# Dynamic System Analysis of Investment in Both Production and R&D with Time Delay

Authors: Debao Gao
Journal: Journal of Mathematics
 (2022)

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## Abstract

This paper, according to the process of capital return, establishes a differential dynamics model of investment with two time delays. When both time delays are zero, it is proved that the model is positively invariant, uniformly bounded, and globally asymptotically stable by using the comparison principle and Bendixson–Dulac theorem. When at least one time delay is not zero, according to Hopf bifurcation theorem, the conditions of local asymptotic stability and existence of periodic solution of investment model are obtained. By using the normal form theory and the center manifold theory, the discriminant formula of periodic solution property of investment model is given. Under the condition of controlled time delay, the model is numerically simulated to verify the correctness of relevant analytical conclusions. Therefore, the investment model describes the dynamic process and development trend of project investment quite closely.


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## Body

## 1. Introduction

In recent years, differential equation theory [1–4] has been widely applied in other disciplines. For example, the integration of mathematics and biology has formed a new discipline: biological mathematics [5–7]. Differential equation theory also has some applications in scientific calculation [8–12]. There are many similar examples, which will not be explained here. This paper mainly studies the application of differential equation theory in commercial investment.Investment is one of the most common business economic activities. Due to the uncertainty of the investment environment, many unexpected situations may occur in the investment process. Through long-term research and analysis on investment, some scholars [13–15] found that most of the investment processes seldom appeared stable and balanced but dynamic, unstable, and unbalanced due to the time lag of investment projects to achieve capital return.Because the purpose of investment projects is to obtain benefits and the return of funds can only be obtained after a period of time, the investment system itself has time lag [1, 2], which makes the system itself more complicated and the stability analysis of the system more difficult. However, the research on the stability [16–20] of investment system, especially on the modelling and analysis of complex investment phenomena, is still in the development stage. The calculation needs to be expanded. It is the complexity of the investment system itself that makes it necessary to combine the investment system with the theory of delay differential equations in order to better understand the essential attributes of the investment system. Therefore, it is of great significance to study the stability of the time-delay dynamic system both theoretically and practically.The method of constructing an investment model with time-delay differential equations proposed in this paper can not only study the influence of parameters with time deviations on the system more accurately but also describe the dynamic relationship of objective things, which is the uniqueness of delay differential equation itself.


## 2. Model Construction and Basic Results

Some investment projects require four steps: financing, R&D of the product, producing the commodity, and supplying the commodity to demand. While it takes time to produce goods and research and develop products, the effective supply of goods to demand is also an important link.When investing in a project, first of all, the funds are pooled together (referred to as the fund link for short). Then, the fund will be used for R&D (R&D link) and production (production link). Finally, the products of the production link will be purchased by the demand link, realizing the return of fund and obtaining investment profits. When the product is purchased by consumers, it can be regarded that the product is effectively supplied to the demand link. The key is that the products produced in the production link can effectively supply demand. Otherwise, the investment will fail. Because the source of investment profit is the demand link, conversely, money flows from the demand link to the production link, part of the money in the production link flows to the R&D link, part of the money in the R&D link flows to the fund link, and part of the money flowing out of the fund link is the investment profit or cost.The flow of funds mentioned above is consistent with the transmission process in the theory of infectious diseases [21, 22] and the predation process in biological mathematics [23, 24], which are the specific applications of differential equation theory in infectious diseases and biology. This paper simply applies differential equation theory to financial investment.Let the amount of funds in the demand link, production link, R&D link, and fund link at timet be Dt,Pt,Rt, and Ft, respectively. Let a be the coefficient of the entry rate of the demand link and b be the coefficient of the exit rate of the demand link. If it is further assumed that the quantity change of the demand link satisfies the logistic equation [2, 5], the change rate of the quantity of the demand link is(1)dDdt=a−bDt1−kDt.Let the functional response function of production effective supply demand becDP. The consumption rates in the production link and the R&D link are uP and vR, respectively. The rate of transfer from production to R&D is dP. The transfer rate from R&D to fund is eR. The capital return rate of the investment project is fF. Based on the above analysis and assumptions, the currency transfer relationships among demand, production, R&D, and fund are shown in Figure 1.Figure 1
Currency transfer relationships among demand, production, R&D, and fund.In addition, this paper assumes that it takes time to produce products and develop products, and the average time isτ1,τ2, respectively. Then, the differential dynamics system of the investment dynamic model is(2)dDdt=a−bDt1−kDt−cDtPt−τ1,dPdt=cDtPt−d+uPt,dRdt=dPt−vRt−eRt−τ2,dFdt=eRt−fFt,where a,b,c,d,e,f,u,v,k,τ1,τ2 are all positive.The first equation of model (2) is equivalent to(3)Dt=D0exp∫0ta−b1−kDs−cPs−τ1ds.Therefore, ifD0>0, then Dt>0. Similarly, if P0>0, then Pt>0.Taking the mobile phone project as an example, if there is no product update, there is no need for investment or research and development, but consumer’s demand and production will still proceed normally and their numbers are positive, which are consistent withDt>0,Pt>0.Practically, it may be necessary to invest in projects intermittently or continuously. During this period, the capital ofRt and Ft flows reversely, so Rt and Ft should be negative.The initial conditions of system (2) are given by(4)Dt=ϕ1θ,Pt=ϕ2θ,Rt=ϕ3θ,Ft=ϕ4θ,where(5)ϕ1θ,ϕ2θ,ϕ3θ,ϕ4θ∈C−τ,0,ℛ4,ϕi0>0,τ=maxτ1,τ2,i=1,2,3,4.C−τ,0,ℛ4,is the Banach space of continuous functions mapping the interval τ,0 into ℛ4; here,(6)ℛ4=D,P,R,F|D∈0,+∞,P∈0,+∞,R∈−∞,+∞,F∈−∞,+∞.According to the existence and uniqueness theorem [3–5] of differential equation solutions, model (2) has a unique solution Dt,Pt,Rt,Ft that satisfies the initial conditions (4).It is easy to know that if the conditionsH1:a−b>0,H2:c−kd+u>0 hold, then model (2) has a unique positive equilibrium E+D∗,P∗,R∗,F∗, where(7)D∗=d+uc,P∗=a−bc−kd+uc2,R∗=de+vP∗,F∗=efR∗.The positive equilibriumE+ means the equilibrium state of the investment model under the interaction of demand, R&D, production, and fund.Theorem 1.
Ifτ1=τ2=0, all solutions of model (2) with the initial conditions (4) are positive for all t≥0.Proof.
Ifτ1=τ2=0, it is easy to know that the first equation of system (2) is equivalent to(8)Dt=D0exp∫0ta−b1−kDs−cPsds.
Therefore, ifD0>0, then Dt>0.
Similarly, ifP0>0, then Pt>0.
Because(9)dRdt|R=0=dP>0,if R0>0, then Rt>0.
Similarly, ifF0>0, then Ft>0. The proof is completed.Theorem 2.
Ifτ1=τ2=0 and a>b, model (2) is uniformly bounded [5] on ℛ4.Proof.
Defining a function(10)Vt=Dt+Pt+Rt+Ft,it has that(11)dVdt=a−bDt1−kDt−uPt−vRt−fFt<a−bDt1−kDt.
Further, we can get(12)limt⟶+∞Vt≤1k.
Considering the constant positivity ofDt,Pt,Rt,Ft on 0,+∞, it can be known that(13)limt⟶+∞Dt≤1k,limt⟶+∞Pt≤1k,limt⟶+∞Rt≤1k,limt⟶+∞Ft≤1k.
To sum up, the conclusion of Theorem2 is true. The proof is completed.
The characteristic equation of model (2) at positive equilibrium E+ is(14)JD∗,P∗,R∗,F∗−λE=−a−bkD∗−λ−cD∗exp−λτ100cP∗−λ000d−v−eexp−λτ2−λ000e−f−λ=−a−bkD∗−λ−cD∗exp−λτ1cP∗−λ−v−eexp−λτ2−λ0e−f−λ=0,that is,(15)λ+fλ+v+e⋅exp−λτ2λ2+a−bkD∗λ+c2D∗P∗exp−λτ1=0.
It is easy to know that equation (15) has a negative real root λ1=−f<0, and other solutions satisfy the following equation:(16)λ+v+e⋅exp−λτ2λ2+a−bkD∗λ+c2D∗P∗exp−λτ1=0.Theorem 3.
Whenτ1=τ2=0 and a>b, the positive equilibrium E+ is globally asymptotically stable if conditions H1 and H2 hold.Proof.
Whenτ1=τ2=0, equation (16) is transformed into the following equation:(17)λ3+e+v+a−bkD∗λ2+akv−bkv+c2P∗+aek−bekD∗λ+c2e+vD∗P∗=0.
Whena>b, it is easy to know that(18)e+v+a−bkD∗>0,e+v+ka−bD∗c2P∗+a−be+vkD∗−c2e+vD∗P∗>0,c2e+vD∗P∗>0.
According to Routh–Hurwitz theorem [3, 5], all roots of equation (17) have negative real parts. Thus, when τ1=τ2=0 and a>b, all roots of equation (15) have negative real parts. According to Hurwitz theorem [5], the positive equilibrium E+ is locally asymptotically stable.
Take Dulac function [5](19)H=1DPRF,and let(20)a−bD1−kD−cDP=G1,cDP−d+uP=G2,dP−e+vR=G3,eR−fF=G4,and then there is(21)∂HG1∂D+∂HG2∂P+∂HG3∂R+∂HG4∂F=−ka−bPRF−1DR2F−1DRF2<0.
According to Bendixson–Dulac theorem [5], when τ1=τ2=0, model (2) has no limit cycles in its feasible region ℛ4. Combined with the local asymptotic stability of the positive equilibrium E+, it can be known that when τ1=τ2=0, the positive equilibrium E+ is globally asymptotically stable. The proof is completed.


## 3. Local Stability and Existence of Hopf Bifurcations

Theorem 4.
Ifτ1>0,τ2=0, and H1−H2 hold, there exists a positive number τ10. The positive equilibrium E+ is locally asymptotically stable when τ1∈0,τ10 and unstable when τ1>τ10. That is, system (2) undergoes a Hopf bifurcation around the positive equilibrium E+ when τ1=τ10.Proof.
Ifτ1>0,τ2=0, equation (16) is transformed into(22)λ+v+eλ2+a−bkD∗λ+c2D∗P∗exp−λτ1=0.
It is easy to know thatλ1=−v+e<0 is a root of equation (22). The other two roots of equation (22) satisfy the following equation:(23)λ2+a−bkD∗λ+c2D∗P∗exp−λτ1=0.
Letλ=m1im1>0 be a root of (23); then, it has(24)−m12+a−bkD∗m1i+c2D∗P∗cosm1τ1−isinm1τ1=0.
Separating the real and imaginary parts from (23), it has(25)cosm1τ1=m12c2D∗P∗,sinm1τ1=a−bkD∗m1c2D∗P∗.
It follows that(26)m14+a−b2k2D∗2m12−c2D∗P∗2=0.
Sincec2D∗P∗2>0, equation (26) has at least one positive root m10. By substituting m10 into (25), we can obtain(27)τ1n=2nπm10+1m10arccosm102c2D∗P∗,n=0,1,2,⋯.
Defineτ10=minτ1n. Taking the derivative of λ with respect to τ1 in equation (23), we can get(28)dλdτ1−1=2λ+a−bkD∗λc2D∗P∗exp−λτ1−τ1λ.
By substitutingλ=m10i into above equation, we can obtain(29)Rdλdτ1−1|m1=m10,τ1=τ10=−a−bkD∗sinm10τ10+2m10cosm10τ10mc2D∗P∗=−a−b2k2D2+2m102c2D∗P∗2<0.
Based on the above analysis and the Hopf bifurcation theorem in [2, 23, 24], Theorem 4 is true. The proof is completed.Theorem 5.
Ifτ1=0,τ2>0 and H1,H2,H3:e−v>0 hold, there exists a positive number τ20. The equilibrium E+ is locally asymptotically stable when τ2∈0,τ20 and unstable when τ2>τ20. That is, system (2) undergoes Hopf bifurcations around the positive equilibrium E+ when τ2=τ20.Proof.
Ifτ1=0,τ2>0, equation (16) is transformed into(30)λ+v+e⋅exp−λτ2λ2+a−bkD∗λ+c2D∗P∗=0.
Letλ1,λ2 be two solutions of equation (30), and we can obtain(31)λ1+λ2=−a−bkD∗<0,λ1λ2=c2D∗P∗>0.
Thus,λ1 and λ2 have negative real parts. Also, the third root of (30) satisfies the equation(32)λ+v+e⋅exp−λτ2=0.
Supposing thatλ=m2im2>0 is a solution of equation (32), one has(33)m2i+v+ecosm2τ2−isinm2τ2=0.
Next, separating the real and imaginary parts of the above equation,(34)cosm2τ2=−ve,sinm2τ2=m2e,one obtains(35)m22=e2−v2=e−ve+v.
IfH3 holds, then equation (35) has at least one positive root m20. From equation (34),(36)τ2n=2nπm20+1m20arccos−ve,n=0,1,2,⋯.
Assume thatτ20 is the minimum positive value in τ2nn=0,1,2,⋯. Taking the derivative of λ with respect to τ2 in equation (32), we can get(37)dλdτ2−1=exp−λτ2λe−τ2λ.
Furthermore,(38)Rdλdτ2−1|m2=m20,τ2=τ20=−sinm20τ20m20e=−1e2<0.
Hence, Theorem5 holds. The proof is completed.
The solution of characteristic equation (16) is that of equation λ+v+e⋅exp−λτ2=0 or λ2+a−bkD∗λ+c2D∗P∗exp−λτ1=0. According to the proof process of Theorems 4 and 5 and the conclusion of Theorem 3, the following theorem holds.Theorem 6.
Suppose that the assumptionsH1−H3 hold. Model (2) is locally asymptotically stable when 0≤τ1<τ10 and 0≤τ2<τ20. Model (2) undergoes Hopf bifurcations at the positive equilibrium E+ when τ1=τ10,0≤τ2<τ20 or 0≤τ1<τ10,τ2=τ20. In other cases, model (2) loses stability.


## 4. Direction of Hopf Branches and Stability of Periodic Solutions

In this section, we shall apply the normal form theory and the center manifold theory [25, 26] to study the direction of Hopf branches and stability of periodic solutions of system (2) at the positive equilibrium E+.Without loss of generality, supposeτ2<τ10. Let t⟶t/τ1,u1t=Dτ1t−D∗,u2t=Pτ1t−P∗,u3t=Rτ1t−R∗,u4t=Fτ1t−F∗,τ1=τ10+μ,μ∈ℛ; then, system (2) can be written as(39)du1tdt=τ10+μb11u1t−cD∗u2t−1−cu1tu2t−1−ka−bu12t,du2tdt=τ10+μb21u1t+b22u2t+cu1tu2t,du3tdt=τ10+μdu2t−vu3t−eu3t−τ2τ1,du4tdt=τ10+μeu3t−fu4t,where(40)b11=a−b1−2kD∗−cP∗,b21=cP∗,b22=cD∗−d+u.In the spaceC=C−1,0,ℛ4, model (39) can be locally expressed as the following delay differential equation:(41)u˙t=Lμut+Fμ,ut,where ut=u1t,u2t,u3t,u4tT,utθ=ut+θ,Lμ:C⟶ℛ4 is a bounded linear operator, and F:R×C⟶ℛ4 is continuously differentiable, respectively, by(42)Lμ,ϕ=τ10+μA0ϕ0+B0ϕ−τ2τ1+C0ϕ−1,Fμ,ϕ=τ10+μ−cϕ10ϕ2−1−ka−bϕ120cϕ10ϕ2000,A0=b11000b21b22000d−v000e−f,B0=0000000000−e00000,C0=0−cD∗00000000000000,where ϕ=ϕ1θ,ϕ2θ,ϕ3θ,ϕ4θ∈C−1,0,ℛ4.By the Riesz representation theorem, there exists a4×4 matrix function ηθ,μ−1≤θ≤0 such that(43)Lμ,ϕ=∫−10dημ,θϕθ.In fact, if it can choose(44)dηθ,μ=τ10+μA0δθ+B0δθ+τ2τ1+C0δθ+1,where δ is the Dirac delta function, equation (43) is obviously true.If it defines(45)Aμϕθ=dϕθdθ,θ∈−1,0,∫−τ00dημ,θϕθ,θ=0,Rμϕθ=0,θ∈−1,0,fμ,ϕ,θ=0,system (41) is equivalent to(46)u˙t=Aμut+Rμut,when θ=0, where utθ=ut+θ,θ∈−1,0. For any ψ∈C0,1,ℛ4∗, where ℛ4∗ is a four-dimensional complex vector space. It defines an operator A∗(47)A∗ψs=−dψsds,s∈0,1,∫−10ψ−ξdηξ,0,s=0,and a bilinear function(48)ψs,ϕθ=ψ¯T0ϕ0−∫−10∫ξ=0θψ¯Tξ−θdηθϕξdξ,where ηθ=ηθ,0. Then, operators A0 and A∗0 are conjugate operators.Letqθ=1,q1,q2,q3Texpim10τ10θ be the eigenvector of A0 corresponding to im10 and q∗s=D1,q1∗,q2∗,q3∗Texpim10τ10s be the eigenvector of A∗0 corresponding to −im10; then,(49)A0+B0exp−im10τ1+C0exp−im10τ10−im10Eq0=0,0,0,0T,q∗0TA0+B0exp−im10τ1+C0exp−im10τ10+im10E=0,0,0,0,where E is the identity matrix.Solving the above two matrix equations, we can obtain(50)q1=−b21b22−im20,q2=dq1v+eexp−im10τ1+im10,q3=eq2f+im10,q1∗=−b11+im20b21,q2∗=0,q3∗=0.If(51)q∗0,q0=D¯q∗0¯Tq0+D¯τ1q∗0¯TCq0exp−im10τ1+D¯τ10q∗0¯TBq0exp−im10τ10=1,then(52)D¯=1+q1q1∗¯+q2q2∗¯+q3q3∗¯−τ10eq2q2∗¯exp−im10τ10−τ1cq1D∗exp−im10τ1−1.According to the above analysis and calculation, it can be seen that(53)q∗s,qθ=1,q∗s,q¯θ=0.In the remainder of this section, we will compute the coordinates describing the center manifoldC0 at μ=0. Let ut be the solution of (46) at μ=0. Define(54)zt=<q∗,ut>.Wz,z¯,θ=utθ−2Reztqθ.On the center manifoldC0,(55)Wt,θ=Wzt,zt¯,θ=12W20θz2+W11θzz¯+12W02θz¯2+⋯,where z and z¯ are local coordinates for center manifold C0 in the direction of q and q¯. Note that W is real if ut is real. We only consider real solutions. For the solution ut∈C0 of (46), since μ=0,(56)z˙t=<q∗,zt>=im10τ10zt+q∗0¯TF0z,z¯def¯¯im10τ10z+gz,z¯,where(57)F0z,z¯=F0,Wz,z¯,θ+2Rztqθ,gz,z¯=g20z22+g11zz¯+g02z¯22+g21z2z¯2+⋯.By (54), we can obtain(58)utθ=Wz,z¯,θ+2Reztqθ=Wz,z¯,θ+ztqθ+ztqθ¯.Then,(59)u10=z+z¯+12W2010z2+W1110zz¯+12W0210z¯2+⋯,u20=q1z+q1¯z¯+12W2020z2+W1120zz¯+12W0220z¯2+⋯,u2−1=q1ze−im10τ10+q1¯z¯eim10τ10+12W202−1z2+W112−1zz¯+12W022−1z¯2+⋯.It follows together with (56) that(60)gz,z¯=q∗¯0F0,ϕ=τ10D¯ϕ10−cϕ2−1−ka−bϕ10+cq1∗¯ϕ20=τ10D¯z+z¯+12W2010z2+W1110zz¯+12W0210z¯2+⋯−cq1ze−im10τ10+q1¯z¯eim10τ10+12W202−1z2+W112−1zz¯+12W022−1z¯2+⋯−ka−bz+z¯+12W2010z2+W1110zz¯+12W0210z¯2+⋯+cq1∗¯q1z+q1¯z¯+12W2020z2+W1120zz¯+12W0220z¯2+⋯.Comparing the coefficients with (57), it has(61)g20=2τ10D¯−cq1e−im10τ10−ka−b+cq1q1∗¯,g11=τ10D¯−cq1e−im10τ10+q1¯eim10τ10−2ka−b+cq1∗¯q1+q1¯,g02=2τ10D¯−cq1¯eim10τ10−ka−b+cq1∗¯q1¯,g21=2τ10D¯12W202−1+W112−1−ka−b12W2010+W1110+cq1∗¯12W2020+W1120.In the expression ofg21, it is necessary to calculate W20θ,W11θ,θ∈−1,0. According to the calculation process in references [27–29], it can be known that(62)W20θ=g20q0−im10τ10expim10τ10θ+g¯20q¯0−3im10τ10exp−im10τ10θ+E1exp2im10τ10θ,W11θ=g11q0im10τ10expim10τ10θ+g¯11q¯0−im10τ10exp−im10τ10θ+F1,where(63)A0+B0+C0e−2im10τ10−2im10EE1=−2−cq1e−im10τ10−ka−bcq100,A0+B0+C0F1=−−cq1e−im10τ10+q1¯eim10τ10−2ka−bcq1+q1¯00.So, there are(64)E1=2b11−2im10−cD∗e−2im10τ1000b21b22−2im10000d−v−2m10i−e−2im10τ10000e−f−2im10−1⋅cq1e−im10τ10+ka−b−cq100,F1=b11−cD∗00b21b22000d−v−e000e−f−1⋅cq1e−im10τ10+q1¯eim10τ10+2ka−b−cq1+q1¯00.In summary, the key parametersg20,g11,g02,g21 can be represented by the parameters in model (2). Then, the following parameters can be calculated as:(65)c10=i2m10τ10g20g11−2g112−13g022+g212,μ2=−Rec10Reλ′τ10,β2=2Rec10,T2=−Imc10+μ2Imλ′τ10m10τ10.Theorem 6.
For system (2), if H1−H3 hold, then(1)
The Hopf bifurcation is supercritical (subcritical) ifμ2>0μ2<0.(2)
The bifurcating periodic solutions are stable (unstable) ifβ2<0β2>0.(3)
The period of the bifurcating periodic solutions increases (decreases) ifT2>0T2<0.


## 5. Numerical Simulations

In this section, in order to support the theoretical results, the dynamic behavior of system (2) with the same coefficient and different delays will be shown.It first considers model (2) with (66)a=4.6,b=4,c=0.3,d=0.7,e=0.6,f=2,k=0.1,u=0.6,v=0.5, and the initial value(67)D0,P0,R0,F0=3.5,3,2,1.By computing, one may obtain that(68)m10=0.64,τ10=0.65,m20=0.33,τ20=7.71,E+D∗,P∗,R∗,F∗=4.33,1.13,0.72,0.22,and H1−H3 hold.The computer simulations (see Figure2) show that E+ is globally asymptotically stable when τ1=τ2=0.Figure 2
Numerical simulation diagram of system (2) when τ1=0,τ2=0.Whenτ1=0.45<τ10,τ2=0, model (2) is locally asymptotically stable at E+. Also, its numerical simulation is shown in Figure 3.Figure 3
Numerical simulation diagram of system (2) when τ1=0.45,τ2=0.If it takesτ1=0.65=τ10,τ2=0, model (2) will undergo Hopf branch. In other words, model (2) has periodic solutions, and its numerical simulation is shown in Figure 4.Figure 4
Numerical simulation diagram of system (2) when τ1=0.65,τ2=0.Whenτ1=0, τ2=5<τ20 or τ1=0.5<τ10,τ2=5<τ20, model (2) is locally asymptotically stable at the positive equilibrium E+. When τ1=0 and τ2=7.71=τ20, model (2) will undergo Hopf branch. Their numerical simulation diagrams are shown in Figures 5–7, respectively.Figure 5
Numerical simulation diagram of system (2) when τ1=0,τ2=5.Figure 6
Numerical simulation diagram of system (2) when τ1=0.5,τ2=5.Figure 7
Numerical simulation diagram of system (2) when τ1=0,τ2=7.71.Whenτ1=0.65=τ10,τ2=7.71=τ20, model (2) has periodic solutions (see Figure 8). If τ1=0.5<τ10,τ2=8.2>τ20, model (2) is unstable (see Figure 9).Figure 8
Numerical simulation diagram of system (2) when τ1=0.65,τ2=7.71.Figure 9
Numerical simulation diagram of system (2) when τ1=0.5,τ2=8.2.In Figures5–8, there are cases where Pt and Ft are negative values. Figures 5 and 6 show that investors need to invest continuously in the early stage of investment, while funds begin to flow back and investors continue to reap benefits in the middle and later stages of investment. Figures 7 and 8 show that investors need to invest capital periodically to obtain periodic returns.Figures2, 3, 5, and 6 show that when τ1<τ10 and τ2<τ20, the amount of currency corresponding to demand, production, R&D, and fund in the process of investment will tend to balance globally or locally. Figures 4, 7, and 8 show that there is phenomenon of circular investment of investment funds, while demand, output, and R&D funds may tend to be balanced or stable. Figure 9 shows a vicious cycle of investment where the demand and output tend to stabilize gradually, the R&D cycle appears in an incremental manner, and the amount of investment increases gradually.As can be seen from Figure2, when the R&D cycle τ1=0 and production cycle τ2=0, there is little dynamic fluctuation in the investment process. It can be seen from Figures 3, 5, and 6 that when the R&D cycle τ1 is less than τ10 and the production cycle τ2 is less than τ20, the investment dynamics will tend to be stable over time. Figures 4, 7, and 8 show that when the R&D cycle τ1=τ10 or the production cycle τ2=τ20, the R&D quantity, the production quantity, and the demand quantity will cyclically fluctuate. Figure 9 shows that when the R&D cycle τ1>τ10 and the production cycle τ2>τ20, the quantity of production and the quantity of demand will fluctuate more and more. Meanwhile, it can be seen that the gap of production funds will also become larger and larger, and such investment will inevitably fail.The above analysis shows that the R&D cycle and production cycle play a decisive role in the dynamic trend of investment projects.


## 6. Conclusions and Future Work

According to the source and flow direction of wealth, an investment differential dynamics model with time delay for both production and R&D was established, which can describe the dynamic change of investment in practice. The numerical simulation clearly shows that the production cycle and the research cycle are the main factors that cause the dynamic changes of the investment system, which is the main advantage of using the time-delay differential equation theory to construct the investment model.Without the influence of other factors, the following conclusions can be obtained through the analysis of model (2).(1)
When the production time and R&D time of commodities are both zero (for example, commodity of seabird dung), the dynamic investment relationship will eventually reach equilibrium as time goes by. They are consistent with the conclusion of Theorem1.(2)
Both the production time and the R&D time will have an important impact on the investment process. Theorems5 and 6 show that the time delay within a certain range will promote the benign process of the investment process. Otherwise, they will push the investment process into a vicious circle.(3)
When the investment process is a cyclic process, the production time and R&D time have the important impact on the stability of the investment process and the length of the investment cycle. Their specific effects are precisely discussed in the content of Theorem6.However, the method in this paper also has its limitations. For example, the coefficients in the model are difficult to determine, and it is difficult to reflect the influence of political environment and human manipulation.This paper only studies the dynamic process and development trend of investment model in theory. In the future, the following exploration still needs to be done.(1)
Determine the coefficients in the model using the relevant data.(2)
How to reflect issues such as cost control in the model.(3)
How do R&D cycle and production cycle interact?This paper constructs and studies the investment dynamic system by using the theory of delay differential equations. It is hoped that the research results of this paper can enrich the financial investment theory and provide valuable reference for scholars and investors.



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*Source: 1116671-2022-04-30.xml*