Hopv bifurcation analysis method in correction research and development model containing time delay

Abstract

The invention relates to a Hopv bifurcation analysis method in a correction research and development model containing time delay, and the method converts a discrete time model into a continuous time model, introduces a time delay parameter, and more accurately describes research and development competition behaviors between enterprises. The local asymptotic stability of a balance point is researched by analyzing a corresponding characteristic equation, and the influence of time lag on the dynamic behavior of the system is disclosed, including stability conversion and Hopf bifurcation phenomenon. And a theoretical basis is provided for development of a competitive strategy and risk management. The method can be used for analyzing research and development competition among high-tech enterprises, provides theoretical support for making research and development strategies, optimizing resource allocation, performing risk management and the like, and has important application value.

Description

Technical Field

[0001] The invention relates to a Hopf bifurcation analysis method in a modified R&D model containing time delay, and belongs to the technical field of game theory. Background Art

[0002] With the rapid development of the economy, business competition has become increasingly fierce. If an enterprise, as an important part of the industry, wants to survive and remain invincible in the market, it must form a competitive advantage that is different from other enterprises. Research and development (R&D) has become the main driving force for corporate growth. Therefore, it is a necessary means for enterprises to achieve core competitiveness. In fact, the R&D activities of enterprises will reduce manufacturing costs, improve product quality, expand market share, and enhance their market competitiveness. However, in modern society, when R&D activities are carried out, R&D spillover effects often occur. Due to the knowledge sharing of R&D between enterprises and the flow of human capital, R&D spillover effects are inevitable.

[0003] Entrepreneurs and economists have been paying attention to the competitiveness and cooperation issues in R&D investment in recent years. Classic research can be traced back to the 1980s, when D'Aspremont and Jacquemin established a popular technology spillover duopoly model. Later, Zhou and Wang established a two-stage duopoly game model based on R&D spillovers, complementary goods and maximization of common interests. In the study of Pansera et al., a continuous-time model with discrete delays was proposed based on the discrete-time nonlinear monopoly model with R&D spillover effects proposed by Zhou and Wang.

[0004] However, the above research and analysis of the R&D competition model assumes that production is immediately available, and there is a time lag between the time when the enterprise calculates its marginal profit and the time when it uses the marginal profit t-τ to produce the final output t. This often affects the observation of the complex dynamic behavior of the system. Therefore, the present invention assumes that the instantaneous change in production is now based on the difference between the target (based on delayed past information) and the current production. If the difference is positive (or negative), the output will tend to increase (or decrease). This difference leads to differences in the model structure and dynamic behavior of the two. Unlike the mechanism detailed previously, it is now necessary to clearly understand the production situation at time t in order to adjust production. Therefore, the present invention can better reveal the complex system dynamic behavior laws of the R&D competition model, and can more effectively guide enterprises to formulate R&D strategies, optimize resource allocation, reduce risks, and enhance corporate competitiveness. Summary of the invention

[0005] The present invention can obtain various conclusions about the local stability of Nash equilibrium by converting the discrete time model into the continuous time model. We can see how equilibrium turns into chaos and how instability turns into stability. Therefore, time delay has both destabilizing and stabilizing effects on the system.

[0006] To achieve the above object, the technical solution specifically adopted by the present invention is: a Hopf bifurcation analysis method in a modified R&D model with time delay, which mainly includes the following steps:

[0007] Step 1: Establish a duopoly game model for R&D competition, introduce a time delay parameter, obtain a new model equation with time delay correction, and obtain the following time delay model:

[0008]

[0009] Among them, x m is the R&D cost of enterprise m (m=1,2), The parameters a>0, b>0 and c>0 reflect the market size, commodity price sensitivity and unit cost of manufactured products without R&D efforts respectively; α m >0 represents the adjustment speed of the enterprise, where m,β∈(0,1), equivalent to R&D spillover, γ>0 is the expenditure parameter of the enterprise's technological progress, and τ is the time delay.

[0010] Step 2: There is a unique Nash-Cunot equilibrium in the previous model system:

[0011]

[0012] E 3 is the equilibrium point of the competition model, x * The same is the R&D investment of the two competitors. 3 It is stable without delay. To ensure the economic significance of E3, the following conditions are assumed

[0013] a>2bc,9γ>8(β+1) 2 b.

[0014] The local stability of the positive equilibrium point is constrained by the roots of the corresponding characteristic equation of the system.

[0015] Step 3: Linearize it at E3, and the relevant characteristic equation can be obtained as the following second-order exponential polynomial equation, namely:

[0016] λ 2 +2λ+1+(pλ+p)e -λτ +qe -2λτ ,

[0017] λ is the root of the characteristic equation, p and q are parameters related to the enterprise R&D spillover effect, R&D investment, adjustment speed and other factors.

[0018]

[0019] and

[0020]

[0021] Step 4: After obtaining the characteristic equation, study the distribution of the characteristic roots and the change law of the roots of the characteristic equation with time delay to determine the stability of the system. First, observe that λ = 0 is not a characteristic value because Assuming that λ=iω(ω>0) is a purely imaginary root of the characteristic equation, we substitute it and obtain:

[0022] -ω 2 +2iω+1+(piω+p)e -iωτ +qe -2iωτ =0.

[0023] ω is the imaginary part of the root when there is a characteristic root or a pure imaginary root. And let θ = tan[(ωτ) / 2], we can get:

[0024]

[0025] Separating the real and imaginary parts of the substituted-in equation yields:

[0026]

[0027] get

[0028]

[0029] and

[0030]

[0031] Solve the previous formula for θ when D(ω)=0, and make E(ω)=F(ω)=0, which means ω satisfies the previous formula. If D(ω)≠0, according to Cramer's rule, this means

[0032]

[0033] This gives the equation D(ω)E(ω)=[F(ω)] 2 By direct calculation, let z = ω 2 We can get the following polynomial equation

[0034] z 4+s 3 z 3 +s 2 z 2 +s 1 z+s 0 =0,

[0035] in

[0036] s 3 =4-p 2 ,s 2 =6-2q 2 -3p 2 +2p 2 q,

[0037] s 1 =-3p 2 +4+3p 2 q-4q 2 ,s 0 =(1-q) 2 [-p 2 +(1+q) 2 ].

[0038] Since the leading term of the polynomial is z 4 , so when z→+∞, the polynomial tends to infinity. If s 0 <0, ie, q≠1 and (1+q) 2 <p 2 , then the existence of positive roots of the above formula can be guaranteed.

[0039] when Due to e -iωτ =-1, according to the characteristic equation, we get ω 2 -1+pq=0and(2-p)ω=0. Therefore, D(ω)=F(ω)=0. In this case, ω 2 It is still the positive root of the above equation after replacing z.

[0040] Step 5: Based on the previous analysis, we get the following result: if the above equation about z has a positive root , then when When , the characteristic equation has a pure imaginary root. And the critical value expression is:

[0041]

[0042] Step 6: Analyze the bifurcation according to the bifurcation condition. Let λ(τ)=u(τ)+iω(τ) be a root of the characteristic equation, satisfying and Then the root is a single root, and the root crossing criterion is:

[0043]

[0044] The calculation formula of Hopf bifurcation parameter is:

[0045]

[0046] According to the calculation formula, the direction of Hopf bifurcation is determined. * ,θ * ) corresponds to the positive sign when τ increases from right to left. The negative sign means crossing from left to right.

[0047] An electronic device comprises a memory, a processor and a computer program stored in the memory and executable on the processor. When the processor executes the program, the Hopf bifurcation analysis method in the modified R&D model with time delay is implemented.

[0048] A computer-readable storage medium stores computer instructions, which, when executed by a processor, implement the Hopf bifurcation analysis method in the modified R&D model with time delay.

[0049] Compared with the prior art, the present invention has the following advantages:

[0050] 1. This invention proposes a new method to convert a discrete-time dynamic system into a continuous-time model with delay, and the resulting R&D model exhibits a Hopf bifurcation phenomenon at the positive equilibrium. Different from the models in existing studies, this model is closer to the reality of corporate R&D competition. At the same time, the introduction of time delay parameters enables the model to more accurately describe the information lag and production lag in the corporate R&D decision-making process, thereby revealing more complex dynamic behaviors, such as stability conversion and Hopf bifurcation. This is of great significance for enterprises to formulate R&D strategies and conduct risk management.

[0051] 2. The present invention has a more in-depth study of the complex dynamic behavior of the system, and the R&D model shows the Hopf bifurcation phenomenon at the positive equilibrium. In addition, the dynamic behavior of the system near the positive equilibrium may change from local asymptotic stability to instability, and vice versa. The general laws of the complex dynamic behavior of the system are summarized through mathematical deduction. The revelation of these laws provides an important theoretical basis for enterprises to understand and predict competitive behavior, which helps enterprises to formulate corresponding competitive strategies earlier in the competition process.

[0052] 3. The present invention conducts a more in-depth study on the impact of time delay on system stability. The conclusion shows that time delay has both positive and negative effects on system stability, which depends on the delay parameter value and model parameters. This provides an important theoretical basis for understanding and controlling corporate competitive behavior. Enterprises can design feedback control strategies based on the rules and adjust parameter values ​​in real time according to the dynamic behavior of the system to maintain system stability, thereby optimizing resource allocation, reducing risks, and enhancing corporate competitiveness. BRIEF DESCRIPTION OF THE DRAWINGS

[0053] Figure 1 For the parameter 0.4=τ<τ * =0.5303, the waveform of the system tends to be stable;

[0054] Figure 2 For the parameter 0.6=τ>τ * =0.5303, the waveform of the system Hopf bifurcation occurs;

[0055] Figure 3 For the parameter 0.4=τ<τ * =0.5303, the system tends to be stable when x 1 (t) Phase diagram;

[0056] Figure 4 For the parameter 0.6=τ>τ * = 0.5303, when the system undergoes Hopf bifurcation, x 1 (t) Phase diagram;

[0057] Figure 5 For the parameter 0.4=τ<τ * =0.5303, the system tends to be stable when x 2 (t) Phase diagram;

[0058] Figure 6 For the parameter 0.6=τ>τ * = 0.5303, when the system undergoes Hopf bifurcation, x 2 (t) Phase diagram. DETAILED DESCRIPTION

[0059] In order to deepen the recognition and understanding of the present invention, the embodiments of the present invention are described below in conjunction with the accompanying drawings.

[0060] Example: See Figure 1-Figure 6 The Hopf bifurcation analysis method in the modified R&D model with time delay mainly includes the following steps:

[0061] Step 1: Establish a duopoly game model for R&D competition, introduce a time delay parameter, obtain a new model equation with time delay correction, and obtain the following time delay model:

[0062]

[0063] Among them, x m is the R&D cost of enterprise m (m=1,2), The parameters a>0, b>0 and c>0 reflect the market size, commodity price sensitivity and unit cost of manufactured products without R&D efforts respectively; α m >0 represents the adjustment speed of the enterprise, where m,β∈(0,1), equivalent to R&D spillover, γ>0 is the expenditure parameter of the enterprise's technological progress, and τ is the time delay.

[0064] Step 2: There is a unique Nash-Kurnot equilibrium in the previous model system:

[0065] E 3 =(x * ,x * ),in

[0066] E 3 is the equilibrium point of the competition model, x * The same is the R&D investment of the two competitors. 3 It is stable without delay. To ensure the economic significance of E3, the following conditions are assumed

[0067] a>2bc,9γ>8(β+1) 2 b.

[0068] The local stability of the positive equilibrium point is constrained by the roots of the corresponding characteristic equation of the system.

[0069] Step 3: Linearize it at E3, and the relevant characteristic equation can be obtained as the following second-order exponential polynomial equation, namely:

[0070] λ 2 +2λ+1+(pλ+p)e -λτ +qe -2λτ ,

[0071] λ is the root of the characteristic equation, p and q are parameters related to the enterprise R&D spillover effect, R&D investment, adjustment speed and other factors.

[0072]

[0073] and

[0074]

[0075] Step 4: After obtaining the characteristic equation, study the distribution of the characteristic roots and the change law of the roots of the characteristic equation with time delay to determine the stability of the system. First, observe that λ = 0 is not a characteristic value because Assuming that λ=iω(ω>0) is a purely imaginary root of the characteristic equation, we substitute it and obtain:

[0076] -ω 2 +2iω+1+(piω+p)e -iωτ +qe -2iωτ =0.

[0077] ω is the imaginary part of the root when there is a characteristic root or a pure imaginary root. Assume (ωτ) / 2≠(π / 2)+kπ, And let θ = tan[(ωτ) / 2], we can get:

[0078]

[0079] Separating the real and imaginary parts of the substituted-in equation yields:

[0080]

[0081] get

[0082]

[0083] and

[0084]

[0085] Solve the previous formula for θ when D(ω)=0, and make E(ω)=F(ω)=0, which means ω satisfies the previous formula. If D(ω)≠0, according to Cramer's rule, this means

[0086]

[0087] This gives the equation D(ω)E(ω)=[F(ω)] 2 By direct calculation, let z = ω 2 We can get the following polynomial equation

[0088] z 4 +s 3 z 3 +s 2 z 2 +s 1 z+s 0 =0,

[0089] in

[0090] s 3 =4-p 2 ,s 2 =6-2q 2 -3p 2 +2p 2 q,

[0091] s 1 =-3p 2 +4+3p 2 q-4q 2 ,s 0 =(1-q) 2 [-p 2 +(1+q) 2 ].

[0092] Since the leading term of the polynomial is z 4 , so when z→+∞, the polynomial tends to infinity. If s 0 <0, ie, q≠1 and (1+q) 2 <p 2 , then the existence of positive roots of the above formula can be guaranteed.

[0093] when Due to e -iωτ =-1, according to the characteristic equation, we get ω 2 -1+pq=0and(2-p)ω=0. Therefore, D(ω)=F(ω)=0. In this case, ω 2 It is still the positive root of the above equation after replacing z.

[0094] Step 5: Based on the previous analysis, we get the following result: if the above equation about z has a positive root Then when When , the characteristic equation has a pure imaginary root. And the critical value expression is:

[0095]

[0096] Step 6: Analyze the bifurcation according to the bifurcation condition. Let λ(τ)=u(τ)+iω(τ) be a root of the characteristic equation, satisfying and Then the root is a single root, and the root crossing criterion is:

[0097]

[0098] The calculation formula of Hopf bifurcation parameter is:

[0099]

[0100] According to the calculation formula, the direction of Hopf bifurcation is determined.* ,θ * ) corresponds to the positive sign when τ increases from right to left. The negative sign means crossing from left to right.

[0101] The present invention is verified by Matlab simulation experiment.

[0102] A numerical example is given in the experiment to illustrate the effectiveness of the general rules summarized in the present invention. Consider a system with parameters a=1.5, b=-2.2, c=1, α 1 =α 2 = 1, γ = 1.2, and β = 0.8, the equilibrium point E3 can be calculated to be 0.6263, and there is a unique solution according to the above equation about z in Then we get θ * =1.3309 and This means that G(ω * ,θ * )=387.6839>0. Based on the conditions in the law, if ign[G(ω * ,θ * )]>0, the system is The waveform is thus as shown in Figure 1 and Figure 2 As shown. Figure 1 and Figure 2 We can find that in When the trajectory converges to the equilibrium point, that is, x 1 (t) and x 2 (t) are stable, they are Loss of stability, then There is a Hopf bifurcation.

[0103] Define the initial condition x 1 (0) = 0.5 and x 2 (0) = 1, for the discrete duopoly game model of R&D competition between two high-tech enterprises, Figure 3 and Figure 4 Show in When the system state x 1 (t) is asymptotically stable and There is a Hopf bifurcation when the state x 1 (t) In cooperation with enterprise x 2 (t) competition still shows periodic changes over time. Figure 5 and Figure 6 Describes the 1(t) Dynamic trajectory phase diagram of the state under competition. Therefore, the two competing states weigh each other and show periodic changes over time. In addition, simulations for other situations can be given similarly.

[0104] It should be noted that the above embodiments are not intended to limit the protection scope of the present invention, and equivalent changes or substitutions made on the basis of the above technical solutions all fall within the protection scope of the claims of the present invention.

Claims

1. The Hopf bifurcation analysis method in the modified R&D model with time delay is applicable to the R&D competition game analysis among high-tech enterprises in monopoly industries with spillover effects. It is characterized by: The method comprises the following steps: S1: Establish a duopoly game model of R&D competition among the research object enterprises, introduce the time delay parameter, and obtain a new model with time delay correction. The model contains parameters such as R&D investment, time delay, adjustment speed, and R&D spillover effect to simulate the R&D competition among enterprises in actual scenarios; S2: Obtain the point where the R&D investment of both competitors reaches a stable state. By calculating the differential equations of the model system, obtain the equilibrium state equation of the system at time t. Solve the equilibrium point by combining the equilibrium equations. By analyzing the equilibrium R&D investment level of the analysis object, analyze the competitive situation of both parties and the adjustment of competitive strategies. S3: Obtain the stability analysis tool for subsequent competitive situation analysis, linearize the model at the equilibrium point, and obtain the characteristic equation, which is an equation about the characteristic root, which includes model parameters and time delay; S4: Use stability analysis tools to analyze the stability of competition between enterprises, analyze the distribution of the roots of the characteristic equation, study the changing rules of the roots of the characteristic equation with time delay to determine the stability of the system, and predict the occurrence of bifurcation phenomena in competition to help enterprises adjust the implementation of competition strategies in advance; S5: Analyze the occurrence of bifurcation phenomenon in competition. If the characteristic equation has pure imaginary roots and its real part changes with the change of τ, it means that the system has Hopf bifurcation phenomenon. According to the characteristic equation, derive the critical time delay value of Hopf bifurcation, analyze the influence of Hopf bifurcation parameters on the dynamic behavior of the system, and obtain the influence of various indicators on the actual enterprise competition strategy and competition pattern; S6: Based on the bifurcation conditions, predict the dynamic behavior of the system when the delay is greater than the critical value. If the delay is greater than the critical value, the system will lose stability and may experience periodic fluctuations or other complex dynamic behaviors. Guide the formulation and optimization of corporate competition strategies, resource allocation, etc. based on the rules.

2. The Hopf bifurcation analysis method in the modified development model of time delay according to claim 1 is characterized in that: In step S1, the mathematical expression of the modified duopoly game model of R&D competition with time delay is: Among them, x m is the R&D cost of enterprise m (m=1,2), represents its derivative, and the parameters a>0, b>0 and c>0 reflect the market size, commodity price sensitivity and unit cost of manufactured products without R&D efforts respectively; α m >0 represents the adjustment speed of enterprise m, where β∈(0,1), equivalent to R&D spillover, γ>0 is the expenditure parameter of enterprise technological progress, and τ is the time delay.

3. The Hopf bifurcation analysis method in the modified development model with time delay according to claim 1 is characterized in that: In step S2, the balance point is: E3=(x * ,x * ),in E3 is the equilibrium point of the competition model, x * The same is the R&D investment of the two competitors. The equilibrium point is stable without delay. In order to ensure that E3 has practical economic significance, the following conditions are assumed a>2bc,9c>8(b+1) 2 b, The local stability of the positive equilibrium point is constrained by the roots of the corresponding characteristic equation of the system.

4. The Hopf bifurcation analysis method in the modified development model with time delay according to claim 1 is characterized in that: In step S3, the linearization process of the model at the equilibrium point is to linearize it at E3, and the relevant characteristic equation can be obtained as the following second-order exponential polynomial equation, namely: l 2 +2λ+1+(pλ+p)e -λτ +qe -2λτ , Where λ is the root of the characteristic equation, p and q are parameters related to the enterprise R&D spillover effect, R&D investment, adjustment speed and other factors.

5. The Hopf bifurcation analysis method in the modified development model with time delay according to claim 1 is characterized in that: In step S4, the distribution of the characteristic roots of the previous characteristic equation is studied. First, it is observed that λ=0 is not an eigenvalue because Assume that λ=iω(ω>0) is a pure imaginary root of the characteristic equation, substitute it and get: -ω 2 +2iω+1+(piω+p)e -iωτ +that -2iωτ =0. Where ω is the imaginary part of the root when there is a characteristic root and a pure imaginary root, assuming And construct the variable θ = tan[(ωτ) / 2], we get: Separating the real and imaginary parts of the substituted-in equation yields: get 6. The Hopf bifurcation analysis method in the modified development model with time delay according to claim 1 is characterized in that: In step S5, the critical value of the Hopf bifurcation is calculated. First, θ in the previous formula is solved when D(ω)=0, and E(ω)=F(ω)=0 is required, which means that ω satisfies the previous formula. If D(ω)≠0, according to Cramer's rule, this means This gives the equation D(ω)E(ω)=[F(ω)] 2 , by direct calculation, and let z = ω 2 The following polynomial equation is obtained With 4 +s3z 3 +s2z 2 +s1z+s0=0, in <h2 style=";text-align:left;direction:ltr">s3=4-p<h2 style=";text-align:left;direction:ltr"> 2 <h2 style=";text-align:left;direction:ltr"> s2 = 6 - 2q<h2 style=";text-align:left;direction:ltr"> 2 <h2 style=";text-align:left;direction:ltr"> -3p<h2 style=";text-align:left;direction:ltr"> 2 <h2 style=";text-align:left;direction:ltr"> +2p<h2 style=";text-align:left;direction:ltr"> 2 <h2 style=";text-align:left;direction:ltr"> q, s1=-3p 2 +4+3p 2 q-4q 2 ,s0=(1-q) 2 [-p 2 +(1+q) 2 ]. Since the leading term of the polynomial is z 4 , so when z→+∞, the polynomial tends to infinity, if s0<0, ie,q≠1 and (1+q) 2 <p 2 , then the existence of positive roots of the above formula is guaranteed. when Due to e -iωτ =-1, according to the characteristic equation, we get ω 2 -1+pq=0and(2-p)ω=0. Therefore, D(ω)=F(ω)=0. In this case, ω 2 It is still the positive root of the above equation after replacing z, According to the above analysis, the following critical value expression is obtained:

7. The Hopf bifurcation analysis method in the modified development model with time delay according to claim 1 is characterized in that: In step S6, the direction of the Hopf bifurcation is determined, and λ(τ)=u(τ)+iω(τ) is a root of the characteristic equation, satisfying and Then the root is a single root, and the root crossing criterion is: The calculation formula of Hopf bifurcation parameter is: G(ω * ,θ * ) corresponds to the positive sign when τ increases from right to left. The negative sign means crossing from left to right. Next, after linearizing the system at the equilibrium point, the roots of the characteristic equation are calculated, and the system stability is analyzed based on the distribution characteristics of the roots of the characteristic equation and the critical conditions of the Hopf bifurcation are obtained. The time delay is selected as the bifurcation parameter, and the appropriate time lag parameter is selected before the critical condition to ensure the stability of the system. Based on this, a feedback control strategy is designed, and the parameter values ​​are adjusted in real time according to the dynamic behavior of the system to maintain the stability of the system.

8. An electronic device comprising a memory, a processor, and a computer program stored in the memory and executable on the processor, wherein: When the processor executes the program, the Hopf bifurcation analysis method in the modified development model with time delay as described in any one of claims 1 to 7 is implemented.

9. A computer-readable storage medium having computer instructions stored thereon, characterized in that: When the computer instructions are executed by a processor, the Hopf bifurcation analysis method in the modified development model with time delay as described in any one of claims 1 to 7 is implemented.

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