# Hopf Bifurcation and Chaos of a Delayed Finance System

Authors: Xuebing Zhang; Honglan Zhu
Journal: Complexity
 (2019)

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## Abstract

In this paper, a finance system with delay is considered. By analyzing the corresponding characteristic equations, the local stability of equilibrium is established. The existence of Hopf bifurcations at the equilibrium is also discussed. Furthermore, formulas for determining the direction of Hopf bifurcation and the stability of the bifurcating periodic solutions are derived by applying the normal form method and center manifold theorem. Finally, numerical simulation results are presented to validate the theoretical analysis. Numerical simulation results show that delay can lead a stable system into a chaotic state.


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## Body

## 1. Introduction

Ever since economist Stutzer first revealed the chaotic phenomena in an economic system in 1980, chaotic dynamics which supports an endogenous explanation of the complexity observed in economic series has become a hot topic, and many economic models have been proposed, e.g., Goodwin’s nonlinear accelerator model [1, 2], the van der Pol model on business cycle [3–5], the IS-LM model [6, 7], and nonlinear dynamical model on finance system [8–11]. In [8, 9], Ma and Chen proposed a simplified financial model as follows:
(1)ẋ=y−ax+z,ẏ=1−by−x2,ż=−x−cz,where x is the interest rate, y is the investment demand, z is the price index, a>0 denotes saving amount, b>0 denotes cost per investment, and c>0 denotes elasticity of demand of commercial markets. The variation of x is not only influenced by the surplus between investment and saving but also structurally adjusted by the price. The changing rate of y is proportional to the rate of investment and inversely proportional to the cost of investment and interest rate. The variation of z is influenced by the contradiction between supply and demand in commercial markets and affected by the inflation rates. The authors studied the focus on bifurcation and topological horseshoe of chaotic financial system (1). Some delay feedback control strategies [12–15] have also been considered for system (1).It is well known that delays are extensively encountered in many fields such as biology [16–18], chemistry [19, 20], and engineering [21–23]. Also, delay is inevitable in economic activities. For example, changes in the money supply do not cause immediate changes in the economy; there is always a lag period. The production cycle has both long and short phases. Price change always has a delay. Therefore, delay differential equations (DDEs) support a realistic economic mathematical modeling than ordinary differential equations (ODEs) [6, 7].In [24], Wang et al. proposed a delayed fractional order financial system as follows:
(2)Dtα1x=yt−τ−ax+z,Dtα2y=1−by−x2t−τ,Dtα3z=−xt−τ−cz,where τ≥0 is the time delay. The authors studied its dynamic behaviors, such as single-periodic, multiple-periodic, and chaotic motions.Based on [24], Chen et al. [25] studied the following delayed financial system:
(3)ẋ=yt−τ−ax+z,ẏ=1−by−x2t−τ,ż=−xt−τ−cz.The authors have studied the asymptotic stability and Hopf bifurcations of the unique equilibrium, and the direction of Hopf bifurcation and the stability of the bifurcating periodic solutions were also considered.According to the above discussions, we consider a delayed finance system as follows:(4)ẋ=y−ax+zt−τ,ẏ=1−by−x2,ż=−x−cz,where τ denotes price change delay, for price change does not immediately affect the interest rate, and it often has a lag period.The main purpose of this paper is to investigate the stability and Hopf bifurcation for system (4) with delay τ as the bifurcation parameter.The structure of this paper is arranged as follows. In Section2, we study the local stability and the existence of Hopf bifurcation. In Section 3, we give the formula determining the direction of Hopf bifurcation and the stability of bifurcating periodic solutions. Finally, to support our theoretical predictions, some numerical simulations are given which support the analysis of Sections 2–3.


## 2. Stability and Hopf Bifurcation

### 2.1. The Existence of Equilibria

In this section, we consider the stability and Hopf bifurcation of the equilibria of system (4). First, we find all possible equilibria of system (4). We make the following hypothesis:
(H1)
(5)c−abc−b>0.According to system (4), equilibria should satisfy
(6)y−ax+z=0,1−by−x2=0,−x−cz=0.Obviously, system (4) has an equilibrium P0=0,1/b,0. For other equilibria, solving for the second and third equations of (6), we have
(7)y=1−x2b,z=−xc.Substitute (7) into the first equation of (6), we obtain
(8)x=±c−abc−bc.So, we have following results.Lemma 1.
If (H1) holds, then system (4) has two other equilibria P1 and P2, where
(9)P1=c−abc−bc,ac+1c,−1cc−abc−bc,P2=−c−abc−bc,ac+1c,1cc−abc−bc.In the following, we consider the stability of the equilibria of system (4) by analyzing the corresponding characteristic equations. Assume that P∗=x∗,y∗,z∗ denotes an arbitrary equilibrium of system (4), then let x¯=x−x∗, y¯=y−y∗, and z¯=z−z∗ and drop the bars for the simplicity of notations. Then by linearizing system (4) around P∗, we have
(10)ẋ=y∗−ax+x∗y+zt−τ,ẏ=−2x∗x−by,ż=−x−cz.The characteristic equation associated with system (10) is
(11)λ3+b1λ2+b2λ+b3+λ+b4e−λτ=0,where
(12)b1=b+c+a−y∗,b2=a−y∗b+c+bc+2x2,b3=a−y∗bc+2x2c,b4=b.


### 2.2. Stability and Hopf Bifurcation of EquilibriumP0

Obviously, the characteristic equation of system (4) at the equilibrium P0=0,1/b,0 has the following form:
(13)λ+bλ2+a+c−1bλ+a−1bc+e−λτ=0.Clearly,λ=−b is negative; we only need to consider the following equation:
(14)λ2+a+c−1bλ+a−1bc+e−λτ=0.For further discussion, we make following hypotheses:(H2)
(15)1−ab−bc<0,
(H3)
(16)abc−c+b>0,Asτ=0, (14) is equivalent to the following equation:
(17)λ2+a+c−1bλ+a−1bc+1=0.Obviously,λ=0 is not a root of (17).Lemma 2.
If (H2) and (H3) hold, then equilibrium P0 of system (4) is locally asymptotically stable with τ=0.Proof.
Letλ1 and λ2 be two roots of (17). Clearly, if (H2) and (H3) hold, then we have
(18)λ1+λ2=−a−1b+c<0,λ1λ2=a−1bc+1>0.
It means that all the roots of (17) have negative real parts. So, equilibrium P0 of system (4) with τ=0 is locally asymptotically stable.Now we discuss the effect of delayτ on the stability of the equilibrium P0 of system (4). Assume that iωω>0 is a root of (11). Then ω should satisfy the following equation:
(19)−ω2+iωa−1b+c+a−1bc+cosωτ−isinωτ=0,which implies that
(20)−ω2+a−1bc=−cosωτ,ωa−1b+c=sinωτ.From (20), adding the squared terms for both equations yields
(21)ω4+a−1b+c2−2a−1bcω2+a−1b2c2−1=0.Make the following assumptions:(H4)
(22)abc−c−b>0,
(H5)
(23)abc−c−b<0,Theorem 1.
If (H2) and (H4) hold, then the equilibrium P0 of system (4) is locally asymptotically stable for all τ≥0.Proof.
Clearly, if (H4) holds, then we have(24)a−1b+c2−2a−1bc=a−1b2+c2>0,a−1b2c2−1=a−1bc+1a−1bc−1>0,which means that (21) has no positive roots. That is to say, all roots of (14) have negative real parts. Combining with Lemma 2, it thus follows from the Routh-Hurwitz criterion that the equilibrium P0 of system (4) is locally asymptotically stable for all τ≥0.Lemma 3.
If (H5) holds, then (21) has a unique positive root.Proof.
(H5) holds, so we have(25)a−1b+c2−2a−1bc=a−1b2+c2>0,a−1b2c2−1=a−1b+1a−1b−1<0.
Hence, (21) has a unique positive root as follows:
(26)ω0=−a−1/b2−c2+a−1/b2−c22+42.According to Lemma3, (21) has a unique positive root ω0. By (20), we have
(27)cosω0τ=ω02−a−1bc.Thus, if we denote(28)τ0j=1ω0arccosω02−a−1bc+2jπ,j=0,1,2,…,then ±iω0 is a pair of purely imaginary roots of (14) with τ=τ0j. Clearly, sequence τ0jj=0∞ is increasing and
(29)limj→+∞τ0j=+∞.Thus, we can define(30)τ0=τ00=minτ0j.Lemma 4.
Let λτ=ατ±iωτ be the root of (14) near τ=τ0j satisfying ατ0j=0 for ωτ0j=ω0. Then the following transversal condition holds:
(31)α′τ−1τ=τ0j>0,j=0,1,2,….Proof.
Differentiating the two sides of (14) with respect to τ yields
(32)dλdτ2λ+a−1b+c−τe−λτ=λe−λτ.
Hence,(33)dλdτ−1=2λ+a−1/b+c−τe−λτλe−λτ=2eλτ+a−1/b+cλeλτ−τλ.
Substitutingτ0j into the above equation, we obtain
(34)α′τ−1τ=τ0j=Redλdττ=τ0j−1=2cosω0τ0j+a−1/b+csinωτ0jω0.
Sincecosω0τ0j=ω02−a−1/bc and sinω0τ0j=a−1/b+cω0, then we have
(35)α′τ−1τ=τ0j=Redλdττ=τ0j−1=2ω02−2a−1bc+a−1b+c2.
By (26), we have
(36)αn′τ−1=Redλdττ=τ0j−1=a−1b2−c22+4>0.On the basis of Lemmas2–4, we have the following result:Theorem 2.
If (H2), (H3), and (H5) hold, then the following statements are true:
(i)
When τ∈0,τ0, the equilibrium P0 of (4) is asymptotically stable
(ii)
The Hopf bifurcation occurs at τ=τ0. That is, system (4) has a branch of periodic solutions bifurcating from P0 near τ=τ0


### 2.3. Stability and Hopf Bifurcation of EquilibriumP1 and P2

In this section, we consider stability and Hopf bifurcation of equilibriaP1 and P2. At the equilibria P1 and P2, the characteristic (11) takes the following form:
(37)λ3+b1λ2+b2λ+b3+λ+b4e−λτ=0,where
(38)b1=b+c−1c,b2=bc+21−ba+1c−bc−1,b3=21−ba+1cc−b,b4=b.Asτ=0, (37) becomes
(39)λ3+b1λ2+b2+1λ+b3+b4=0.Make the following assumptions:(H6)
(40)1c<minb,c,
(H7)
(41)c−abc−2b<0.Lemma 5.
Based on Lemma 1, if (H6) holds, then equilibria P1 and P2 are both locally asymptotically stable with τ=0.Proof.
As (H1) and (H6) hold, we have(42)b1=b+c−1c>0,b3=2c−2abc−2b>0,b1b2+1−b3=bb−1cc−1c+2x∗2+bcc−1c>0.
By the Routh-Hurwitz criteria, all the roots of (39) have negative real parts. Therefore, P1 and P2 are both locally asymptotically stable with τ=0.Now we discuss the effect of delayτ on the stability of the equilibria P1 and P2 of system (4). Assume that iωω>0 is a root of (37). Then ω should satisfy the following equation:
(43)−iω3−b1ω2+ib2ω+b3+iω+b4cosωτ−isinωτ=0,which implies that
(44)−ω3+b2ω=b4sinωτ−ωcosωτ,−b1ω2+b3=−ωsinωτ−b4cosωτ.From (44), adding up the squares of both equations, we have
(45)ω6+b12−2b2ω4+b22−2b1b3−1ω2+b32−b42=0.Letz=ω2, then (45) can be rewritten into the following form:
(46)z3+b12−2b2z2+b22−2b1b3−1z+b32−b42=0.Denote(47)hz=z3+R0z2+Q0z+V0.Lemma 6.
If (H7) holds, then (46) has at least a root.Proof.
Obviously,(48)b32−b42=b3−b4b3+b4=2c−abc−2bc−abc−b<0.
Therefore, (46) has at least a positive root.According to Lemma6, (46) has a positive root, denoted by z0, and thus, (45) has a positive root ω0=z0. By (44), we have
(49)cosω0τ=ω03−b2ω0ω0−−ω02b1+b3b4ω02+b42.Thus, if we denote(50)τ0j=1ω0arccosω03−b2ω0ω0−−ω02b1+b3b4ω02+b42+2jπ,j=0,1,2,…,then ±iω0 is a pair of purely imaginary roots of (21) with τ=τ0j. Clearly, sequence τ0jj=0∞ is increasing and
(51)limj→+∞τ0j=+∞.Thus, we can define(52)τ0=τ00=minτ0j.Lemma 7.
Let λτ=ατ±iωτ be the root of (37) near τ=τ0j satisfying ατ0j=0, ωτ0j=ω0. Suppose that h′z0≠0, where hz is defined by (47). Then the following transversal condition holds:
(53)dReλτdττ=τ0j≠0,and the sign of dReλτ/dτ∣τ=τ0j is consistent with that of h′z0.Proof.
Denote(54)Rλ=λ3+b1λ2+b2λ+b3,Qλ=λ+b4.
Then (37) can be written as
(55)Rλ+Qλe−λτ=0,and (45) can be transformed into the following form:
(56)RiωR¯iω−QiωQ¯iω=0.
Thus, together with (46) and (47), we have
(57)hω2=RiωR¯iω−QiωQ¯iω.
Differentiating both sides of (57) with respect to ω, we obtain
(58)2ωh′ω2=iR′iωR¯iω+RiωR′¯iω−Q′iωQ¯iω+QiωQ′¯iω.
Ifiω0 is not simple, then ω0 must satisfy
(59)ddλRλ+Qλe−λτ0λ=iω0=0,that is, ω0 must satisfy
(60)R′iω+Q′iωe−iω0τ0−τ0Qiω0e−iω0τ0=0.
With (55), we have
(61)τ0=Q′iω0Qiω0−R′iω0Riω0.
Thus, by (56) and (57), we obtain
(62)Imτ0=ImQ′iω0Qiω0−R′iω0Riω0=ImQ′iω0Q¯iω0Qiω0Q¯iω0−R′iω0R¯iω0Riω0R¯iω0=ImQ′iω0Q¯iω0−R′iω0R¯iω0Riω0R¯iω0=−iQ′iω0Q¯iω0−R′iω0R¯iω0+Q′¯iω0Qiω0+R′¯iω0Riω02Riω0R¯iω0=ω0h′ω02Riω02.
Sinceτ0 is real, i.e., Imτ0=0, we have h′ω02=0. We get a contradiction to the condition h′ω02≠0. This proves the first conclusion. Differentiating both sides of (55) with respect to τ, we obtain
(63)R′λ+Q′λe−λτ−τQλe−λτdλdτ−λQλe−λτ=0,which implies
(64)dλdτ=λQλR′λeλτ+Q′λ−τQλ=λQλR′¯λeλτ+Q′¯λ−τQ¯λR′λeλτ+Q′λ−τQλ2=λ−RλR′¯λeλτ+QλQ′¯λ−τQλ2R′λeλτ+Q′λ−τQλ2.
It follows together with (58) that
(65)dReλτdττ=τ0,λ=iω0=Reλ−RλR′¯λeλτ+QλQ′¯λ−τQλ2τ=τ0,λ=iω0R′λeλτ+Q′λ−τQλ2=iω0−Riω0R′¯iω0+Qiω0Q′¯iω0+R′iω0R¯iω0−Q′iω0Q¯iω0R′λeλτ+Q′λ−τQλ2=ω02h′ω02R′λeλτ+Q′λ−τQλ2=ω02h′z0R′λeλτ+Q′λ−τQλ2≠0.
Clearly, the sign ofdReλτ/dτ∣τ=τ0 is determined by that of h′z0.On the basis of Lemma1 and Lemma 5–Lemma 7, we have the following result.Theorem 3.
If (H1), (H6), and (H7) hold, and h′z0>0, then the following statements are true:
(i)
When τ∈0,τ0, the equilibria P1 and P2 of system (4) are both locally asymptotically stable
(ii)
The Hopf bifurcation occurs at τ=τ0, i.e., system (4) has a branch of periodic solutions that bifurcates from P1 and P2 near τ=τ0, respectively


## 2.1. The Existence of Equilibria

In this section, we consider the stability and Hopf bifurcation of the equilibria of system (4). First, we find all possible equilibria of system (4). We make the following hypothesis:
(H1)
(5)c−abc−b>0.According to system (4), equilibria should satisfy
(6)y−ax+z=0,1−by−x2=0,−x−cz=0.Obviously, system (4) has an equilibrium P0=0,1/b,0. For other equilibria, solving for the second and third equations of (6), we have
(7)y=1−x2b,z=−xc.Substitute (7) into the first equation of (6), we obtain
(8)x=±c−abc−bc.So, we have following results.Lemma 1.
If (H1) holds, then system (4) has two other equilibria P1 and P2, where
(9)P1=c−abc−bc,ac+1c,−1cc−abc−bc,P2=−c−abc−bc,ac+1c,1cc−abc−bc.In the following, we consider the stability of the equilibria of system (4) by analyzing the corresponding characteristic equations. Assume that P∗=x∗,y∗,z∗ denotes an arbitrary equilibrium of system (4), then let x¯=x−x∗, y¯=y−y∗, and z¯=z−z∗ and drop the bars for the simplicity of notations. Then by linearizing system (4) around P∗, we have
(10)ẋ=y∗−ax+x∗y+zt−τ,ẏ=−2x∗x−by,ż=−x−cz.The characteristic equation associated with system (10) is
(11)λ3+b1λ2+b2λ+b3+λ+b4e−λτ=0,where
(12)b1=b+c+a−y∗,b2=a−y∗b+c+bc+2x2,b3=a−y∗bc+2x2c,b4=b.


## 2.2. Stability and Hopf Bifurcation of EquilibriumP0

Obviously, the characteristic equation of system (4) at the equilibrium P0=0,1/b,0 has the following form:
(13)λ+bλ2+a+c−1bλ+a−1bc+e−λτ=0.Clearly,λ=−b is negative; we only need to consider the following equation:
(14)λ2+a+c−1bλ+a−1bc+e−λτ=0.For further discussion, we make following hypotheses:(H2)
(15)1−ab−bc<0,
(H3)
(16)abc−c+b>0,Asτ=0, (14) is equivalent to the following equation:
(17)λ2+a+c−1bλ+a−1bc+1=0.Obviously,λ=0 is not a root of (17).Lemma 2.
If (H2) and (H3) hold, then equilibrium P0 of system (4) is locally asymptotically stable with τ=0.Proof.
Letλ1 and λ2 be two roots of (17). Clearly, if (H2) and (H3) hold, then we have
(18)λ1+λ2=−a−1b+c<0,λ1λ2=a−1bc+1>0.
It means that all the roots of (17) have negative real parts. So, equilibrium P0 of system (4) with τ=0 is locally asymptotically stable.Now we discuss the effect of delayτ on the stability of the equilibrium P0 of system (4). Assume that iωω>0 is a root of (11). Then ω should satisfy the following equation:
(19)−ω2+iωa−1b+c+a−1bc+cosωτ−isinωτ=0,which implies that
(20)−ω2+a−1bc=−cosωτ,ωa−1b+c=sinωτ.From (20), adding the squared terms for both equations yields
(21)ω4+a−1b+c2−2a−1bcω2+a−1b2c2−1=0.Make the following assumptions:(H4)
(22)abc−c−b>0,
(H5)
(23)abc−c−b<0,Theorem 1.
If (H2) and (H4) hold, then the equilibrium P0 of system (4) is locally asymptotically stable for all τ≥0.Proof.
Clearly, if (H4) holds, then we have(24)a−1b+c2−2a−1bc=a−1b2+c2>0,a−1b2c2−1=a−1bc+1a−1bc−1>0,which means that (21) has no positive roots. That is to say, all roots of (14) have negative real parts. Combining with Lemma 2, it thus follows from the Routh-Hurwitz criterion that the equilibrium P0 of system (4) is locally asymptotically stable for all τ≥0.Lemma 3.
If (H5) holds, then (21) has a unique positive root.Proof.
(H5) holds, so we have(25)a−1b+c2−2a−1bc=a−1b2+c2>0,a−1b2c2−1=a−1b+1a−1b−1<0.
Hence, (21) has a unique positive root as follows:
(26)ω0=−a−1/b2−c2+a−1/b2−c22+42.According to Lemma3, (21) has a unique positive root ω0. By (20), we have
(27)cosω0τ=ω02−a−1bc.Thus, if we denote(28)τ0j=1ω0arccosω02−a−1bc+2jπ,j=0,1,2,…,then ±iω0 is a pair of purely imaginary roots of (14) with τ=τ0j. Clearly, sequence τ0jj=0∞ is increasing and
(29)limj→+∞τ0j=+∞.Thus, we can define(30)τ0=τ00=minτ0j.Lemma 4.
Let λτ=ατ±iωτ be the root of (14) near τ=τ0j satisfying ατ0j=0 for ωτ0j=ω0. Then the following transversal condition holds:
(31)α′τ−1τ=τ0j>0,j=0,1,2,….Proof.
Differentiating the two sides of (14) with respect to τ yields
(32)dλdτ2λ+a−1b+c−τe−λτ=λe−λτ.
Hence,(33)dλdτ−1=2λ+a−1/b+c−τe−λτλe−λτ=2eλτ+a−1/b+cλeλτ−τλ.
Substitutingτ0j into the above equation, we obtain
(34)α′τ−1τ=τ0j=Redλdττ=τ0j−1=2cosω0τ0j+a−1/b+csinωτ0jω0.
Sincecosω0τ0j=ω02−a−1/bc and sinω0τ0j=a−1/b+cω0, then we have
(35)α′τ−1τ=τ0j=Redλdττ=τ0j−1=2ω02−2a−1bc+a−1b+c2.
By (26), we have
(36)αn′τ−1=Redλdττ=τ0j−1=a−1b2−c22+4>0.On the basis of Lemmas2–4, we have the following result:Theorem 2.
If (H2), (H3), and (H5) hold, then the following statements are true:
(i)
When τ∈0,τ0, the equilibrium P0 of (4) is asymptotically stable
(ii)
The Hopf bifurcation occurs at τ=τ0. That is, system (4) has a branch of periodic solutions bifurcating from P0 near τ=τ0


## 2.3. Stability and Hopf Bifurcation of EquilibriumP1 and P2

In this section, we consider stability and Hopf bifurcation of equilibriaP1 and P2. At the equilibria P1 and P2, the characteristic (11) takes the following form:
(37)λ3+b1λ2+b2λ+b3+λ+b4e−λτ=0,where
(38)b1=b+c−1c,b2=bc+21−ba+1c−bc−1,b3=21−ba+1cc−b,b4=b.Asτ=0, (37) becomes
(39)λ3+b1λ2+b2+1λ+b3+b4=0.Make the following assumptions:(H6)
(40)1c<minb,c,
(H7)
(41)c−abc−2b<0.Lemma 5.
Based on Lemma 1, if (H6) holds, then equilibria P1 and P2 are both locally asymptotically stable with τ=0.Proof.
As (H1) and (H6) hold, we have(42)b1=b+c−1c>0,b3=2c−2abc−2b>0,b1b2+1−b3=bb−1cc−1c+2x∗2+bcc−1c>0.
By the Routh-Hurwitz criteria, all the roots of (39) have negative real parts. Therefore, P1 and P2 are both locally asymptotically stable with τ=0.Now we discuss the effect of delayτ on the stability of the equilibria P1 and P2 of system (4). Assume that iωω>0 is a root of (37). Then ω should satisfy the following equation:
(43)−iω3−b1ω2+ib2ω+b3+iω+b4cosωτ−isinωτ=0,which implies that
(44)−ω3+b2ω=b4sinωτ−ωcosωτ,−b1ω2+b3=−ωsinωτ−b4cosωτ.From (44), adding up the squares of both equations, we have
(45)ω6+b12−2b2ω4+b22−2b1b3−1ω2+b32−b42=0.Letz=ω2, then (45) can be rewritten into the following form:
(46)z3+b12−2b2z2+b22−2b1b3−1z+b32−b42=0.Denote(47)hz=z3+R0z2+Q0z+V0.Lemma 6.
If (H7) holds, then (46) has at least a root.Proof.
Obviously,(48)b32−b42=b3−b4b3+b4=2c−abc−2bc−abc−b<0.
Therefore, (46) has at least a positive root.According to Lemma6, (46) has a positive root, denoted by z0, and thus, (45) has a positive root ω0=z0. By (44), we have
(49)cosω0τ=ω03−b2ω0ω0−−ω02b1+b3b4ω02+b42.Thus, if we denote(50)τ0j=1ω0arccosω03−b2ω0ω0−−ω02b1+b3b4ω02+b42+2jπ,j=0,1,2,…,then ±iω0 is a pair of purely imaginary roots of (21) with τ=τ0j. Clearly, sequence τ0jj=0∞ is increasing and
(51)limj→+∞τ0j=+∞.Thus, we can define(52)τ0=τ00=minτ0j.Lemma 7.
Let λτ=ατ±iωτ be the root of (37) near τ=τ0j satisfying ατ0j=0, ωτ0j=ω0. Suppose that h′z0≠0, where hz is defined by (47). Then the following transversal condition holds:
(53)dReλτdττ=τ0j≠0,and the sign of dReλτ/dτ∣τ=τ0j is consistent with that of h′z0.Proof.
Denote(54)Rλ=λ3+b1λ2+b2λ+b3,Qλ=λ+b4.
Then (37) can be written as
(55)Rλ+Qλe−λτ=0,and (45) can be transformed into the following form:
(56)RiωR¯iω−QiωQ¯iω=0.
Thus, together with (46) and (47), we have
(57)hω2=RiωR¯iω−QiωQ¯iω.
Differentiating both sides of (57) with respect to ω, we obtain
(58)2ωh′ω2=iR′iωR¯iω+RiωR′¯iω−Q′iωQ¯iω+QiωQ′¯iω.
Ifiω0 is not simple, then ω0 must satisfy
(59)ddλRλ+Qλe−λτ0λ=iω0=0,that is, ω0 must satisfy
(60)R′iω+Q′iωe−iω0τ0−τ0Qiω0e−iω0τ0=0.
With (55), we have
(61)τ0=Q′iω0Qiω0−R′iω0Riω0.
Thus, by (56) and (57), we obtain
(62)Imτ0=ImQ′iω0Qiω0−R′iω0Riω0=ImQ′iω0Q¯iω0Qiω0Q¯iω0−R′iω0R¯iω0Riω0R¯iω0=ImQ′iω0Q¯iω0−R′iω0R¯iω0Riω0R¯iω0=−iQ′iω0Q¯iω0−R′iω0R¯iω0+Q′¯iω0Qiω0+R′¯iω0Riω02Riω0R¯iω0=ω0h′ω02Riω02.
Sinceτ0 is real, i.e., Imτ0=0, we have h′ω02=0. We get a contradiction to the condition h′ω02≠0. This proves the first conclusion. Differentiating both sides of (55) with respect to τ, we obtain
(63)R′λ+Q′λe−λτ−τQλe−λτdλdτ−λQλe−λτ=0,which implies
(64)dλdτ=λQλR′λeλτ+Q′λ−τQλ=λQλR′¯λeλτ+Q′¯λ−τQ¯λR′λeλτ+Q′λ−τQλ2=λ−RλR′¯λeλτ+QλQ′¯λ−τQλ2R′λeλτ+Q′λ−τQλ2.
It follows together with (58) that
(65)dReλτdττ=τ0,λ=iω0=Reλ−RλR′¯λeλτ+QλQ′¯λ−τQλ2τ=τ0,λ=iω0R′λeλτ+Q′λ−τQλ2=iω0−Riω0R′¯iω0+Qiω0Q′¯iω0+R′iω0R¯iω0−Q′iω0Q¯iω0R′λeλτ+Q′λ−τQλ2=ω02h′ω02R′λeλτ+Q′λ−τQλ2=ω02h′z0R′λeλτ+Q′λ−τQλ2≠0.
Clearly, the sign ofdReλτ/dτ∣τ=τ0 is determined by that of h′z0.On the basis of Lemma1 and Lemma 5–Lemma 7, we have the following result.Theorem 3.
If (H1), (H6), and (H7) hold, and h′z0>0, then the following statements are true:
(i)
When τ∈0,τ0, the equilibria P1 and P2 of system (4) are both locally asymptotically stable
(ii)
The Hopf bifurcation occurs at τ=τ0, i.e., system (4) has a branch of periodic solutions that bifurcates from P1 and P2 near τ=τ0, respectively


## 3. Direction and Stability of Hopf Bifurcation

In the previous section, we have shown that system (4) admits a series of periodic solutions bifurcating from the equilibrium at the critical value τ0j j∈N0. In this section, we derive explicit formulae to determine the properties of the Hopf bifurcation at the critical value τ0j by using the normal form theory and center manifold reduction developed by [26].Denoteτ0j by τ∗ and introduce the new parameter μ=τ−τ∗. Normalizing the delay τ by the time-scaling t→t/τ, (4) can then be rewritten as
(66)dUtdt=Lτ∗Ut+FUt,μ,where
(67)Lμφ=μy∗−aφ10+φ3−1−2x∗φ10−bφ20−φ10−cφ30,Fφ,μ=Lμφ+fφ,μ,(68)fφ,μ=τ∗+μφ10φ20φ1200+h.o.t.,forφ=φ1,φ2,φ3T∈C.Then the linearized system of (66) at 0,0,0 is
(69)dUtdt=Lτ∗Ut.Based on the discussion in Section2, we can easily know that for τ=τ∗, the characteristic equation of (11) has a pair of simple purely imaginary eigenvalues Λ0=iω0τ∗,−iω0τ∗.LetC≔C−1,0,ℝ3, considering the following FDE on C:
(70)ż=Lτ∗zt.Obviously,Lτ∗ is a continuous linear function mapping C−1,0,ℝ3 into ℝ3. By the Riesz representation theorem, there exists a 3×3 matrix function θ,τ −1≤θ≤0, whose elements are of bounded variation such that
(71)Lτ∗φ=∫−10dηθ,τ∗φθ,forφ∈C.In fact, we can choose(72)ηθ,τ∗=τ∗y∗−ax∗0−2x∗−b0−10−cδθ−τ∗001000000δθ+1,where δ is the Dirac delta function.LetAτ∗ denote the infinitesimal generator of the semigroup induced by the solutions of (70) and A∗ be the formal adjoint of Aτ∗ under the bilinear pairing
(73)ψ,ϕ=ψ0,ϕ0−∫−10∫ξ=0θψξ−θdηθϕξdξ=ψ0,ϕ0+τ∗∫−10ψθ+1001000000ϕθdθ,for ϕ∈C and ψ∈C∗=C0,1,R3. Then Aτ∗ and A∗ are a pair of adjoint operators. From the discussion in Section 2, we know that Aτ∗ has a pair of simple purely imaginary eigenvalues ±iω0τ∗, and they are also eigenvalues of A∗ since Aτ∗ and A∗ are a pair of adjoint operators. Let P and P∗ be the center spaces, that is, the generalized eigenspaces of Aτ∗ and A∗, respectively, associated with Λ0. Then P∗ is the adjoint space of P and dimP=dimP∗=2. Direct computations give the following results.Lemma 8.
Let
(74)α=−−2x∗iω0+b,β=−1iω0+c,α∗=x∗iω0+b,β∗=1iω0+c.
Then,
(75)p1θ=eiω0τ∗θ1,α,βT,p2θ=p¯1θ,−1≤θ≤0,
is a basis ofP associated with Λ0 and
(76)q1s=1,α∗,β∗e−iω0τ∗s,q2s=q¯1s,0≤s≤1,
is a basis ofQ associated with Λ0.LetΦ=Φ1,Φ2 and Ψ∗=Ψ1∗,Ψ2∗T with
(77)Φ1θ=p1θ+p2θ2=Reeiω0τ∗θReαeiω0τ∗θReβeiω0τ∗θ=cosω0τ∗θ−2x∗bcosω0τ∗θ+ω0sinω0τ∗θb2+ω02−ω0sinω0τ∗θ−ccosω0τ∗θc2+ω02,Φ2θ=p1θ−p2θ2i=Imeiω0τ∗θImαeiω0τ∗θImβeiω0τ∗θ=sinω0τ∗θ2x∗ω0cosω0τ∗θ−bsinω0τ∗θb2+ω02ω0cosω0τ∗θ−csinω0τ∗θc2+ω02,forθ∈−1,0, and
(78)Ψ1∗s=q1s+q2s2=Ree−iω0τ∗sReα∗e−iω0τ∗sReβ∗e−iω0τ∗s=cosω0τ∗s−x∗bcossτω0+ω0sinsτω0b2+ω02−ccossτω0+ω0sinsτω0c2+ω02,Ψ2∗s=q1s−q2s2i=Ime−iω0τ∗sImαe−iω0τ∗sImβeiω0τ∗s=−sinω0τ∗s−bsinsτω0−ω0cosτω0sb2+ω02−csinsτω0−ω0cosτω0sc2+ω02,for s∈0,1. From (73), we can obtain Ψ1∗,Φ1 and Ψ1∗,Φ2, noting that
(79)q1,p1=Ψ1∗,Φ1−Ψ2∗,Φ2+iΨ1∗,Φ2+Ψ2∗,Φ1,(80)q1,p1=1+αα∗+ββ∗−ββ∗τ∗e−iω0τ∗≔D∗.Therefore, we have(81)Ψ1∗,Φ1−Ψ2∗,Φ2=ReD∗,Ψ1∗,Φ2+Ψ2∗,Φ1=ImD∗.Now, we defineΨ∗,Φ=Ψj∗,Φkj,k=1,2 and construct a new basis ψ for Q by
(82)Ψ=Ψ1,Ψ2T=Ψ∗,Φ−1Ψ∗.Obviously,Ψ,Φ=I2×2, the second-order identity matrix. In addition, define f0=ξ01,ξ02,ξ03, where
(83)ξ01=100,ξ02=010,ξ02=001.Letc·f0 be defined by
(84)c·f0=c1ξ01+c2ξ02+c3ξ03,forc=c1,c2,c3T and cj∈Rj=1,2,3.Then the center space of linear Equation (69) is given by PCNC, where
(85)PCNφ=ΦΨ,φ,f0·f0,φ∈c,and C=PCNC⊕PSC; here PSC denotes the complementary subspace of PCNC.LetAτ∗ be defined by
(86)Aτ∗φθ=φ̇θ+X0θLτ∗φθ−φ̇0,φ∈BC,where X0:−1,0→BX,X is given by
(87)X0θ=0,−1≤θ<0,I,θ=0.ThenAτ∗ is the infinitesimal generator induced by the solution of (69) and (66) and can be rewritten as the following operator differential equation:
(88)U̇t=Aτ∗Ut+X0FUt,μ.Using the decompositionC=PCNC⊕PSC and (85), the solution of (66) can be rewritten as
(89)Ut=Φx1tx2t·f0+hx1,x2,μ,where
(90)x1tx2t=Ψ,<Ut,f0>,and hx1,x2,μ∈Psc with h0,0,0=Dh0,0,0=0. In particular, the solution of (66) on the center manifold is given by
(91)Ut∗=Φx1tx2t·f0+hx1,x2,0.Settingz=x1−ix2 and noticing that p1=Φ1+iΦ2, then (91) can be rewritten as
(92)Ut∗=12Φz+z¯iz−z¯·f0+wz,z¯=12p1z+p1¯z¯·f0+Wz,z¯,where Wz,z¯=hz+z¯/2,−z−z¯/2i,0. Moreover, by [26], z satisfies
(93)ż=iω0τ∗z+gz,z¯,where
(94)gz,z¯=Ψ10−iΨ20<FUt∗,0,f0>.Let(95)Wz,z¯=W20z22+W11zz¯+W02z¯22+⋯,(96)gz,z¯=g20z22+g11zz¯+g02z¯22+⋯.From (92), we have
(97)<FUt∗,0,f0>=τ∗z241α0+τ∗zz¯42α+α¯0+τ∗z¯241α20+τ∗44w1110+2w2010,12w1120+w2020+2αw1110+α¯w2010,10z2z¯+⋯,where
(98)<Wijnθ,1>=1π∫0πWijnθxdx,i+j=2,n∈ℕ.Letψ1,ψ2,ψ3=Ψ10−iΨ20. Then by (94), (95), and (96), we can obtain the following quantities:
(99)g20=τ∗2ψ1+αψ2,g11=τ∗42ψ1+α+α¯ψ2,g02=τ∗2ψ1¯+α2ψ2¯,g21=τ∗24w1110+2w2010,1ψ1+2w1120+w2020+2αw1110+α¯w2010,1ψ2.SinceW20θ and W11θ for θ∈−1,0 appear in g21, we still need to compute them. It follows easily from (95) that
(100)Ẇz,z¯=W20zż+W11żz¯+zż¯+W02z¯ż¯+⋯,(101)Aτ∗W=Aτ∗W20z22+Aτ∗W11zz¯+Aτ∗W02z¯22+⋯.In addition, by [26], Wzt and z¯t satisfy
(102)Ẇ=Aτ∗W+Hz,z¯,where
(103)Hz,z¯=H20z22+H11zz¯+H02z¯22+⋯=X0FUt∗,0−ΦΨ,<X0FUt∗,0,f0>·f0,with Hij∈PSC,i+j=2. Thus, from (92), (100), (101), and (102), we can obtain that
(104)2iω0τ∗−Aτ∗W20=H20,−Aτ∗W11=H11.Noticing thatAτ∗ has only two eigenvalues ±iω0τ∗ with zero real parts, (102), therefore, has a unique solution Wiji+j=2 in PSC given by
(105)W20=2iω0τ∗−Aτ∗−1H20,W11=−Aτ∗−1H11.From (103), we know that for −1≤θ<0,
(106)Hz,z¯=−ΦθΨ0<FUt∗,0,f0>·f0=−p1θ+p2θ2,p1θ−p2θ2iΨ10Ψ20×<FUt∗,0,f0>·f0=−12p1θΨ10−iΨ20+p2θΨ10+iΨ20×<FUt∗,0,f0>·f0=−14g20p1θ+g¯02p2θz2·f0−12g11p1θ+g¯11p2θzz¯·f0+⋯.Therefore, for−1≤θ<0,
(107)H20θ=−12g20p1θ+g¯02p2θ·f0,(108)H11θ=−12g11p1θ+g¯11p2θ·f0,(109)Hz,z¯0=FUt∗,0−ΦΨ,<FUt∗,0,f0>·f0,H200=τ∗21α0−12g20p10+g¯02p20·f0,H110=τ∗42α+α¯0−12g11p10+g¯11p20·f0.By the definition ofAτ∗, we get from (105) that
(110)Ẇ20θ=2iω0τ∗W20θ+12g20p1θ+g¯02p2θ·f0,−1≤θ<0.Noting thatp1θ=p10eiω0τ∗, −1≤θ≤0. Hence,
(111)W20θ=i2g20ω0τ∗p1θ+g¯023ω0τ∗p2θ·f0+Ee2iω0τ∗θ,(112)E=W200−i2g20ω0τ∗p10+g¯023ω0τ∗p20·f0.Using the definition ofAτ∗ and combining (105) and (112) we get
(113)2iω0τ∗ig202ω0τ∗p10·f0+ig¯026ω0τ∗p20·f0+E−Lτ∗ig202ω0τ∗p1θ·f0+ig¯026ω0τ∗p2θ·f0+Ee2iω0τ∗θ=τ∗21α0−12g20p10+g¯02p20·f0.Notice that(114)Lτ∗p1θ·f0=iω0τ∗p10·f0,Lτ∗p2θ·f0=−iω0τ∗p20·f0.Then, we have(115)2iω0τ∗E−τ∗DΔE−Lτ∗Ee2iω0τ∗θ=τ∗21α0.From the above expression, we can see easily that(116)E=122iω0−y∗+a−x∗e−2iω0τ∗2x∗2iω0+b0102iω0+c−1×1α0.By the similar way, we have(117)Ẇ11θ=12g11p1θ+g¯11p2θ·f0,−1≤θ<0,(118)W11θ=i2ω0τ∗−g11p1θ+g¯11p2θ·f0+F.Similar to the above, we can obtain that(119)F=14a−y∗−x∗12x∗b010c−1×2α+α¯0.So far,W20θ and W11θ have been expressed by the parameters of system (4). Therefore, g21 can be expressed explicitly.Theorem 4.
System (4) has the following Poincaré normal form
(120)ξ̇=iω0τ∗ξ+c10ξξ2+oξ5,where
(121)c10=i2ω0τ∗g20g11−2g112−g0223+g212,so we can compute the following results:
(122)σ2=−Rec10Reλ′τ∗,β2=2Rec10,T2=−Imc10+σ2Imλ′τ∗ω0τ∗,which determine the properties of bifurcating periodic solutions at the critical values τ∗, i.e., σ2 determines the directions of the Hopf bifurcation: if σ2>0σ2<0, then the Hopf bifurcation is supercritical (subcritical) and the bifurcating periodic solutions exist for τ>τ∗; β2 determines the stability of the bifurcating periodic solutions: the bifurcating periodic solutions on the center manifold are stable (unstable), if β2<0β2>0; and T2 determines the period of the bifurcating periodic solutions: the periodic increase (decrease), if T2>0T2<0.


## 4. Numerical Simulation

In this section, we present numerical simulations of some examples to illustrate our theoretical results.


### 4.1. Stability of EquilibriumP0 for All τ≥0

Consider system (4) with the following parameters: a=3, b=0.7, and c=0.9. By a direct calculation, we obtain that system (4) has only an equilibrium P0=0,1.4286,0 and the parameters satisfy the conditions of (H2)–(H4). According to Theorem 1, the system is locally asymptotically stable at P0 for all τ≥0; see Figure 1. However, in this case, the interest rate and the price index are all zero; this is impractical.Figure 1
The equilibriumP0 is locally asymptotically stable with τ=10.
(a)
(b)


### 4.2. Hopf Bifurcation at EquilibriumP0

Consider system (4) with the following parameters: a=0.5, b=0.8, and c=0.8. By a simple calculation, we obtain that system (4) has only an equilibrium P0=0,1.25,0. Obviously, (H2), (H3), and (H5) are satisfied. By (28), we obtain τ0=0.05. According to Theorem 2, system (4) is locally asymptotically stable at P0 for τ=0.03∈0,τ0 (see Figure 2) and Hopf bifurcation occurs at τ=0.06>τ0, as shown in Figure 3.Figure 2
The equilibriumP0 is locally asymptotically stable for τ=0.03.
(a)
(b)Figure 3
The periodic solutions emerge from the equilibriumP0 with τ=0.06.
(a)
(b)Whenτ=τ0=0.05, we can compute c10=−0.0023+0.0042i, σ2=−Rec10/Reλ′τ∗=0.0047>0, and β2=2Rec10=−0.0047<0. Therefore, from the discussions in Section 3, we know that the bifurcated periodic solutions are orbitally asymptotically stable on the center manifold. In addition, from Theorem 4, we know that system (4) has a stable center manifold near the equilibrium P0 for τ near τ0=0.05. Therefore, the center manifold theory implies that the bifurcated periodic solutions of system (4) when τ0=0.05 in the whole phase space are orbitally asymptotically stable, and the Hopf bifurcation is supercritical for σ2>0.


### 4.3. Hopf Bifurcation at EquilibriaP1 and P2

Choose the parameters of system (4) as a=0.5, b=0.8, and c=1.5. By a simple calculation, it is easy to obtain that P1=0.2582,1.1667,−0.1721 and P2=−0.2582,1.1667,0.1721. Obviously, the parameters satisfy (H6) and (H7). By (50), we obtain the critical value τ0=0.7307. According to Theorem 3, P1 and P2 are both stable with τ=0.68∈0τ0, and with τ=0.71>τ0, two limit cycles emerge from the equilibria P1 and P2, as shown in Figure 4.Figure 4
(a)P1 and P2 are both stable with τ=0.7; (b) the periodic solutions emerge from the equilibria P1 and P2 with τ=0.71.
(a)
(b)In addition, whenτ=τ0=0.7307, at equilibrium P1, we get c10=−0.0316+0.056i, σ2=−Rec10/Reλ′τ∗=0.0898>0, and β2=2Rec10=−0.0632<0. At equilibrium P2, we get c10=−0.0366+0.0675i, σ2=−Rec10/Reλ′τ∗=0.0104>0, and β2=2Rec10=−0.0733<0. According to Theorem 4 in Section 3, the bifurcated periodic solutions of system (4) when τ0=0.7307 in the whole phase space are both orbitally asymptotically stable, and the Hopf bifurcations are supercritical for σ2>0.However, with increasing delayτ, the two limit cycles emerging form the equilibria P1 and P2 and appear to overlap, as shown in Figure 5. Figure 5(a) shows that the maximum and minimum of x varies with τ under two groups of different initial values. It shows that two lines about maxima and minima appear to overlap with increasing delay τ, which mean that two limit cycles overlap; see Figure 5(b).Figure 5
(a) Maximum and minimum ofx; (b) the two limit cycles emerging from the equilibria P1 and P2 appear to overlap with τ=1.
(a)
(b)


### 4.4. Chaos Vanishes by Delayτ

According to [10, 15, 27], system (4) is chaotic for appropriate parameters. Figure 6 shows the Lyapunov exponents’ spectrum of system (4) with the increasing of parameter c, where a=2 and b=0.1. Figure 7 shows the bifurcation diagram of system (4) in the c−y plane. Let c=1.1, and the chaotic attractor of system (4) is shown in Figure 8.Figure 6
Lyapunov exponent spectrum of system (4) when the parameter value of c is changed continuously with τ=0.Figure 7
Bifurcation diagrams of system (4) when the parameter value of c is changed continuously with τ=0.Figure 8
(a) Time trajectories of system (4) with τ=0; (b) phase diagram of system (4) with τ=0.
(a)
(b)In the following, in order to investigate the effect of delayτ on system (4), we fix a=2, b=0.1, and c=1.1, and choosing τ as a parameter, the Lyapunov exponent spectrum and the detailed bifurcation scenarios of system (4) are shown in Figures 9 and 10. It can be seen that chaos disappears through a cascade of inverse period-doubling; see Figure 11. This observation indicates that the delay is a sensitive factor for system bifurcation and chaos and that chaos can be suppressed by delay τ.Figure 9
Lyapunov exponent spectrum of system (4) when the parameter value of τ is changed continuously.Figure 10
Bifurcation diagrams of system (4) when the parameter value of τ is changed continuously.Figure 11
Phase diagrams showing chaos disappearing via inverse period-doubling: (a)τ=0.4; (b) τ=0.44; (c) τ=0.45; (d) τ=0.5; (e) τ=0.54; (f) τ=0.56.
(a)
(b)
(c)
(d)
(e)
(f)


### 4.5. Chaos Induced by Delayτ

Consider system (4) with the following parameters a=3, b=0.2, and c=6. Obviously, parameters satisfy condition (H6). Therefore, according to Lemma 5, P1,2=±0.6055,3.1667,∓0.1009 are both locally stable with τ=0. However, with increasing delay τ, system (4) presents strong nonlinear phenomena such as periodic motion, double-periodic motion, and chaotic motion and the bifurcation diagram of system (4) with increasing delay τ, which can be seen from the bifurcation diagram and maximum Lyapunov exponent with the parameter value of τ changed continuously, as shown in Figures 12 and 13. We list dynamic behaviors of system (4) corresponding to different delays in Table 1 and Figure 14. The route to chaos in finance system (4) was shown to be via classical period-doubling bifurcations (see Figures 14(e)–14(i)).Figure 12
Bifurcation diagram of system (4) with the parameter value of τ is changed continuously.Figure 13
Maximum Lyapunov exponent of system (4) when the parameter value of τ is changed continuously.Table 1
The dynamic behaviors of system (4) for different τ.
τ
Dynamics of system (4)
Figure
[0, 1.52)
P1 and P2 are both stable
Figure14(a)
(1.52, 3.18]
System (4) is chaotic
Figure14(b)
(3.18, 5.52]
System (4) exhibits period 1 motion
Figure14(c)
(5.52, 5.61]
System (4) is chaotic
Figure14(d)
(5.61, 8.32]
P1 and P2 are both stable
Figure14(e)
(8.32, 9.46]
System (4) exhibits period 1 motion
Figure14(f)
(9.46, 9.68]
System (4) exhibits period 2 motion
Figure14(g)
(9.68, 9.71]
System (4) exhibits period 4 motion
Figure14(h)
(9.71, 12]
System (4) is chaotic
Figure14(i)Figure 14
Nonlinear phenomena of system (4) induced by delay τ: (a) τ=1 (stable); (b) τ=2 (chaos); (c) τ=4.2 (period 1); (d) τ=5.58 (chaos); (e) τ=7 (stable); (f) τ=8.5 (period 1); (g) τ=9.5 (period 2); (h) τ=9.69 (period 4); (i) τ=12 (chaos).
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)


## 4.1. Stability of EquilibriumP0 for All τ≥0

Consider system (4) with the following parameters: a=3, b=0.7, and c=0.9. By a direct calculation, we obtain that system (4) has only an equilibrium P0=0,1.4286,0 and the parameters satisfy the conditions of (H2)–(H4). According to Theorem 1, the system is locally asymptotically stable at P0 for all τ≥0; see Figure 1. However, in this case, the interest rate and the price index are all zero; this is impractical.Figure 1
The equilibriumP0 is locally asymptotically stable with τ=10.
(a)
(b)


## 4.2. Hopf Bifurcation at EquilibriumP0

Consider system (4) with the following parameters: a=0.5, b=0.8, and c=0.8. By a simple calculation, we obtain that system (4) has only an equilibrium P0=0,1.25,0. Obviously, (H2), (H3), and (H5) are satisfied. By (28), we obtain τ0=0.05. According to Theorem 2, system (4) is locally asymptotically stable at P0 for τ=0.03∈0,τ0 (see Figure 2) and Hopf bifurcation occurs at τ=0.06>τ0, as shown in Figure 3.Figure 2
The equilibriumP0 is locally asymptotically stable for τ=0.03.
(a)
(b)Figure 3
The periodic solutions emerge from the equilibriumP0 with τ=0.06.
(a)
(b)Whenτ=τ0=0.05, we can compute c10=−0.0023+0.0042i, σ2=−Rec10/Reλ′τ∗=0.0047>0, and β2=2Rec10=−0.0047<0. Therefore, from the discussions in Section 3, we know that the bifurcated periodic solutions are orbitally asymptotically stable on the center manifold. In addition, from Theorem 4, we know that system (4) has a stable center manifold near the equilibrium P0 for τ near τ0=0.05. Therefore, the center manifold theory implies that the bifurcated periodic solutions of system (4) when τ0=0.05 in the whole phase space are orbitally asymptotically stable, and the Hopf bifurcation is supercritical for σ2>0.


## 4.3. Hopf Bifurcation at EquilibriaP1 and P2

Choose the parameters of system (4) as a=0.5, b=0.8, and c=1.5. By a simple calculation, it is easy to obtain that P1=0.2582,1.1667,−0.1721 and P2=−0.2582,1.1667,0.1721. Obviously, the parameters satisfy (H6) and (H7). By (50), we obtain the critical value τ0=0.7307. According to Theorem 3, P1 and P2 are both stable with τ=0.68∈0τ0, and with τ=0.71>τ0, two limit cycles emerge from the equilibria P1 and P2, as shown in Figure 4.Figure 4
(a)P1 and P2 are both stable with τ=0.7; (b) the periodic solutions emerge from the equilibria P1 and P2 with τ=0.71.
(a)
(b)In addition, whenτ=τ0=0.7307, at equilibrium P1, we get c10=−0.0316+0.056i, σ2=−Rec10/Reλ′τ∗=0.0898>0, and β2=2Rec10=−0.0632<0. At equilibrium P2, we get c10=−0.0366+0.0675i, σ2=−Rec10/Reλ′τ∗=0.0104>0, and β2=2Rec10=−0.0733<0. According to Theorem 4 in Section 3, the bifurcated periodic solutions of system (4) when τ0=0.7307 in the whole phase space are both orbitally asymptotically stable, and the Hopf bifurcations are supercritical for σ2>0.However, with increasing delayτ, the two limit cycles emerging form the equilibria P1 and P2 and appear to overlap, as shown in Figure 5. Figure 5(a) shows that the maximum and minimum of x varies with τ under two groups of different initial values. It shows that two lines about maxima and minima appear to overlap with increasing delay τ, which mean that two limit cycles overlap; see Figure 5(b).Figure 5
(a) Maximum and minimum ofx; (b) the two limit cycles emerging from the equilibria P1 and P2 appear to overlap with τ=1.
(a)
(b)


## 4.4. Chaos Vanishes by Delayτ

According to [10, 15, 27], system (4) is chaotic for appropriate parameters. Figure 6 shows the Lyapunov exponents’ spectrum of system (4) with the increasing of parameter c, where a=2 and b=0.1. Figure 7 shows the bifurcation diagram of system (4) in the c−y plane. Let c=1.1, and the chaotic attractor of system (4) is shown in Figure 8.Figure 6
Lyapunov exponent spectrum of system (4) when the parameter value of c is changed continuously with τ=0.Figure 7
Bifurcation diagrams of system (4) when the parameter value of c is changed continuously with τ=0.Figure 8
(a) Time trajectories of system (4) with τ=0; (b) phase diagram of system (4) with τ=0.
(a)
(b)In the following, in order to investigate the effect of delayτ on system (4), we fix a=2, b=0.1, and c=1.1, and choosing τ as a parameter, the Lyapunov exponent spectrum and the detailed bifurcation scenarios of system (4) are shown in Figures 9 and 10. It can be seen that chaos disappears through a cascade of inverse period-doubling; see Figure 11. This observation indicates that the delay is a sensitive factor for system bifurcation and chaos and that chaos can be suppressed by delay τ.Figure 9
Lyapunov exponent spectrum of system (4) when the parameter value of τ is changed continuously.Figure 10
Bifurcation diagrams of system (4) when the parameter value of τ is changed continuously.Figure 11
Phase diagrams showing chaos disappearing via inverse period-doubling: (a)τ=0.4; (b) τ=0.44; (c) τ=0.45; (d) τ=0.5; (e) τ=0.54; (f) τ=0.56.
(a)
(b)
(c)
(d)
(e)
(f)


## 4.5. Chaos Induced by Delayτ

Consider system (4) with the following parameters a=3, b=0.2, and c=6. Obviously, parameters satisfy condition (H6). Therefore, according to Lemma 5, P1,2=±0.6055,3.1667,∓0.1009 are both locally stable with τ=0. However, with increasing delay τ, system (4) presents strong nonlinear phenomena such as periodic motion, double-periodic motion, and chaotic motion and the bifurcation diagram of system (4) with increasing delay τ, which can be seen from the bifurcation diagram and maximum Lyapunov exponent with the parameter value of τ changed continuously, as shown in Figures 12 and 13. We list dynamic behaviors of system (4) corresponding to different delays in Table 1 and Figure 14. The route to chaos in finance system (4) was shown to be via classical period-doubling bifurcations (see Figures 14(e)–14(i)).Figure 12
Bifurcation diagram of system (4) with the parameter value of τ is changed continuously.Figure 13
Maximum Lyapunov exponent of system (4) when the parameter value of τ is changed continuously.Table 1
The dynamic behaviors of system (4) for different τ.
τ
Dynamics of system (4)
Figure
[0, 1.52)
P1 and P2 are both stable
Figure14(a)
(1.52, 3.18]
System (4) is chaotic
Figure14(b)
(3.18, 5.52]
System (4) exhibits period 1 motion
Figure14(c)
(5.52, 5.61]
System (4) is chaotic
Figure14(d)
(5.61, 8.32]
P1 and P2 are both stable
Figure14(e)
(8.32, 9.46]
System (4) exhibits period 1 motion
Figure14(f)
(9.46, 9.68]
System (4) exhibits period 2 motion
Figure14(g)
(9.68, 9.71]
System (4) exhibits period 4 motion
Figure14(h)
(9.71, 12]
System (4) is chaotic
Figure14(i)Figure 14
Nonlinear phenomena of system (4) induced by delay τ: (a) τ=1 (stable); (b) τ=2 (chaos); (c) τ=4.2 (period 1); (d) τ=5.58 (chaos); (e) τ=7 (stable); (f) τ=8.5 (period 1); (g) τ=9.5 (period 2); (h) τ=9.69 (period 4); (i) τ=12 (chaos).
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)


## 5. Conclusions

In this study, we have investigated dynamical behaviors such as stability, Hopf bifurcation, and chaos for a delayed finance system.Firstly, we took delayτ as the bifurcation parameters to study the Hopf bifurcation of system (4). We have proved theoretically that the discrete delay is responsible for the stability switch of the model and that a Hopf bifurcation occurs as the delays increase through a certain threshold.Secondly, by the normal form method and center manifold theorem, we have derived the normal forms of Hopf bifurcation.Finally, by numerical simulations, we have given the Hopf bifurcation (Figures3 and 4) that was induced by delay. We have also given the bifurcation diagram (Figures 10 and 12) and the corresponding Lyapunov exponents’ spectrum (Figures 6 and 13). All these show that delay τ can cause the system to exhibit strong nonlinear phenomena such as periodic motion, double-periodic motion, and chaotic motion (Figure 14).The study will help in understanding the role of financial policies and interpreting economics phenomena in theory.



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