# P D ϑ Control Strategy for a Fractional-Order Chaotic Financial Model

Authors: Changjin Xu; Maoxin Liao; Peiluan Li; Qimei Xiao; Shuai Yuan
Journal: Complexity
 (2019)

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## Abstract

In this article, based on the previous works, a new fractional-order financial model is put up. The chaotic behavior of the fractional-order financial model is suppressed by designing an appropriatePDϑ controller. By choosing the delay as the bifurcation parameter, we establish the sufficient condition to guarantee the stability and the existence of Hopf bifurcation of fractional-order financial model. Also, the influence of the delay and the fractional order on the stability and the existence of Hopf bifurcation of fractional-order financial model is revealed. An example is given to confirm the effectiveness of the analysis results. The main findings of this article play an important role in maintaining economic stability.


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## Body

## 1. Introduction

Establishing financial models to investigate the complex dynamical behavior of economic society has attracted more attention of scholars in numerous areas. To grasp the law of operation accurately, various financial models have been established to reveal the inherent characteristics of economic development and numerous fruitful results are achieved. For instance, Zhang et al. [1] discussed the stability of a financial hyperchaotic model, Serletic [2] investigated the chaos in economic system, Lin et al. [3] made an detailed analysis on chaotic behavior of a financial complex model, and Gao and Ma [4] studied the chaotic phenomenon and bifurcation of a finance model. For more information on financial models, one can see [5–9].In many cases, chaotic behavior often happens in financial models. Chaotic phenomenon will have a serious effect on man’s everyday life. Thus the research on chaos control of financial models becomes a hot issue in financial community. The appearance of chaotic phenomenon in economic system implies that the macroeconomic operation has its inherent indefiniteness and complexity. Of course, government departments can take measures for control or interference, but the effect is very limited [10]. Thus, it is worthwhile to deal with the control of chaos in financial systems by theoretical analysis.In 2001, Ma and Chen [11, 12] studied the following financial model:(1)du1dt=u3+u2-p1u1,du2dt=1-p2u2-u12,du3dt=-u1-p3u3,where p1≥0 denotes the saving amount, p2≥0 denotes the cost per investment, p3≥0 denotes the elasticity of demand of commercial markets, u1 represents interest rate, u2 represents investment demand, and u3 represents price index. In 2008, Chen [13] designed a suitable time delayed feedback controller to control the chaotic phenomena of model (1); computer simulations are presented to illustrate the effectiveness of designed controller. In 2011, Son and Park [14] further dealt with the chaos control issue of model (1) by applying delayed feedback method. Detailed theoretical analysis and numerical simulations are carried out to check the correctness of the controller. In 2009, Gao and Ma [4] discussed the chaos control of model (1) by adding a time delay feedback term to the second equation of system (1). Also the sufficient condition to guarantee the stability and the existence of Hopf bifurcation of involved controlled financial model is established. Considering the effect of time delay on the financial phenomena, Zhang and Zhu [15], Chen et al. [16], Mircea et al. [17], and Zhang [18] established different delayed financial models and analyzed their Hopf bifurcation or chaos control issue. In 2014, Zhao et al. [19] investigated the the anticontrol of Hopf bifurcation and chaos control of model (1) by applying delayed washout filters. For details, one can see [5–7, 20–30].Here we would like to point out that all the above works are only concerned with the integer-order differential systems. Nowadays, numerous scholars have found that fractional calculus, which is a generalization of ordinary differentiation and integration, has potential applications in numerous fields such as economics, physics, heat transfer, and chemical engineer [31–38]. Many researchers argued that it is more reasonable to describe the object natural phenomena by fractional-order differential equations than integer-order differential equations, since fractional-order differential equations can better describe the memory characteristics and historical dependence. Noticing that financial coefficients possess very long memory and the variation of financial coefficients has close connection with previous and current time, we think that it is important for us to establish fractional-order financial systems. In recent years, there are numerous articles that investigate the fractional-order financial systems. One can see [11, 12, 39–47].In view of the above analysis and based on system (1), we modify system (1) as the following fractional-order financial model:(2)dϑu1dtϑ=u3+u2-p1u1,dϑu2dtϑ=1-p2u2-u12,dϑu3dtϑ=-u1-p3u3,where 0<ϑ<1 stands for the fractional order. The study reveals that chaotic phenomenon will appear if ϑ=0.73 and p1=3,p2=0.1,p3=1. The results can be shown in Figures 1–10.Figure 1
The relation oft and u1 in model (2) when ϑ=0.73, p1=3, p2=0.1, and p3=1.Figure 2
The relation oft and u2 in model (2) when ϑ=0.73, p1=3, p2=0.1, and p3=1.Figure 3
The relation oft and u3 in model (2) when ϑ=0.73, p1=3, p2=0.1, and p3=1.Figure 4
The relation ofu1 and u2 in model (2) when ϑ=0.73, p1=3, p2=0.1, and p3=1.Figure 5
The relation ofu1 and u3 in model (2) when ϑ=0.73, p1=3, p2=0.1, and p3=1.Figure 6
The relation ofu2 and u3 in model (2) when ϑ=0.73, p1=3, p2=0.1, and p3=1.Figure 7
The relation ofu1, u2, and u3 in model (2) when ϑ=0.73, p1=3, p2=0.1, and p3=1.Figure 8
The relation oft, u1, and u2 in model (2) when ϑ=0.73, p1=3, p2=0.1, and p3=1.Figure 9
The relation oft, u2, and u3 in model (2) when ϑ=0.73, p1=3, p2=0.1, and p3=1.Figure 10
The relation oft, u2, and u3 in model (2) when ϑ=0.73, p1=3, p2=0.1, and p3=1.Our key task is concerned with two topics: (i) designing a suitablePDϑ controller to suppress the chaotic behavior of system (2) and (2) seeking the influence of delay and fractional order on the stability and bifurcation phenomenon of controlled system.The superiority of this article is stated as follows:(I)
A new fractional-order financial model is established.(II)
APDϑ controller is designed to suppress the chaos of the fractional-order financial model. Also, a set of new sufficient conditions to guarantee the stability and the existence of Hopf bifurcation of fractional-order financial model are obtained. In addition, the effect of the delay and fractional order on the dynamics of fractional-order financial model is displayed.(III)
To the best of our knowledge, it is the first time to control chaos of fractional-order financial model by applyingPDϑ controller.We organize this article as follows. In Section2, some basic knowledge on fractional calculus is presented. In Section 3, PDϑ controller is designed to control chaos of fractional-order financial model. In Section 4, an example is given to show the effectiveness of the main findings. A conclusion is presented in Section 5.


## 2. Basic Results

In this section, we give some basic results on fractional calculus.Definition 1 (see [48]).
Define Caputo fractional-order derivative as follows:(3)Dϑpι=1Γn-ϑ∫ι0ιpmsι-sϑ-m+1ds,where p(ι)∈([ι0,∞),R),Γ(s)=∫0∞‍ιs-1e-ιdι.  ι≥ι0 and m is a positive integer such that m-1≤ϑ<m. If 0<ϑ<1, then(4)Dϑpι=1Γ1-ϑ∫ι0ιp′sι-sϑds.Definition 2 (see [49]).
The point(u1∗,u2∗,u3∗) is called an equilibrium point if the equations(5)u3∗+u2∗-p1u1∗=0,1-p2u2∗-u1∗2=0,-u1∗-p3u3∗=0hold.Lemma 3 (see [50]).
Letλi(i=1,2,…,n) be the root of the characteristic equation of the autonomous system Dϑu=Bu,u(0)=u0, where 0<ϑ<1,u∈Rn,B∈Rn×n. Then system Dϑu=Bu is asymptotically stable if and only if argλi>ϑπ/2(i=1,2,…,n). The system Dϑu=Bu is stable if and only if argλi>ϑπ/2(i=1,2,…,n) and those critical eigenvalues that satisfy argλi=ϑπ/2(i=1,2,…,n) have geometric multiplicity one.Lemma 4 (see [51]).
The systemDϑv(t)=A1v(t)+A2v(t-ρ), where v(t)=ϕ(t),t∈[-ρ,0],ϑ∈(0,1],v∈Rn,A1,A2∈Rn×n,ϑ∈R+(n×n). Then the characteristic equation of the system is det⁡sϑI-A1-A2e-sρ=0. If all the roots of the characteristic equation of the system have negative real roots, then the zero solution of the system is asymptotically stable.


## 3. Chaos Control byPDϑ Controller

Ifp2+p1p2p3-p3>0, then system (2) has a unique equilibrium point:(6)u10,u20,u30=0,1p2,0.If p2+p1p2p3-p3<0, then model (2) has three equilibrium points:(7)u10,u20,u30=0,1p2,0,u1∗,u2∗,u3∗=p3-p2-p1p2p3p3,1+p1p3p3,-p3-p2-p1p2p3p323,u1∗∗,u2∗∗,u3∗∗=-p3-p2-p1p2p3p3,1+p1p3p3,p3-p2-p1p2p3p323.During the past several decades, many different control strategies have been applied to control the chaos and bifurcation behavior. But they only involve the integer-order differential systems. Applying PDϑ control strategy to control the chaotic behavior of fractional-order chaotic system is rare. To make up for the deficiency, we try to design a PDϑ feedback controller to suppress the chaotic phenomenon of model (2). In this paper, we only consider the equilibrium point (u1∗,u2∗,u3∗). The other equilibrium points can be discussed in the same way. Here we omit it. Throughout this paper, we always assume that p2+p1p2p3-p3<0.Add the followingPDϑ feedback controller to the first equation of system (2):(8)ϱt=κpu1t-ρ-u1∗+κddϑdtϑu1t-u1∗,where κp and κd are the proportional control parameter and the derivative control parameter, respectively, and ρ is time delay; then system (2) becomes(9)dϑu1dtϑ=u3+u2-p1u1+ϱt,dϑu2dtϑ=1-p2u2-u12,dϑu3dtϑ=-u1-p3u3.That is,(10)dϑu1dtϑ=u3+u2-p1u1+κpu1t-ρ-u1∗+κddϑdtϑu1t-u1∗,dϑu2dtϑ=1-p2u2-u12,dϑu3dtϑ=-u1-p3u3.Let vi(t)=ui(t)-ui(t)(i=1,2,3); then linear system of system (10) takes the form(11)dϑv1dtϑ=11-κdu2∗-p1v1+v2+u1∗v3+κpv1t-ρ,dϑv2dtϑ=2u1∗v1-p2v2,dϑv3dtϑ=-v1-p3v3.The corresponding characteristic equation of (11) is given by(12)det⁡sϑ-u2∗-p11-κd-κp1-κde-sρ-11-κd-u1∗1-κd2u1∗sϑ+p2010sϑ+p3.Then(13)Q1s+Q2se-sρ=0,where (14)Q1s=s3ϑ+p2+p3-u2∗-p11-κds2ϑ+p2p3-u2∗-p1p2+p31-κdsϑ-u2∗-p1p2+p31-κd,Q2s=-κp1-κds2ϑ+p2+p3sϑ+p2p3.Assume that s=iζ=ζcos⁡π/2+isin⁡π/2 is a root of (13); then one has(15)Q2Rζcos⁡ζρ+Q2Iϕsin⁡ζρ=-Q1Rζ,Q2Iζcos⁡ζρ-Q2Rϕsin⁡ζρ=-Q1Iζ,where(16)Q1Rζ=ζ3ϑcos⁡3ϑπ2+p2+p3-u2∗-p11-κdζ2ϑcos⁡ϑπ+p2p3-u2∗-p1p2+p31-κdζϑcos⁡ϑπ2-u2∗-p1p2+p31-κd,Q1Iζ=ζ3ϑsin⁡3ϑπ2+p2+p3-u2∗-p11-κdζ2ϑsin⁡ϑπ+p2p3-u2∗-p1p2+p31-κdζϑsin⁡ϑπ2,Q2Rζ=-κp1-κdϑ2ϑcos⁡ϑπ+p2+p3ζϑcos⁡ϑπ2+p2p3,Q2Iζ=-κp1-κdϑ2ϑsin⁡ϑπ+p2+p3ζϑsin⁡ϑπ2.By (15), one has(17)cos⁡ζρ=-Q1RζQ2Rζ+Q1IζQ2IζQ2R2ζ+Q2I2ζ,sin⁡ζρ=-Q1RζQ2Iζ-Q1IζQ2RζQ2R2ζ+Q2I2ζ.Let(18)a1=cos⁡3ϑπ2,a2=p2+p3-u2∗-p11-κdcos⁡ϑπ,a3=p2p3-u2∗-p1p2+p31-κdcos⁡ϑπ2,a4=-u2∗-p1p2+p31-κd,a5=sin⁡3ϑπ2,a6=p2+p3-u2∗-p11-κdsin⁡ϑπ,a7=p2p3-u2∗-p1p2+p31-κdsin⁡ϑπ2,a8=-κpcos⁡ϑπ1-κd,a9=-κpp2+p31-κd,a10=-κpp2p31-κd,a11=-κpsin⁡ϑπ1-κd,a12=-κpp2+p3sin⁡ϑπ/21-κd.Then(19)Q1Rζ=a1ζ3ϑ+a2ζ2ϑ+a3ζϑ+a4,Q1Iζ=a5ζ3ϑ+a6ζ2ϑ+a7ζϑ,Q2Rζ=a8ζ2ϑ+a9ζϑ+a10,Q2Iζ=a11ζ2ϑ+a12ζϑ.It follows from (17) that(20)Q1RQ2R+Q1IQ2I2+Q1RQ2I-Q1IQ2R2=Q2R2ϕ+Q2I2ϕ2.Notice that(21)Q1RζQ2Rζ+Q1IζQ2Iζ2=b1ϕ5ϑ+b2ϕ4ϑ+b3ζ3ϑ+b4ζ2ϑ+b5ζϑ+b62,Q1RζQ2Iζ-Q1IζQ2Rζ2=b7ϕ5ϑ+b8ϕ4ϑ+b9ϕ3ϑ+b10ζ2ϑ+b11ζϑ2,Q2R2ζ+Q2I2ζ2=b12ζ4q+b13ζ3ϑ+b14ζ2ϑ+b15ζϑ2,where(22)b1=a1a8+a5a11,b2=a2a8+a1a9+a6a11+a5a12,b3=a3a8+a2a9+a1a10+a7a11+a6a12,b4=a4a8+a3a9+a2a10+a7a12,b5=a4a9+a3a10,b6=a4a10,b7=a1a11-a5a8,b8=a2a11+a1a12-a5a9-a6a8,b9=la3a11+a2a12-a5a10-a6a9-a7a8,b10=a4a11+a3a12-a6a10-a7a9,b11=a4a12-a7a10,b12=a82+a112,b13=2a8a9+a11a12,b14=2a8a10+a122,b15=2a9a10.By (20), one has(23)ϱ1ζ10ϑ+ϱ2ζ9ϑ+ϱ3ζ8ϑ+ϱ4ζ7ϑ+ϱ5ζ6ϑ+ϱ6ζ5ϑ+ϱ7ζ4ϑ+ϱ8ζ3ϑ+ϱ9ζ2ϑ+ϱ10ζϑ+ϱ11=0,where(24)ϱ1=b12+b72,ϱ2=2b1b2+2b7b8,ϱ3=b22+b82-b122+2b1b3+2b7b9,ϱ4=2b2b3+b1b2+b7b10+b8b9-b12b13,ϱ5=b32+2b1b5+2b2b4+b92+2b7b11+2b8b10-b132-2b12b14,ϱ6=2b1b6+2b2b5+2b3b4+2b9b10-2b12b15-2b13b14,ϱ7=b42+2b3b5+b102+2b8b11+2b9b11-b142-2b13b15,ϱ8=2b3b6+2b4b5+2b10b11-2b14b15,ϱ9=b52+2b4b6+b112-b152,ϱ10=2b5b6,ϱ11=b62.Let(25)Φζ=ϱ1ζ10ϑ+ϱ2ζ9ϑ+ϱ3ζ8ϑ+ϱ4ζ7ϑ+ϱ5ζ6ϑ+ϱ6ζ5ϑ+ϱ7ζ4ϑ+ϱ8ζ3ϑ+ϱ9ζ2ϑ+ϱ10ζϑ+ϱ11and(26)γν=ϱ1ν10+ϱ2ν9+ϱ3ν8+ϱ4ν7+ϱ5ν6+ϱ6ν5+ϱ7ν4+ϱ8ν3+ϱ9ν2+ϱ10ν+ϱ11.Lemma 5.
(a) Ifϱi>0(i=1,2,3,…,11), then (13) has no root with zero real parts; (b) if ϱ11>0 and there exists a constant ν0>0 such that γ(ν0)<0, then (13) has at least two pairs of purely imaginary roots.Proof.
(a) It follows from (25) that(27)dΦζdζ=10ϑϱ1ζ10ϑ-1+9ϑϱ2ζ9ϑ-1+8ϑϱ3ζ8ϑ-1+7ϑϱ4ζ7ϑ-1+6ϑϱ5ζ6ϑ-1+5ϑϱ6ζ5ϑ-1+4ϑϱ7ζ4ϑ-1+3ϑϱ8ζ3ϑ-1+2ϑϱ9ζ2ϑ-1+ϑϱ10ζϑ-1.Notice that ϱi>0(i=1,2,3,…,10); then dΦ(ζ)/dζ>0,∀ζ>0. By Φ(0)=ϱ11>0, one knows that (26) has no positive real root. In addition, s=0 is not the root of (13). We complete the proof of (a).
(b) Notice thatγ(0)=ϱ11>0,γ(η0)<0(η0>0) and limμ→+∞γ(μ)/dμ=+∞; then ∃η01∈(0,η0) and ε02∈(η0,+∞) such that γ(η01)=γ(η02)=0, and then (25) has at least two positive real roots. Thus (13) has at least two pairs of purely imaginary roots. We complete the proof of (b).Without loss of generality, one can assume that (23) has ten positive real roots ζj(j=1,2,…,10). By (13), we obtain(28)ρjl=1ζjarccos⁡-Q1RζjQ2Rζj+Q1IζjQ2IζjQ2R2ζj+Q2I2ζj+2lπ,where l=0,1,2,…,j=1,2,…,10. Then ±iζj is a pair of purely imaginary roots of (10) when ρ=ρjl. Let(29)ρ0=minj=1,2,…,11ρj0,ζ0=ζρ=ρ0.Now we give the following assumption:( A 1 )   U 1 V 1 + U 2 V 2 > 0, where (30)U1=3ϑζ03ϑ-1cos⁡3ϑ-1π2+2ϑp2+p3-u2∗-p11-κdζ02ϑ-1cos⁡2ϑ-1π2+ϑp2p3-u2∗-p1p2+p31-κdζ0ϑ-1cos⁡ϑ-1π2-κp1-κd2ϑζ02ϑ-1cos⁡2ϑ-1π2+ϑp2+p3ζ0ϑ-1cos⁡ϑ-1π2cos⁡ζ0ρ0-κp1-κd2ϑζ02ϑ-1sin⁡2ϑ-1π2+ϑp2+p3ζ0ϑ-1sin⁡ϑ-1π2sin⁡ζ0ρ0,U2=3ϑζ03ϑ-1sin⁡3ϑ-1π2+2ϑp2+p3-u2∗-p11-κdζ02ϑ-1sin⁡2ϑ-1π2+ϑp2p3-u2∗-p1p2+p31-κdζ0ϑ-1sin⁡ϑ-1π2+κp1-κd2ϑζ02ϑ-1cos⁡2ϑ-1π2+ϑp2+p3ζ0ϑ-1cos⁡ϑ-1π2sin⁡ζ0ρ0-κp1-κd2ϑζ02ϑ-1sin⁡2ϑ-1π2+ϑp2+p3ζ0ϑ-1sin⁡ϑ-1π2cos⁡ζ0ρ0,V1=-κp1-κdζ02ϑcos⁡ϑπ+p2+p3ζ0ϑcos⁡ϑπ2+p2p3ζ0sin⁡ζ0ρ0+κp1-κdζ02ϑsin⁡ϑπ+p2+p3ζ0ϑsin⁡ϑπ2+p2p3ζ0cos⁡ζ0ρ0,V2=-κp1-κdζ02ϑcos⁡ϑπ+p2+p3ζ0ϑcos⁡ϑπ2+p2p3ζ0cos⁡ζ0ρ0-κp1-κdζ02ϑsin⁡ϑπ+p2+p3ζ0ϑsin⁡ϑπ2+p2p3ζ0sin⁡ζ0ρ0.Lemma 6.
Ifs(ρ)=ξ(ρ)+iψ(ρ) is the root of (13) near ρ=ρ0 which satisfies ξ(ρ0)=0,ψ(ρ0)=ψ0, then Reds/dρρ=ρ0,ψ=ψ0>0.Proof.
By (13), one has(31)Q1sdρ+Q2sdρe-sρ-e-sρdsdρρ+sQ2s=0.Notice that (32)Q1sdρ=3ϑs3ϑ-1dsdρ+2ϑp2+p3-u2∗-p11-κds2ϑ-1dsdρ+ϑp2p3-u2∗-p1p2+p31-κdsϑ-1dsdρ=3ϑs3ϑ-1+2ϑp2+p3-u2∗-p11-κds2ϑ-1+ϑp2p3-u2∗-p1p2+p31-κdsϑ-1dsdρ,Q2sdρ=-κp1-κd2ϑs2ϑ-1+ϑp2+p3sϑ-1dsdρ,and then one has(33)dsdρ-1=F1ρF2ρ-ρs,where(34)F1s=3ϑs3ϑ-1+2ϑp2+p3-u2∗-p11-κds2ϑ-1+ϑp2p3-u2∗-p1p2+p31-κdsϑ-1-κp1-κd2ϑs2ϑ-1+ϑp2+p3sϑ-1e-sρ,F2s=Q2sse-sρ.Then(35)Redsdρ-1=ReF1ρF2ρ.Thus(36)Redsdρρ=ρ0,ψ=ψ0=ReF1ρF2ρρ=ρ0,ψ=ψ0=U1V1+U2V2V12+V22.It follows from (A1) that(37)Redsdρ-1ρ=ρ0,ψ=ψ0>0.Th proof of Lemma 6 is completed.Let(38)ϵ1=p2+p3-u2∗-p11-κd-κp1-κd,ϵ2=p2p3-u2∗-p1-κpp2+p31-κd,ϵ3=-κpp2p3+u2∗-p1p2+p31-κd.Next we give an assumption as follows:( A 2 )   ϵ 1 > 0 ,   ϵ 1 ϵ 2 > 2 ϵ 3 ,   ϵ 3 > 0.Lemma 7.
Ifρ=0 and (A2) holds true, then system (9) is locally asymptotically stable.Proof.
Ifρ=0, then (13) takes the form(39)λ3+ϵ1λ2+ϵ2λ+ϵ3=0.By (A2), all roots λi of (39) satisfy argλi>ϑπ/2(i=1,2,3). By Lemma 3, we know that the conclusion of Lemma 4.3 holds. The proof of Lemma 3.4 is completed.According to the analysis above, we have the following conclusion.Theorem 8.
In addition to condition (b) of Lemma6. If (A1) and (A2) are fulfilled, then the equilibrium point (u1∗,u2∗,u3∗) of system (9) is locally asymptotically stable when ρ∈[0,ρ0) and a Hopf bifurcation will appear around the equilibrium point E1 when ρ=ρ0.Remark 9.
In [4–7, 13–20], the authors studied the various dynamics of integer-order financial models. They did not involve the fractional-order forms. In [39], Bhalekar and Gejji considered the chaos of fractional-order financial model by predictor-corrector method. In this article, we control the chaos of fractional-order financial model by applying PDϑ control strategy. All the derived results and analysis ways of [4–7, 13–20, 39] can not be transferred to (2) to control the chaotic behavior. Based on these viewpoints, the fruits of this paper are entirely innovative and supplement the previous publications.


## 4. An Example

We give the following controlled financial system:(40)dϑu1dtϑ=u3+u2-3u1+κpu1t-ρ-u1∗+κddϑdtϑu1t-u1∗,dϑu2dtϑ=1-0.1u2-u12,dϑu3dtϑ=-u1-u3.Clearly, system (40) has an equilibrium point (0.6831,2.6667,-0.4554). Let κp=0.07,κp=0.32,ϑ=0.672. Then the critical frequency ζ0=0.8653 and the bifurcation point ρ0=0.5262. We can check that the assumptions in Theorem 8 hold true. Figures 11–20 indicate that when ρ∈[0,0.5262), the equilibrium point (0.6831,2.6667,-0.4554) of model (40) is locally asymptotically stable. From the financial point of view, it means that as time goes on, the interest rate will tend to the constant 0.6831, investment demand will tend to the constant 2.6667, and price index will tend to the constant -0.4554. Figures 21–30 indicate that ρ∈[0.5262,+∞), system (40) becomes unstable, and a Hopf bifurcation emerges. From the financial point of view, it means that as time goes on, the interest rate, investment demand, and price index will keep a periodic cycle. In addition, we show the relation of parameters ϑ and ρ0 of (40) with Table 1.Table 1
The relation of parametersϑ and ρ0 of model (40).
ϑ ρ 0 0.195 0.1968 0.267 0.2569 0.384 0.3450 0.558 0.4593 0.672 0.5262 0.736 0.5615 0.829 0.6104Figure 11
The relation oft and u1 in model (40) when ρ=0.4237<ρ0=0.5262.Figure 12
The relation oft and u2 in model (40) when ρ=0.4237<ρ0=0.5262.Figure 13
The relation oft and u3 in model (40) when ρ=0.4237<ρ0=0.5262.Figure 14
The relation ofu1 and u2 in model (40) when ρ=0.4237<ρ0=0.5262.Figure 15
The relation ofu1 and u3 in model (40) when ρ=0.4237<ρ0=0.5262.Figure 16
The relation ofu2 and u3 in model (40) when ρ=0.4237<ρ0=0.5262.Figure 17
The relation ofu1, u2, and u3 in model (40) when ρ=0.4237<ρ0=0.5262.Figure 18
The relation oft, u1, and u2 in model (40) when ρ=0.4237<ρ0=0.5262.Figure 19
The relation oft, u2, and u3 in model (40) when ρ=0.4237<ρ0=0.5262.Figure 20
The relation oft, u2, and u3 in model (40) when ρ=0.4237<ρ0=0.5262.Figure 21
The relation oft and u1 in model (40) when ρ=0.8345>ρ0=0.5262.Figure 22
The relation oft and u2 in model (40) when ρ=0.8345>ρ0=0.5262.Figure 23
The relation oft and u3 in model (40) when ρ=0.8345>ρ0=0.5262.Figure 24
The relation ofu1 and u2 in model (40) when ρ=0.8345>ρ0=0.5262.Figure 25
The relation ofu1 and u3 in model (40) when ρ=0.8345>ρ0=0.5262.Figure 26
The relation ofu2 and u3 in model (40) when ρ=0.8345>ρ0=0.5262.Figure 27
The relation ofu1, u2, and u3 in model (40) when ρ=0.8345>ρ0=0.5262.Figure 28
The relation oft, u1, and u2 in model (40) when ρ=0.8345>ρ0=0.5262.Figure 29
The relation oft, u2, and u3 in model (40) when ρ=0.8345>ρ0=0.5262.Figure 30
The relation oft, u2, and u3 in model (40) when ρ=0.8345>ρ0=0.5262.


## 5. Conclusions

In this paper, we propose a new fractional-order financial system. To control the chaotic behavior of the fractional-order financial system, we successfully design aPDϑ controller to achieve our goal. By adjusting the proportional and derivative parameters, we can change the stability and Hopf bifurcation character of the considered fractional-order financial system. By regarding the time delay as bifurcation parameter, we have established a new sufficient condition to ensure the stability and the existence of Hopf bifurcation of the fractional-order financial model. Also, the effect of the fractional order and delay on the stability and Hopf bifurcation is revealed. The research idea and the obtained theoretical results of this article enrich and develop the bifurcation and control theory of fractional-order differential equations. The obtained results can provide useful guidance to people in financial community. We can properly adjust the parameter of the PDϑ controller to apply the suggested fractional-order financial models to deal with financially chaotic problems. In addition, we point out that although the PDϑ controller can effectively control the chaos of the fractional-order financial model, it involves multiple parameters: proportional control parameter κp, the derivative control parameter κd, time delay ρ, and fractional order ϑ. In the future, we will seek some more simple controllers with less parameters to suppress the chaotic behavior.



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*Source: 2989204-2019-04-22.xml*