# Hopf Bifurcation and Dynamic Analysis of an Improved Financial System with Two Delays

Authors: G. Kai; W. Zhang; Z. Jin; C. Z. Wang
Journal: Complexity
 (2020)

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## Abstract

The complex chaotic dynamics and multistability of financial system are some important problems in micro- and macroeconomic fields. In this paper, we study the influence of two-delay feedback on the nonlinear dynamics behavior of financial system, considering the linear stability of equilibrium point under the condition of single delay and two delays. The system undergoes Hopf bifurcation near the equilibrium point. The stability and bifurcation directions of Hopf bifurcation are studied by using the normal form method and central manifold theory. The theoretical results are verified by numerical simulation. Furthermore, one feature of the proposed financial chaotic system is that its multistability depends extremely on the memristor initial condition and the system parameters. It is shown that the nonlinear dynamics of financial chaotic system can be significantly changed by changing the values of time delays.


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## Body

## 1. Introduction

It is widely recognized that chaos can be obtained in some mathematically simple systems of nonlinear differential equations. With the advent of computers, it is now possible to study the entire parameter space of these systems that result in some desired characteristics of the system. Recently, there has been increasing attention to some unusual examples and application of such systems [1–8]. The financial system is an extremely complex nonlinear dynamical system composed of many elements. The study of the complex nonlinear dynamics behavior of the financial system is an important problem in the fields of micro- and macroeconomy [9]. The uncertain factors bring very important influence to the description of the financial system and make analysis of the financial systems become a very important problem.Researchers try to explain the core characteristics of economic data: irregular microeconomic fluctuations, instable macroeconomic fluctuations, irregular growth, and syntax changes [10–13]. However, some inappropriate combination of parameters in the financial system may lead to financial markets in trouble or out of control. Therefore, it is necessary to make a systematic and deep study on the internal syntax characteristics of the complicated financial system. The results will reveal the bifurcation phenomena under different parameter combinations, probe into the causes of the complicated nonlinear dynamics phenomena, and predict and control the complicated financial systems [14, 15]. In addition, multistability is a critical property of nonlinear dynamical systems when coexisting attractors can be obtained for the same parameters, but different initial conditions [16–19]. The flexibility in the systems’ performance can be archived without changing parameters.Wang et al. [20] studied the bifurcation topology and the global complexity of a class of nonlinear financial systems. Ishiyama and Saiki [21] established the macroeconomic growth cycle model and solved the qualitative- and quantitative-related unstable periodic solutions embedded in the chaotic attractor. By using Lyapunov stability theory and Routh–Hurwitz criterion, Zhao et al. [22] studied the global synchronization of the three-dimensional chaotic financial system. Yu et at. [23] used numerical simulation to analyze the Lyapunov exponents and bifurcation diagram of chaotic financial system. Cantore and Levine [24] studied the reparameterized model with evaluation parameters. Gao et al. [25] gave the final bounded estimator set and chaotic synchronization analysis of the financial risks system.Until now, one of the nonlinear economic and financial dynamic models confirmed by economists comes from a financial system model composed of four subblocks (production, money, security, and labor) [15, 24, 26].(1a)x˙=z+y−ax,(1b)y˙=1−by−x2,(1c)z˙=−x−cz,where x represents the interest rate, y represents the investment demand, z represents the price index, a represents the saving amount, b represents the unit investment cost, and c represents the elasticity of commodity demand; a,b,andc are all normal numbers.In real life, with the development of economy, there are more and more factors that restrict the development of economy. Some classical chaotic financial systems can not reflect the laws and changes of economic development well. For example, the factors that affect the change of interest rate are related to the average profit rate besides investment demand and price index, and the average profit rate is proportional to the interest rate. Therefore, we construct the following improved chaotic financial system model:(2a)x˙=z+y−ax,(2b)y˙=1−by−x2−bxy,(2c)z˙=−x−cz,where x is the interest rate, y is the investment demand, z is the price index, a is the saving amount, b is the unit investment cost, and c is the elasticity of commodity demand; a,b,andc are all normal numbers. When the parameters a=3, b=0.1, and c=1 and initial values are at points 0.1,2,0.1, system (2a)–(2c) generates chaotic attractors, as shown in Figure 1.Figure 1
Chaotic attractor for the 3D chaotic system ((2a)–(2c)).With the development and innovation of financial markets, scholars have found that it will be better to add time delay factor for describing the actual economic markets [27–30]. Chen [31] analyzed the complex nonlinear dynamics, such as periodicity, quasiperiodicity, and chaotic behavior in the delayed feedback of financial systems. Ma and Tu [32] established a class of complex dynamic macroeconomic systems and studied the effect of time delay on savings rate and dynamic financial stability. Holyst and Urbanowicz [33] have shown that the chaotic attractor of the financial model can be stabilized in a periodic track by using Pyragas delayed feedback control. In addition, Ma and Chen [14] added the delayed feedback to the three variables of financial system and gave some results on the existence of Hopf bifurcation and the effect of delayed feedback. Based on political events and other human factors, some scholars have considered the impact of delay and feedback items (see [28, 34]). In practice, financial behaviour is not only affected by a single time delay but often seems to be affected by multiple external shocks. These various external influences embody multiple delays and can be reflected in all variables, i.e., by introducing various delayed feedback items into interest rates x of change of interest rate, they will also have a significant impact on system (2a)–(2c). Therefore, we further consider the double-delay system.(3a)x˙=z+y−ax+k1xt−xt−τ1+k2xt−xt−τ2,(3b)y˙=1−by−x2−bxy,(3c)z˙=−x−cz.where τ1 and τ2 are the two time delays and k1 and k2 are the feedback control intensities.In this paper, we study the Hopf bifurcation and nonlinear dynamics of an improved financial system with two delays. Firstly, we study the distribution of the roots of the characteristic equations at the equilibrium point. Sufficient conditions for the local stability of the equilibrium point and the existence of Hopf bifurcation are obtained. Secondly, taking two delays as bifurcation parameters and using the canonical form method and the central manifold theorem, we determine the bifurcation direction of the periodic solution and the explicit algorithm for the stability of the bifurcation periodic solution. Under the premise of the existence of local bifurcation, the existence of the bifurcation periodic solution of this system is discussed by using the theory of functional differential equations. Finally, the correctness of the conclusion is verified by numerical simulation.


## 2. Existence of Hopf Bifurcation in Financial System

In order to study the influence of time delays on nonlinear dynamic system, the three equilibrium points of the system are obtained as follows:(4a)−1,1+acc,1c,(4b)0,1b,0,(4c)−b+c−abcc,1+acc,b−c+abcc2.Here, we only analyze the following equilibrium point:(5)x0,y0,z0=0,1b,0.Linear transformations are given as(6a)u1=x−x0,(6b)u2=y−y0,(6c)u3=z−z0.System (2a)–(2c) becomes the following equations:(7a)u˙1=u3+u2−a+1bu1+k1u1t−u1t−τ1+k2u1t−u1t−τ2,(7b)u˙2=−u1−bu2−u12−bu1,(7c)u˙3=−u1−cu3.The characteristic equation of equations (7a) and (7b) at 0,0,0 is(8)b−c+abc+abλ+acλ−cλb+bcλ+aλ2−λ2b+bλ2+cλ2+λ3+e−λτ1−bck1−bk1λ−ck1λ−k1λ2+e−λτ2−bck2−bk2λ−ck2λ−k2λ2=0.Since the dynamics of differential equations with two delays are very complex, we first discuss the case ofτ1=τ2=0, then discuss the case of τ1>0, τ2=0 with single delay, and finally discuss the case of τ1>0, τ2>0.Case 1.
τ
1
=
τ
2
=
0.
The characteristic equation (5) becomes(9)b−c+abc+ab+ac−cb+bcλ+a−1b+b+cλ2+λ3=0.
Let(10a)p1=a−1b+b+c,(10b)p2=ab+ac−cb+bc,(10c)p3=b−c+abc,and assume(11)H1:p1>0,p3>0,p1p2−p3>0.
According to Routh– Hurwitz criterion, ifH1 holds, the equilibrium point 0,0,0 of system (7a)–(7c) is locally asymptotically stable.Case 2.
τ
1
>
0, τ2=0.
The characteristic equation (5) becomes(12)b−c+abc−bck1+abλ+acλ−cλb+bcλ−bk1λ−ck1λ+aλ2−λ2b+bλ2+cλ2−k1λ2+λ3+e−λτ1bck1+bk1λ+ck1λ+k1λ2=0.
Ifλ=iw is a solution of equation (12), then the real part and imaginary part are separated and made equal to zero. We can obtain(13a)w2+b21+ac+ck1−1+m+k1nw−w2+b−a+k1−1+mw2+c−1+k1nw−w2=0,(13b)−cw+bwac+ck1−1+m+k1n−ww+b2a+k1−1+mw+c−k1n+w=0,where m=coswτ1 and n=sinwτ1.
From formulas (13a) and (13b),(14)b2w4+1−2ab−2b2+a2b2+b2c2+2bk1−2ab2k1w2+b2−2bc+2ab2c+c2−2abc2+a2b2c2−2b2ck1+2bc2k1−2ab2c2k1=0.
If all the parameters in system (3a)–(3c) are given, it is easy to calculate the numerical solution of equation (14) by computer. Thus, the following assumptions are given.
SupposeH2 that equation (14) has at least one positive real root.
IfH2 is assumed to be true and equation (14) has two positive real roots ωkk=1,2, we have(15a)τ1k,j=1ωkarccosp+2jπ,Q≥0,(15b)τ1k,j=1ωk2π−arccosp+2jπ,Q≤04,k=1,2,j=0,1…,where(16a)P=−bc−c2+abc2−bc2k1−w2+abw2−bk1w2bk1c2+w2,(16b)Q=−w−c2w−w3k1c2+w2.±iωk is a pair of pure virtual root under τ1=τ1k,j of equation (12). Let us take(17)τ10=minτ1k,j,k=1,2;j=0,1,….
Letλτ=vt+iωτ be the virtual root of equation (12) near τ1=τ1k,j. By differential degeneracy of equation (12) with respect to τ, we can obtain(18)dλdτ1−1=eλτ1ab2−c+abc+b2c−b2−bck1+2ab+b2+bc−1λbk1λb+λc+λ+3bλ2+b2k1+bck1−b2ck1τ1−b2k1λτ1−bck1λ2τ1bk1λb+λc+λ.
Whenλ=iω is substituted into equation (18), we have(19)dReλdτ1−1=fωkΛ.Among them,(20a)fωk=a2b2c2+b4c2+2b3k1−2ab4k1−2b2ck1+2bc2k1+2−4ab−4b2+2a2b2+2b4+2b2c2+4bk1−4ab2k1wk2+2b2−2ab3−2b4+a2b4−2bc+2ab2c+c2−2abc2−2ab2c2k1+3b2wk4,(20b)Λ=b2kl2b2+wk2c2+wk2.Hypothesis 1.
H
3:(21)dReλdτ1−1≠0.
Therefore, there are the following theorems.Theorem 1.
IfH1, H2, and H3 are assumed to be true, then the equilibrium 0,1/b,0 is locally stable for τ1∈0,τ10. When τ1>τ10 equilibrium 0,1/b,0 is unstable and Hopf bifurcation occurs in system (3a)–(3c) at τ1=τ10.Case 3.
τ
1
>
0, τ2>0.
The corresponding characteristic equation of system (3a)–(3c) is (8). Now, let delay τ1∈0,τ10, and τ2 is taken as a parameter. Assuming λ=iσ is the characteristic root under the two delays in the characteristic equation (5), then(22)λ3+a−1b+b+c−k1−k2λ2+ab+ac−cb+bc−bk1−ck1−bk2−ck2λ+b−c+abc−bck1−bck2+cosστ1−sinστ1bck1+bk1λ+ck1λ+k1λ2+cosστ2−sinστ2bck2+bk2λ+ck2λ+k2λ2=0.
From (22),(23a)cosστ2=−1bk2c2+σ2bc−c2+abc2−bc2k1−bc2k2+bc2k1cosστ1−σ2+abσ2−bk1σ2−bk2σ2+bk1cosστ1σ2,(23b)sinστ2=−c2k1sinστ1+σ−c2σ+k1sinστ1σ2−σ3k2c2+σ2.
Therefore, the equation aboutσ can be obtained:(24)b2−2bc+2ab2c+c2−2abc2+a2b2c2−2b2ck1+2bc2k1−2ab2c2k1+c2b2k1−2b2ck2+2bc2k2−2ab2c2k2+2b2c2k1k2+2b2ck1cosστ1−2bc2k1cosστ1+2ab2c2k1cosστ1−2b2c2k12cosστ1−2b2c2k1k2cosστ1+b2c2k12+2b2k1sinστ11−c2σ+1−2ab−2b2+a2b2+b2c2+2bk1−2ab2k1+b2k12+2bk2−2ab2k2+2b2k1k2−2bk1cosστ1+2ab2k1cosστ1−2b2k12cosστ1−2b2k1k2cosστ1+b2k12σ2−2b2k1sinστ1σ3+b2σ4=0.
Obviously, equation (24) has at most a positive real root of NN≤4, which is denoted as σhh=1,2,…N. Similarly, the following can be obtained:(25a)τ2h,j=1σharccosp¯+2jπ,Q¯≥0,(25b)τ2h,j=1σh2π−arccosp¯+2jπ,Q¯<0,where(26a)p¯=−1bk2c2+σ2bc−c2+abc2−bc2k1−bc2k2+bc2k1cosστ1−σ2+abσ2−bk1σ2−bk2σ2+bk1cosστ1σ,(26b)Q¯=−c2k1sinστ1+σ−c2σ+k1sinστ1σ2−σ3k2c2+σ2,h=1,2,…N;j=0,1….
Then, there is a point columnτ2=τ2h,j satisfying equation (26a) and (26b),(27)τ20=minτ2h,j,h=1,2,…,N;j=0,1….
Correspondingly,h=h0∈1,2,…N and σ=σhji.
Whenτ1∈0,τ10 and τ2∈τ20, equation (8) has a pair of pure virtual root.Hypothesis 2.
(28)
H
4
:
d
Re
λ
d
τ
2
−
1
≠
0.
In this way, by using the general Hopf bifurcation theorem [35–40] for functional differential equations, the results on the stability and bifurcation of system (3a)–(3c) are obtained.Theorem 2.
AssumingH4 holds and τ1∈0,τ12,0 holds, the equilibrium 0,1/b,0 of system (3a)–(3c) is asymptotically stable at τ2∈0,τ20. When τ2∈τ20 occurs, Hopf bifurcation occurs in system (3a)–(3c).


## 3. Period Solution and Stability of Hopf Bifurcation

In this section, we study the relevant properties of Hopf bifurcation in financial system (3a)–(3c) under the condition of delays τ1>0, τ2>0.Using the ideas of Hassard et al. [36], the exact expression of Hopf bifurcation property of system (3a)–(3c) is considered by using central manifold theorem. Here, we consider the Hopf bifurcation of system (4a)–(4c) at the equilibrium point 0,0,0 for τ2=τ20. The financial system can be converted to the following equation:(29)u˙t=Luut+fμ,ut,where(30)ut=u1t,u2t,u3tT∈R3,(31a)Lμ:C⟶R3,f:R×C⟶R3,utθ=ut+θ∈C,(31b)Lμϕ=A0+B1−τ1+B2−τ20+μ.According to systems (3a)–(3c) and (7a)–(7c), it can be seen that(32a)A=1b−a+k1=k201−1−b0−10−c,(32b)B1=−k100000000,(32c)B2=−k200000000.Let(33)ϕt=ϕ1t,ϕ2t,ϕ3tΤ.Then,(34)fμ,ut=μ+τkφ10φ20−φ1200.When the equilibrium point0,0,0 of system (29) passes Hopf bifurcation at μ=0, the characteristic equation has a pair of pure virtual root iσh and −iσh. According to Ritz representation theorem, there is a matrix function of bounded variation(35)Lμϕ=∫−τ10dηθ,μϕθ,ϕ∈C.In fact, we can choose(36)ηθ,μ=A+B1+B2,θ=0,B1+B2,θ∈−τ2,0,B1,θ∈−τ1,−τ2,0,θ∈−τ1..Forϕ=C−τ1,0,R3, we define(37a)Aμϕ=dϕθdθ,θ∈−τ1,0,(37b)Aμϕ=∫−τ10dηξμϕξ,θ=0,(38a)Rμϕ=0,θ∈−τ1,0,(38b)Rμϕ=fμ,ϕ,θ=0.To simplify, equation (29) can be written in the following form:(39)u˙t=Aμut+Rμut,in which(40)ut=ut+θ,θ∈−τ1,0.The adjoint operatorA∗ that defines A for ψ∈C10,τ1,R3∗ is as follows:(41a)A∗ψs=−dψsds,s∈0,τ1,(41b)A∗ψs=∫−τ10dηTt,0ψ−t,s=0.In addition, we define a bilinear form(42a)ψ,ϕ=ψ¯0ϕ0−∫−τ10∫ς−0θψ¯ξ−θdηθϕξdξ,(42b)ηθ=ηθ,0.According to the above analysis,σh and −iσh0 are the eigenvalues of A0 and A∗0. Let qθ be the eigenvector corresponding to the eigenvalue iσh0 of A0 and q∗θ be the eigenvector corresponding to the eigenvalue −iσh0 of A∗0. There are(43a)A0qθ=iσh0qθ,(43b)A∗0qθ=−iσh0qθ.Through simple calculation, we can get(44a)q0=1,α,βT=1,1−b−σh0i,1−c−σh0iT,(44b)q∗s=D1,α,βeisσh0=D1,0,1−c−σh0ieisσh0=D¯1,α¯∗,β¯∗1,α,βT−∫−τ10∫ξ=0θD¯1,α¯∗,β¯∗e−iξ−θσh0dηθ1,α,βTeisσh0dξ=D¯1+αα¯∗+ββ¯∗−k1τ1e−iσh0−k2τ20e−iσh0,where D is a constant, making q∗s,qθ=1 valid.Therefore, we have(45)D=11+αα¯∗+ββ¯∗−k1τ1eiσh0−k2τ20eiσh0.Using the same notation of Ruan et al. [37], we can calculate the center popularity of μ=0. Let μ1 be the solution of equation (29) when μ=0, defining(46a)zt=q∗,ut,(46b)Wt,θ=Wzt,z¯t,θ=W20θz22+W11θzz¯+W02θz¯22+W30θz36+⋯,where z and z¯ are the local coordinates of the central epidemic. When μ=0,(47)z¯t=iσkτkz+q∗θ,f0,Wzt,z¯t,θ+2Reztqθ=iσh0z+q¯∗0f0,Wzt,z¯t,0+2Reztq0.Make(48)f0,Wzt,z¯t,0+2Reztq0=f0z,z¯.Then,(49a)z˙t=iσh0z+q¯∗0f0z,z¯=iσh0z+gz,z¯,(49b)gz,z¯=g20z22+g11zz¯+g02θz¯22+g21θz2z¯2+⋯.Because(50a)qθ=1,α,βTeiθσh0,(50b)utθ=u1tθ,u2tθ,u3tθ=Wt,θ+ztqθ+z¯tq¯θ,we can obtain(51a)u1t0=z+z¯+W201z22+W111zz¯+W021z¯22+⋯,(51b)u2t0=αz+α¯z¯+W202z22+W112zz¯+W022z¯22+⋯,(51c)u3t0=βz+β¯z¯+W203z22+W113zz¯+W023z¯22+⋯.Based on the above formula (49b), it can be seen that(52)gz,z¯=q∗0f0z,z¯=D¯1,α¯∗,β¯∗u1t0−u1t20−bu1t0u2t00=Dz+z¯+W201z22+W111zz¯+W021z¯22+⋯=αz+α¯z+W202z22+W112zz¯+W021z¯22+⋯.By using the method of comparison coefficient, we obtain(53a)g20=2αD¯,(53b)g21=2D¯12W2010α¯+12W2010+W1120+W1110α,(53c)g02=2α¯D¯,(53d)g11=α+α¯D¯.In order to calculateW20θ and W11θ, we use(54)W˙=u˙t−z˙q−z¯˙q¯=A0W−2Req¯∗0f0qθ,θ∈−τ1,0,A0W−2Req¯∗0f0qθ+f0,θ=0,.Make(55)Hz,z¯,θ=2Req¯∗0f0qθ,θ∈−τ1,0,2Req¯∗0f0qθ+f0,θ=0,.We rewrite (54):(56)W˙=A0W+Hz,z¯,θ,(57)Hz,z¯,θ=H20θz¯22+H11θzz¯+H02θz¯22+⋯.Using (54) and (55), one can obtain(58a)A0−2iσh0W20θ=−H,20(58b)A0W11θ=−Hθ11,(59a)θ∈−τ1,0,Hz,z¯,θ=−q¯∗0f0qθ−q∗0f¯0q¯θ=−gz,z¯qθ−g¯z,z¯q¯θ.The combination formula (57) is obtained:(60a)H20θ=−g20qθ−g¯02q¯θ,(60b)H11θ=−g11qθ−g¯11q¯θ.Using equations (58a) and (58b) and (60a) and (60b), it is easy to obtain(61a)W20=2iσh0W20θ+g20qθ+g¯02q¯θ,(61b)W20=ig20σh0q0eiθσh0+ig¯023σh0q¯0e−iθσh0+Ee2iθσh0,(62)W11=−ig11σh0q0eiθσh0+ig¯11σh0q¯0e−iθσh0+E2.Make(63a)E1=E11,E12,E13T∈R3,(63b)E2=E21,E22,E23T∈R3.Combine formulas (58a) and (58b) again to obtain(64a)W˙20θ=∫−τ10dηθW20θ=2iθσh0W20θ−H20θ,(64b)W˙11θ=∫−τ10dηθW11θ=−H11θ,(65a)H200=−g20q0−g¯02q¯0+2α,−1−bα,0,(65b)H110=−g11q0−g¯11q¯0+α+α¯,−2−bα¯+α,0T,(65c)iσh0−∫−τ10eiθσh0dηθq0=0,(65d)−iσh0−∫−τ10e−iθσh0dηθq¯0=0.Substituting equations (62) and (65a)–(65d) into (64a) and (64b), there is(66)2iσh0I−∫−τ10eiθσh0dηθE1=2α,−1−bα,0T.Therefore,(67)2iσh0+Π0−112iσh0+b0102iσh0+cE1=2α−1−bα0,where(68a)Π=a−1b−k1−k2+k2e−2iσh0τ20,(68b)fτ1,τ20=k1e−2iσh0τ1+k2e−2iσh0τ20.Let(69)E11=Δ11Δ1,E12=Δ12Δ1,E13=Δ13Δ1,in which(70a)Δ11=2α0−1−2−2bα2iσh00002iσh0+c,(70b)Δ12=2iσh0+Π2α−11−2−2bα0102iσh0+c,(70c)Δ13=2iσh0+Π02α12iσh0+b−b−2bα100,(70d)Δ1=2iσh0+a−1b−k1−k2+fτ1,τ200−112iσh0+b0102iσh0+c.Similarly, we can also have(71)a−1b−k1−k20−11b01bcE2=α+α¯−2−bα¯+α0.Let(72)E21=Δ21Δ2,E22=Δ22Δ2,E23=Δ23Δ2,where(73a)Δ21=α+α¯0−1−2−bα¯+αb010c,(73b)Δ22=a−1b−k1−k2α+α¯01−2−bα¯+α010c,(73c)Δ23=a−1b−k1−k20α+α¯1b−2−bα¯+α100,(73d)Δ2=a−1/b−k1−k20−11b010c.Furthermore,gij can be determined by the coefficients and time delays of system (7a)–(7c). Thus, the following values can be calculated using the method in [37]:(74a)C10=i2σh0g20g11−2g112−13g022+g212,(74b)μ2=−ReC10edλ/dτ20,(74c)T2=−ImC10+μ2Imdλ/dτ20σh0,(74d)β2=2ReC10.Formulas (74a)–(74d) determine the critical point above the central flow τ20. Now, the properties of periodic solutions of system (3a)–(3c) could be obtained. Therefore, we obtain the following theorem.Theorem 3.
For system (3a)–(3c), when τ1∈0,τ10,(1)
The symbolμ2 determines the orientation of Hopf bifurcation. If μ2>0 μ2<0, then the Hopf bifurcation is supercritical (subcritical). If τ2>τ20 τ2<τ20, the bifurcation period solution exists.(2)
The sign ofβ2 determines the stability of the bifurcation period solution: if β2<0 β2>0, the period solution is stable (unstable).(3)
The symbolT2 determines the period of bifurcation period solution. If T2>0 T2<0, the period solution is increased (decreased).


## 4. Numerical Results and Analysis

In this section, we will give numerical simulations on the theoretical results of Hopf bifurcation with two delays.Given the parametersa=3, b=0.51, c=1.0, k1=1, and k2=5, it is verified that at that time τ1=τ2=0, the parameters satisfy the assumption H1 that the equilibrium point 0,1/b,0 of system (3a)–(3c) without time delay is asymptotically stable.(1)
τ1>0,τ2=0If equation (8) has a pair of pure virtual root iw and iw, there are two positive roots w1=0.20211 and w2=1.39915. Substituting w1 and w2 into equations (15a) and (15b), we can obtain(75a)τ11,j=15.5044+31.0872j,(75b)τ12,j=1.39921+4.49073j,j=0,1,2….That is,τ10=1.39921, and we have the following result for τ1=τ10:(76)H3:dReλdτ1=0.23975>0.According to Theorem1, equilibrium point 0,1/b,0 of system (3a)–(3c) is asymptotically stable for τ1∈0,τ10. Hopf bifurcation occurs when τ1=τ10, τ2=0, and a stable period solution is obtained. The time series diagram and phase diagram are shown in Figure 2.(2)
τ1>0,τ2>0Letτ1=0.45<τ10 and consider τ2>0. Suppose that equation (22) has a pair of pure virtual root iσ and −iσ. Then, according to equation (24), it is found that N=1. There is σ1=3.05932. Substituting σ1 into equations (25a) and (25b), we can obtain(77)τ22,j=0.11917+2.05378j,j=0,1,2….That is,τ20=0.11917 and(78)dReλdτ2=0.227516>0.According to Theorem2, we know when τ2∈0,τ20, the equilibrium point 0,1/b,0 is asymptotically stable (as shown in Figure 3: τ1=0.45<τ10 and τ2=0.1<τ20).Whenτ1=0.45 and τ2=τ20, Hopf bifurcation occurs and we can have(79a)C10=−2.29734+4.10025i,(79b)μ2=0.92535>0,(79c)β2=−4.59468<0.Therefore, the bifurcation direction isτ2>τ20, and the period solution is stable, as shown in Figure 4: τ1=0.45 and τ2=0.15. In addition, the complex dynamics of system (3a)–(3c) could be shown from the bifurcation diagram in Figure 5.Figure 2
The phase diagram (a) and the time series diagram (b) of system (3a)–(3c) with initial value 0.1,2,0.1 for τ1=1.5>τ10 and τ2=0.
(a)
(b)Figure 3
Phase diagram (a) and time sequence diagram (b) of system (3a)–(3c) with initial value 0.1,2,0.1 for τ1=0.45<τ10 and τ2=0.1<τ20.
(a)
(b)Figure 4
Phase space (a) and time series (b) of system (3a)–(3c) with initial value 0.1,2,0.1 for τ1=0.45<τ10 and τ2=0.1<τ20.
(a)
(b)Figure 5
Bifurcation diagram of system (3a)–(3c) with initial value 0.1,2,0.1 and τ1=0.45<τ10 and τ2∈1,4.


## 5. Multistability in the Improved Financial System (3a)–(3c) with Two Delays

Whenτ1=0.45 and τ2=3.8, we will find the multistability in the systems without changing parameters. Given the parameters a=3, b=0.51, c=1.0, k1=1, and k2=5, we have obtained the periodic attractor with initial value 0.01,1,0.01 in Figure 6(a). However, we also obtain chaos for same parameters’ values but different initial values 0.1, 0, 0 in Figure 6(b). Therefore, when different initial conditions are taken, the coexisting and different attractors are exhibited. We know multistability is a critical property of nonlinear dynamical systems [41–44]. Since the crisis of the financial system is subject to various factors, the nature of the multi-steady state plays an important role in making correct decisions for government workers.Figure 6
Time series of system (3a)–(3c) with τ1=0.45<τ10 and τ2=3.8: (a) initial value 0.01,1, 0.01 and periodic orbit; (b) initial value 0.1, 0,0 and means chaos.
(a)
(b)


## 6. Conclusion

Time delay is a very sensitive factor in financial systems with multistability. Financial systems with multiple time delays have richer dynamic characteristics than those with single time delay. Two-delay feedback can effectively control the unstable behavior of financial markets. In this paper, Hopf bifurcation of an improved financial model with two time delays is studied in detail. The existence of the bifurcation period solution of this system is discussed by using the theory of functional differential equations. Complexity of the proposed financial chaotic system is studied from the bifurcation diagram that its multistability depends extremely on the memristor initial condition and the system parameters. In summary, time delay is one of the effective methods to control the stability of the financial market, so it can provide a theoretical reference for relevant departments to regulate economic behavior.



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