T2=−ImC10+ℰ2Imλ′τ20ϖ0,ℬ2=2ReC10.Hence, we have the following result.Theorem 5.
Hopf bifurcation is supercritical (subcritical) ifℰ2>0<0. The bifurcation periodic solutions are orbitally stable (unstable) if ℬ2<0>0. The period increase (decrease) if T2>0<0.


## 4. Crossing Curve Method

The results in Theorem4 clearly show that the stability of system (2) changes depending on the parameters of system. However, the τ1,τ2 plane analysis results for bifurcation generation are not obtained by this method in Section 2. Gu et al. [52] gave an effective approach to separate the stable and unstable regions in the τ1,τ2 plane by using the stability crossing curves. In this part, we carry out the method. on the basis of equation (5), and it can define the following polynomials about λ:(52)p0λ=λ3+a2λ2+a1λ+a0,p1λ=k1λ2+b1λ+b0,p2λ=k2k1p1λ,satisfying(53)idegp0λ≥maxdegp1λ,degp2λiip00+p10+p20≠0iiiThe polynomialsp0λ,p1λandp2λdo not have any common zerosivlimλ⟶∞p1λp0λ+p2λp0λ<1The following discussions will follow the continuity of the zeros with respect to the delay parameters as stated in the following lemma [52].Lemma 4.
As the delaysτ1,τ2 continuously vary within ℝ+2, the number of zeros (counting multiplicity) of Δλ,τ1,τ2 on ℂ+ can change only if a zero appears on or across the imaginary axis.
The characteristic equation (5) has the same zeros with the zeros of(54)Δλ,τ1,τ2=1+δ1λe−λτ1+δ2λe−λτ2=0,where δsλ=psλ/p0λ, s=1,2. Therefore, in general, we may obtain all the crossing points and directions of crossing from the solutions of Δλ,τ1,τ2=0 instead of ∇λ,τ1,τ2=0. Now, based on the procedure proposed by [52], the procedure is comprised of the following steps.
The first step is to determine the crossing setΩ of ω that satisfies the feasibility condition so that the purely imaginary root exists, and geometrically, the vectors that satisfy (54) form a triangle (see Figure 2).
From Figure2, the crossing set Ω can be represented as(55)L1ω=δ1iω+δ2iω≥1,(56)L2ω=δ1iω−δ2iω≤1.
The second step is to determine the inner anglesθ1,θ2∈0,π of the triangle in Figure 2. From the cosine law, it has(57)cosθ1=1+δ1iω2−δ2iω22δ1iω,cosθ2=1+δ2iω2−δ1iω22δ2iω.
For anyω∈Ω, one can obtain τ1,τ2 from (54) as follows:(58)τ1u±ω=1ωargδ1iω±θ1+2u−1π≥0,u=u0±,u0±+1,u0±+2,…,and(59)τ2b±ω=1ωargδ2iω∓θ2+2b−1π≥0,b=b0±,b0±+1,b0±+2,…,where u0± and b0± are the smallest integers so that the right sides of (58) and (59) are nonnegative.
Let(60)Su,b±=τ1u±ω,τ2b±ω=1ωargδ1iω+2u−1π±θ1,1ωargδ2iω+2b−1π∓θ2,then(61)Tω=∪u≥u0+,b≥b0+Su,b+∪∪u≥u0−,b≥b0−Su,b−,which is the set of all τ1,τ2 such that Δλ,τ1τ2 has a zero at λ=iω.
T=Tω:ω∈Ω identifies the stability crossing curves in τ1,τ2 plane, and the crossing set Ω is composed by a finite number of intervals with finite length. Let these intervals be Ωk, k=1,2,…,N, arranged in such an order that the left endpoint of Ωk increases with increasing k. Then, Ω=∪k=1NΩk, and the left endpoints of the intervals ωkl and the right endpoints ωkr must only satisfy one of the three equations: L1ω=1 and L2ω=±1.
Let(62)Tu,b,k±=∪ω∈ΩkSu,b±,Tk=∪u=−∞+∞∪b=−∞+∞Tu,b,k+∪Tu,b,k−∩R+2.
Then,T=∪k=1NTk.
Hence, we can divide these endpoints into three types according to the conditions satisfied by the equationωkl or ωkr. If ω1l=0, then Ω1 may have a special type. As stated by [52], the possible shapes of Tk must belong to one of the following three types:(i)
A series of closed curves.(ii)
A series of spiral-like curves oriented along horizontally, vertically, or diagonally.(iii)
A series of open-ended curves whose ends approach∞.
If the left endpoint ofΩk is of Type l and its right endpoint is of Type r, we call an interval Ωk is of Type lr. There are a total of 12 possible types, whereType 1:L2ω=1 is satisfied. Su,b,k+ links Su,b−1,k− at the end.Type 2:L2ω=−1 is satisfied. Su,b,k+ links Su+1,b,k− at the end.Type 3:L1ω=1 is satisfied. Su,b,k+ links Su,b,k− at the end.Type 0:ωkl=0. As ω⟶0, Su,b,k+ and Su,b,k− approach ∞.
In 12 possible types, Type 11, Type 22, and Type 33 form a series of closed curves. Type 12 and Type 21, Type 13 and Type 31, and Type 23, and Type 32 form series spiral-like curves oriented along diagonally, vertically, and horizontally, respectively. Type 01, Type 02, and Type 03 form a series of open-ended curves.
Next, to determine the existence of Hopf bifurcation, we consider the direction of the root of (5) through the imaginary axis by the method given in [52]. By (54) and the implicit function theorem, τ1 and τ2 can be expressed as the function of λ=iω. As λ moves along the imaginary axis, τ1,τ2=τ1u±ω,τ2b±ω moves along Tk. For a fixed ω∈Ωk, let(63)Reiλ∂Δλ,τ1,τ2∂λλ=iω=ℜ0,Imiλ∂Δλ,τ1,τ2∂λλ=iω=ℑ0,−Re1λ∂Δλ,τ1,τ2∂τsλ=iω=ℜs,−Im1λ∂Δλ,τ1,τ2∂τsλ=iω=ℑs,where s=1,2.
The direction in which theω increases is called the positive direction of the curve, which is reversed when the curve passes the point corresponding to the Ωk endpoint. When we move in the positive direction of the curve, we also call the region on the left-hand side the region on the left. The following results come from [52].Figure 2
Triangle formed by 1,δ1iω, and δ2iω.Lemma 5.
Letω∈ωkl,ωkr and τ1,τ2∈Tk so that iω is a simple root of (5) and for any ω′≠ω, Δiω′,τ1,τ2≠0. Then, as τ1,τ2 moves from the right-side region to the left-side region of the corresponding curve in Tk, a pair of roots of (54) cross the imaginary axis to the right side if ℜ2ℑ1−ℜ1ℑ2>0. If the inequality is reversed, the crossing direction is opposite.Theorem 6.
Letω,τ1,andτ2 satisfy the conditions in Lemma 5. Then, when τ1,τ2 crosses the curve along the direction ℓ1,ℓ2, a pair of roots of (54) cross the imaginary axis to the right side if(64)ℓ1ℜ0ℑ1−ℜ1ℑ0+ℓ2ℜ0ℑ2−ℜ2ℑ0>0.
If the inequality is reversed, the crossing direction is opposite.


## 5. Numerical Simulations

In this part, we will carry out some numerical simulations by using Matlab Microsoft to confirm the theoretical analyses.Firstly, as an example, we investigate the following system:(65)x˙t=y−0.9x+z−xt−xt−τ1−2xt−xt−τ2,y˙t=1−0.2y−x2,z˙t=−x−1.2z,and the initial functions are φ1θ≡2,φ2θ≡3, and φ3θ≡2. With these parameters, condition (H2) holds. When τ2=0, (10) has two positive roots ω1≐0.9752 and ω2≐1.9997. Substituting them into (14) gives, respectively,(66)τ11j=0.3795+6.4430j,τ12i=2.5811+3.1421i,j,i=0,1,2,….Furthermore,dReλτ11j/dτ1<0anddReλτ12i/dτ1>0. By Theorem 4, E+∗ is unstable when τ1∈0,0.3795∪2.5811,+∞, and LAS when τ1∈0.3795,2.5811. The numerical simulation results are shown in Figures 3–5.Figure 3
E+∗ is unstable, and chaos phenomenon still exists for system (65) when τ1=0.1∈0,0.3795. (a) Time series of the solutions. (b) Three-dimensional phase diagram.
(a)(b)Figure 4
E+∗ is stable and the chaos phenomenon disappears for system (65) when τ1=0.8∈0.3795,2.5811. (a) Time series of the solutions. (b) Three-dimensional phase diagram.
(a)(b)Figure 5
E+∗ is unstable and a stable periodic solution exists for system (65) when τ1=2.77>2.5811. (a) Time series of the solutions. (b) Three-dimensional phase diagram.
(a)(b)Fixτ1=2.2∈0.3795,2.5811, and it computes τ20≐2.4692. By Theorem 4, we know that E+∗ is LAS for τ2∈0,2.4692. Choosing τ2=1, E+∗ is stable (see Figure 6). Furthermore, by Section 3, it has C10=−10.1592+0.7794i,ℬ2<0 and ℰ2>0 when τ2=2.4692, and the bifurcating periodic solution is stable, which is illustrated in Figure 7.Figure 6
E+∗ is stable for system (65) with τ1=2.2 and τ2=1. (a) Time series of the solutions. (b) Three-dimensional phase diagram.
(a)(b)Figure 7
E+∗ is unstable and there exists a stable periodic solution for system (65), where τ1=2.2 and τ2=τ20. (a) Time series of the solutions. (b) Three-dimensional phase diagram.
(a)(b)Next, one gives some examples for the crossing curve method using Matlab Microsoft. Firstly, it still chooses the parameters in system (65). It can obtain the crossing set based on the equation of L1ω and L2ω. In equations (58) and (59), we regard τ1 and τ2 as the function of ω, by drawing the parametric equation curves in τ1,τ2 plane, and it obtains the crossing curves. The crossing set has only a interval Ω1 and ω1l=0.9893andω1r=2.6715, satisfying L1ω=1 with ω=ω1l and ω=ω1r (see Figure 8(a)). So, the interval Ω1 is Type 33 and the crossing curves form a series of closed curves (see Figures 8(b) and 8(c)). Firstly, it chooses τ1=10>2.5811 and τ2=0, and it can obtain that E+∗ is unstable (see Figure 9). In the following, it chooses, respectively, τ2=1,3.35,3.7 for fixed τ1=10, and it can find that E+∗ is stable when τ2=1 (see Figure 10) and τ2=3.7 (see Figure 11), unstable when τ2=3.35, and there exists a stable periodic solution (see Figure 12). Furthermore, it can fix a=0.9,b=0.2,and c=1.2, and let k1andk2 change, and the crossing curves can produce different shapes (see Figures 13–15). When k1=1 and k2=−5, the crossing sets have two intervals Ω1=0.3696,0.9733 and Ω2=2.9688,5.4793. Here, ω1r and ω2l satisfy L2ω=−1 while ω1l and ω2r satisfy L1ω=1 (see Figure 13(a)). So, the interval Ω1 belongs to Type 32 and the interval Ω2 belongs to Type 23. The crossing curves are spatial-like curves, as shown in Figures 13(b) and 13(c). Choosing k1=−4 and k2=2, the crossing sets include two intervals Ω1=0.2097,1.0151 and Ω2=2.3341,6.2418 (see Figure 14(a)). The crossing curves belongs to Type 31 and Type 13 with the spiral-like shape (see Figure 14(b)). Here, the stability crossing curves in Ω1=0.2097,1.0151 belonging to type 31 are not drawn. If k1=3 and k2=1, the crossing set includes a interval Ω1=0,0.7486. Since ω1l=0 and L1ω1r=0 (see Figure 15(a)), the stability crossing curves belongs to Type 03 with the open-ended shapes (see Figure 15(b)). These show that the changes of the feedback strengths k1 and k2 can alter the stable region of the system in τ1,τ2 plane, and it has an important effect on the stability of the financial system.Figure 8
The feedback strength sets:k1=−1andk2=−2. The arrow directions point to an unstable region. (a) The crossing set Ω = (0.9893, 2.6715). (b) Stability crossing curves τ1 in τ1,τ2 plane. (c) The zoom-in T 1 plot.
(a)(b)(c)Figure 9
E+∗ for system (2) is unstable and there exists a stable periodic solution when τ1=10andτ2=0 with feedback strength sets: k1=−1andk2=0. (a) Time series of the solutions. (b) Three-dimensional phase diagram.
(a)(b)Figure 10
E+∗ for system (2) is stable when τ1=10 and τ2=1 with feedback strength sets: k1=−1andk2=−2. (a) Time series of the solutions. (b) Three-dimensional phase diagram.
(a)(b)Figure 11
E+∗ for system (2) is stable when τ1=10 and τ2=3.7 with feedback strength sets: k1=−1andk2=−2. (a) Time series of the solutions. (b) Three-dimensional phase diagram.
(a)(b)Figure 12
E∗ for system (2) is unstable and there exists a stable periodic solution when τ1=10 and τ2=3.35 with feedback strength sets: k1=−1andk2=−2. (a) Time series of the solutions. (b) Three-dimensional phase diagram.
(a)(b)Figure 13
The feedback strength sets:k1=1andk2=−5. (a) The crossing set Ω = 0.3696,0.9733∪2.9688,5.4793. (b) Stability crossing curves τ1 in τ1,τ2 plane. Stability crossing curves are spiral-like curves along the horizontal axis belonging to type 32 for the crossing set Ω1 = (0.3696, 0.9733). (c) Stability crossing curves τ2 in τ1,τ2 plane. Stability crossing curves are spiral-like curves along the horizontal axis belonging to type 23 for the crossing set Ω2 = (2.9688; 5.4793).
(a)(b)(c)Figure 14
The feedback strength sets:k1=−4andk2=2. (a) The crossing set Ω = 0.2097,1.0151∪2.3341,6.2418. (b) Stability crossing curves τ2 in τ1,τ2 plane. Stability crossing curves are spiral-like curves along the vertical axis belonging to type 13 for the crossing set Ω2 = (2.3341, 6.2418).
(a)(b)Figure 15
The feedback strength sets:k1=3andk2=1. (a) The crossing set Ω = (0, 0.7486). (b) Stability crossing curves τ1 in τ1,τ2 plane. Stability crossing curves are open-ended curves belonging to type 03.
(a)(b)In addition, for the following two systems:(67)x˙t=y−ax+z,y˙t=1−by−x2+k1yt−yt−τ1+k2yt−yt−τ2,z˙t=−x−cz,and(68)x˙t=y−ax+z,y˙t=1−by−x2,z˙t=−x−cz+k1zt−zt−τ1+k2zt−zt−τ2,that is, delayed feedback terms appear on the investment demand or the price index, respectively. Systems (67) and (68) can be investigated as system (2) and can also obtain similar results to system (2).The time-delay feedback controllerke−dτut−ut−τ with delay correlation coefficients can also be designed to control system (1) which can modify the bifurcation characteristics of a nonlinear system to obtain some specific dynamical behaviors. Note that the strength of feedback control is in the form of ke−dτ, and the function decreases exponentially with delay τ. This means that the feedback effects of past states diminish over time. Hence, it can carry out the feedback with time-delay correlation coefficients in system (1). The systems with coefficient dependent delay increase the complexity of the analysis and are challenging, especially those with two time delays. The research is set aside for future consideration.


## 6. Conclusion

This paper analyzes a class of chaotic financial systems with two feedback delays. System (1) exists in chaos under some parameters. The purpose of this study is to control the chaos of the system. For controlling chaos, we improve the DFC method and introduce the double-delay feedback control method in system (1). We introduce the control term in the equation of the interest rate. The system may exist in three equilibria, and we choose one of these equilibria as the research target. It finds that the single delay feedback control can make the system stable and produce the stable switches, i.e., when τ1 changes with τ2=0, system (2) exists stable switches and chaos may disappear. Furthermore, fixing τ1 in a stability interval and taking the delay τ2 as a parameter, proves the existence of the first critical value τ2. At this critical value, the equilibrium loses stability and Hopf bifurcation occurs. The properties of Hopf bifurcation are also studied by using central manifold theory and normal form method for determining the direction of Hopf bifurcation and the stability of bifurcating periodic solution. The abovementioned results are obtained under the condition fixed τ1 in a stability interval; however, if we choose the τ1 in the unstable interval, then there may exist no the critical value τ20 such that τ20 is the first Hopf bifurcation value. Hence, for obtaining the complete result separating the stable and unstable regions in the τ1,τ2, using the stability crossing curve methods in [52], it obtains the curve sets in which Hopf bifurcation occurs in τ1,τ2 plane for fixed a,b,andc. By numerical simulations, it can find that the different shape crossing curves, and crossing sets can produce by changing k1 and k2. Theoretical analysis and numerical simulation results show that, for chaotic financial systems, chaotic oscillation can be controlled by delays. In other words, the multiple delay financial system we study has chaotic oscillations when τ1=τ2=0. When the delay increases, the chaos disappears, the equilibrium point gains stability or the system appears periodic oscillation, and the periodic solution is generated by the Hopf bifurcation. The DDFC method can control the chaotic behavior of the system more effectively than the DFC method. When τ1 cannot change the chaos behavior of system (2), system (2) can be stabilized by varying τ2 value. These show that the effectiveness of the DDFC method.



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