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.
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 time be , and , respectively. Let be the coefficient of the entry rate of the demand link and 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
Let the functional response function of production effective supply demand be . The consumption rates in the production link and the R&D link are and , respectively. The rate of transfer from production to R&D is . The transfer rate from R&D to fund is . The capital return rate of the investment project is . Based on the above analysis and assumptions, the currency transfer relationships among demand, production, R&D, and fund are shown in Figure 1.
In addition, this paper assumes that it takes time to produce products and develop products, and the average time is , respectively. Then, the differential dynamics system of the investment dynamic model iswhere are all positive.
The first equation of model (2) is equivalent to
Therefore, if , then . Similarly, if then .
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 with .
Practically, it may be necessary to invest in projects intermittently or continuously. During this period, the capital of and flows reversely, so and should be negative.
The initial conditions of system (2) are given bywhereis the Banach space of continuous functions mapping the interval into ; here,
According to the existence and uniqueness theorem [3–5] of differential equation solutions, model (2) has a unique solution that satisfies the initial conditions (4).
It is easy to know that if the conditions hold, then model (2) has a unique positive equilibrium , where
The positive equilibrium means the equilibrium state of the investment model under the interaction of demand, R&D, production, and fund.
Theorem 1. If , all solutions of model (2) with the initial conditions (4) are positive for all .
Proof. If , it is easy to know that the first equation of system (2) is equivalent toTherefore, if , then .
Similarly, if then .
Becauseif then
Similarly, if then . The proof is completed.
Theorem 2. If and , model (2) is uniformly bounded [5] on .
Proof. Defining a functionit has thatFurther, we can getConsidering the constant positivity of on , it can be known thatTo sum up, the conclusion of Theorem 2 is true. The proof is completed.
The characteristic equation of model (2) at positive equilibrium isthat is,It is easy to know that equation (15) has a negative real root , and other solutions satisfy the following equation:
Theorem 3. When and , the positive equilibrium is globally asymptotically stable if conditions and hold.
Proof. When , equation (16) is transformed into the following equation:When , it is easy to know thatAccording to Routh–Hurwitz theorem [3, 5], all roots of equation (17) have negative real parts. Thus, when and , all roots of equation (15) have negative real parts. According to Hurwitz theorem [5], the positive equilibrium is locally asymptotically stable.
Take Dulac function [5]and letand then there isAccording to Bendixson–Dulac theorem [5], when , model (2) has no limit cycles in its feasible region . Combined with the local asymptotic stability of the positive equilibrium , it can be known that when , the positive equilibrium is globally asymptotically stable. The proof is completed.
3. Local Stability and Existence of Hopf Bifurcations
Theorem 4. If , and hold, there exists a positive number . The positive equilibrium is locally asymptotically stable when and unstable when . That is, system (2) undergoes a Hopf bifurcation around the positive equilibrium when .
Proof. If , equation (16) is transformed intoIt is easy to know that is a root of equation (22). The other two roots of equation (22) satisfy the following equation:Let be a root of (23); then, it hasSeparating the real and imaginary parts from (23), it hasIt follows thatSince , equation (26) has at least one positive root . By substituting into (25), we can obtainDefine . Taking the derivative of with respect to in equation (23), we can getBy substituting into above equation, we can obtainBased on the above analysis and the Hopf bifurcation theorem in [2, 23, 24], Theorem 4 is true. The proof is completed.
Theorem 5. If and hold, there exists a positive number . The equilibrium is locally asymptotically stable when and unstable when . That is, system (2) undergoes Hopf bifurcations around the positive equilibrium when .
Proof. If , equation (16) is transformed intoLet be two solutions of equation (30), and we can obtainThus, and have negative real parts. Also, the third root of (30) satisfies the equationSupposing that is a solution of equation (32), one hasNext, separating the real and imaginary parts of the above equation,one obtainsIf holds, then equation (35) has at least one positive root . From equation (34),Assume that is the minimum positive value in . Taking the derivative of with respect to in equation (32), we can getFurthermore,Hence, Theorem 5 holds. The proof is completed.
The solution of characteristic equation (16) is that of equation or . 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 assumptions hold. Model (2) is locally asymptotically stable when and . Model (2) undergoes Hopf bifurcations at the positive equilibrium when or . 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 .
Without loss of generality, suppose . Let ; then, system (2) can be written aswhere
In the space , model (39) can be locally expressed as the following delay differential equation:where is a bounded linear operator, and is continuously differentiable, respectively, bywhere
By the Riesz representation theorem, there exists a matrix function such that
In fact, if it can choosewhere is the Dirac delta function, equation (43) is obviously true.
If it definessystem (41) is equivalent towhen , where . For any , where is a four-dimensional complex vector space. It defines an operator and a bilinear functionwhere . Then, operators and are conjugate operators.
Let be the eigenvector of corresponding to and be the eigenvector of corresponding to ; then,where is the identity matrix.
Solving the above two matrix equations, we can obtain
Ifthen
According to the above analysis and calculation, it can be seen that
In the remainder of this section, we will compute the coordinates describing the center manifold at . Let be the solution of (46) at . Define
On the center manifold ,where and are local coordinates for center manifold in the direction of and . Note that is real if is real. We only consider real solutions. For the solution of (46), since ,where
By (54), we can obtain
Then,
It follows together with (56) that
Comparing the coefficients with (57), it has
In the expression of , it is necessary to calculate . According to the calculation process in references [27–29], it can be known thatwhere
So, there are
In summary, the key parameters can be represented by the parameters in model (2). Then, the following parameters can be calculated as:
Theorem 6. For system (2), if hold, then(1)The Hopf bifurcation is supercritical (subcritical) if .(2)The bifurcating periodic solutions are stable (unstable) if .(3)The period of the bifurcating periodic solutions increases (decreases) if .
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 and the initial value
By computing, one may obtain thatand hold.
The computer simulations (see Figure 2) show that is globally asymptotically stable when .
When , model (2) is locally asymptotically stable at . Also, its numerical simulation is shown in Figure 3.
If it takes , model (2) will undergo Hopf branch. In other words, model (2) has periodic solutions, and its numerical simulation is shown in Figure 4.
When , or , model (2) is locally asymptotically stable at the positive equilibrium . When and , model (2) will undergo Hopf branch. Their numerical simulation diagrams are shown in Figures 5–7, respectively.
When , model (2) has periodic solutions (see Figure 8). If , model (2) is unstable (see Figure 9).
In Figures 5–8, there are cases where and 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.
Figures 2, 3, 5, and 6 show that when and , 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 Figure 2, when the R&D cycle and production cycle , 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 is less than and the production cycle is less than , the investment dynamics will tend to be stable over time. Figures 4, 7, and 8 show that when the R&D cycle or the production cycle , the R&D quantity, the production quantity, and the demand quantity will cyclically fluctuate. Figure 9 shows that when the R&D cycle and the production cycle , 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 Theorem 1.(2)Both the production time and the R&D time will have an important impact on the investment process. Theorems 5 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 Theorem 6.
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.
Data Availability
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The author declares that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
This study was supported by the Philosophy and Social Science Program Funds of Heilongjiang Province (21JYE400).