# Dynamic Analysis and Chaos Control for a NEV Enterprise and a Competitive TEV Enterprise under the CAFC-NEV Mandate

Authors: Liuwei Zhao; Hongyun Jiang
Journal: Discrete Dynamics in Nature and Society
 (2022)

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## Abstract

This paper studies the complexity analysis in production between competition and R&D for a traditional energy vehicles enterprise and a new energy vehicles enterprise under the introduction of corporate average fuel consumption credits and new energy vehicle credits mandate. From the internal perspective of market complexity evolution, this paper analyzes the internal mechanism of market competition fluctuation and discusses the influence of different corresponding speeds of enterprises on market stability through the nonlinear dynamics’ theory. The research results show that the adjustment coefficient of R&D investment and the price coefficient of credit transactions are the key factors affecting the competitive strategies of the two companies. The difference between these two factors will even change the relative R&D investment intensity, relative profitability, and market stability of the two companies in a balanced state. In addition, the numerical analysis showed that the time-delay feedback control method could control the unstable behavior of the dynamic system efficiently and quickly and make the market quickly recover a stable and orderly state, which provided a scientific basis for decision makers to effectively solve the market instability.


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## Body

## 1. Introduction

Stimulated by the multiple policies of fossil energy, environmental pollution, and new energy vehicles (NEV) subsidies, China’s NEV industry has developed rapidly. In 2019, China’s NEV sales reached 1.24 million, ranking first in the world and becoming the world’s largest NEV market [1, 2]. However, as the consumption of NEVs continues to grow, the Chinese government’s expenditure on NEV subsidies has also become problematic. In addition, with the frequent occurrence of fraudulent subsidies, the efficiency of subsidy policies for NEVs is greatly reduced, and the introduction of the corporate average fuel consumption credits and new energy vehicle credits (CAFC-NEVC) mandate will also have a significant impact on the future development of China’s auto industry. At that time, traditional car companies will not only have to meet fuel consumption standards but also must complete a certain percentage of NEVs, which will inevitably force traditional car companies to accelerate their transition to the NEV market. On the other hand, due to the constraints of battery cost and convenience, NEVs obviously cannot completely replace traditional energy vehicles (TEV) in the short term. The CAFC-NEVC mandate will inevitably intensify the integration and competition in the auto industry and increase the complexity of automakers’ production decisions. The advantage of new energy vehicles compared with traditional cars is that new energy vehicles are more environmentally friendly. It has higher fuel efficiency, which is conducive to easing the oil crisis, and will also change the energy structure of society. When working, it will not emit harmful gases to the outside world. With the same fuel consumption, the hybrid electric car can drive a longer distance. Traditional fuel vehicles are more convenient to use; gas stations are everywhere; there is no need to worry about endurance problems, safety, and stability; fuel vehicles are relatively safer; moreover, the value of traditional fuel vehicles is higher than that of new energy vehicles; and fuel car power, horsepower, and late acceleration ability are better. Therefore, how to make production decisions for TEVs and NEVs under the CAFC-NEVC mandate will become a brand-new challenge and topic for automobile manufacturers.The CAFC-NEVC mandate originated from the US CAFÉ fuel economy credits and California zero emission vehicle mandate. Scholars have conducted some analysis on the role of the CAFC-NEVC mandate in promoting enterprise R&D investment. From the research point of view, it is mainly to research the implementation effect of the CAFC-NEVC mandate. Scholars generally believe that the CAFC-NEVC mandate has an active role in promoting the development and innovation of NEVs. For example, Bakker and Farla believe that the CAFC-NEVC mandate can encourage NEV companies to reduce costs through R&D and innovation and keep market products consistent with consumer preferences [3]. The research of Vooren et al. shows that the CAFC-NEVC mandate “labels” energy-saving and NEVs, provides NEV companies with a new dimension of product positioning, and helps to stimulate NEV companies’ technological innovation and large-scale production [4]. Research by Wesseling et al. believes that the CAFC-NEVC mandate can induce NEV companies to shift from a “defensive” response to a “proactive” strategy such as R&D innovation and support for regulations, and the effect of R&D patent growth and commercialization is significant [5]. The research of Melton et al. believes that the CAFC-NEVC mandate effectively guides and stimulates the R&D and innovation activities of new energy vehicle companies and provides more models, production, and sales [6]. The research of Syke and Axsen indicates that the mandatory CAFC-NEVC mandate effectively avoids the free-riding phenomenon in the spillover of R&D innovation [7]. The research of Whistance et al. believes that the CAFC-NEVC mandate also plays an active role in promoting R&D and innovation in the fields of energy supply and infrastructure [8]. The research of Stokes and Breetz showed that although new energy vehicle technology was underestimated in the early stages, the promulgation and proliferation of the CAFC-NEVC mandate. It can greatly promote the R&D investment and the technological maturity of the automotive industry [9].CAFC-NEVC mandate and carbon emission rights trading both aim at reducing greenhouse gas emissions, and they have a certain degree of similarity (National Development and Reform Commission’s “NEV Carbon Allowance Management Measures” (Development and Reform Commission Industry (2016) No. 1768)), and there are relatively many production decision-making documents based on carbon emission rights trading. Ji et al. comparatively analyzed the impact of three trading mechanisms— unlimited trading, grandfather method limit and benchmark method limit, trading—on the decision-making, profit, and social welfare of supply chain enterprises [10]. He et al. studied the production decision-making problems of producers under carbon quota trading [11]. Diabat et al. studied the location of multi-stage and multiproduct companies under the carbon emissions trading mechanism [12]. But these models are not suitable for the CAFC-NEVC mandate. First, these models cannot distinguish between the two different carbon quota formation mechanisms in CAFC-NEVC, fuel consumption credits were obtained through energy saving and emission reduction, and NEV credits in proportion to production and sales. Fuel consumption credits can only be carried forward and cannot be traded, so they can only be redeemed in one direction but cannot be sold for profit. Second, the existing carbon trading model is only for single-product emission reduction decisions, while the CAFC-NEVC mandate involves the joint decision-making of two related products, such as TEVs and NEVs. Finally, the existing models generally assume that carbon allowances can be fully traded; especially in the automotive industry, credit acquisition and trading face greater uncertainty, leading most auto companies such as BAIC and Great Wall to adopt corporate shareholding methods to intervene in the NEV market by means of enterprise shares and reach an internal agreement in advance to ensure a certain amount of credits and a certain credit price [13, 14]. This production decision-making model means that interests can be shared according to the shareholding ratio, but decision-making remains relatively independent (for example, financial investment rather than equity investment in reality), which is essentially different from the decentralized decision-making model in which interests and decisions are completely independent in the existing supply chain [15].In addition, the production decision of automobile manufacturers is closely related to the credit prices, and the credits price has the characteristics of large volatility and complex volatility rules [16]. Therefore, the volatility of credit prices must be considered when making production plans. Brink et al. used the CGE model to analyze the robustness of the carbon price in the EU carbon market (EUETS) [17]. Kockar et al. analyzed the impact of carbon prices on power industry operations and electricity prices [18]. These studies only do a sensitivity analysis of the carbon price. Huang et al. studied the production decision-making and coordination issues of the supply chain under the disturbance of production costs [19]. Zhang et al. studied the supply chain coordination problem under the disturbance situation [20]. The above studies deal with the irregular fluctuations of decision-making elements from the perspective of disturbance management, which is more suitable for credit production decision-making for automobile manufacturers than [21] probability models such as the newsboy model. If the NEV manufacturer only plays a single role in the credit transaction, such as GWM and Yogomo, then its relationship with the traditional energy vehicle manufacturer is relatively simple. In this case, both parties only need to determine the production quantity and the credit transactions price. If the two parties are competitors with product substitution, according to the traditional oligopoly theory, the party that first determines the output can obtain a larger auto market share and obtain higher profits, thus showing a first-mover advantage [22]. However, in addition to being competitors in business, the two parties are also participants in credit transactions and manufacturing technology, such as BYD and Toyota. The results of the Cournot competition between TEV manufacturers and NEV manufacturers and the incentives for the two parties to choose a cooperative and competitive production strategy under the CAFC-NEVC mandate are still unclear, which motivates this study.The complexity and diversity of nonlinear characteristics can be used to describe the characteristics of many actual systems. For example, multiple isolated equilibrium points can represent different states in the actual system, and limit cycles can be used to describe nonlinear oscillators. Among all nonlinear characteristics, chaos and bifurcation are the most typical and common nonlinear characteristics. Chaos has been considered as a harmful and uncontrollable phenomenon since its discovery. However, for the needs of practical application and in-depth study of nonlinear system control theory, scientists continue to study how to control chaos. In 1989, Hubler applied the adaptive control method to realize chaos control for the first time [23]. Subsequently, Ott et al. proposed the OGY method for chaos control [24], which is named after the initials of Ot, Grebogi, and Yorke. Also from this time on, chaos control has really attracted great attention in academia. In this paper, the time-delay feedback control method proposed by Pyragas will be used to control the chaotic behavior of discrete dynamic systems [25]. See for more research [21, 26–31].In summary, the existing research has laid a good foundation for analyzing the role of the CAFC-NEVC mandate in promoting the R&D investment of NEV enterprises. Based on the existing research, this paper will explore the complex dynamics of automobile manufacturers’ production decisions under the CAFC-NEVC mandate. The possible innovations are as follows: first, based on the principle of limited cooperation, benefit sharing, and independent decision-making, the R\&D and production decision-making model of automobile manufacturers under the CAFC-NEVC mandate is established; second, the model with limited transactions is solved, and the equilibrium stability of the integral agreement price is given according to the system complexity and stability theory; third, under limited trading, in response to market instability arising from price fluctuations in points and fluctuations in demand for TEVs and NEVs, the strategy for adjusting R&D and production plans is given.The remainder of the paper is organized as follows: in Section1, we introduce the problem’s characteristics and assumptions. In Section 2, we study the existence and local stability of the equilibrium points for the system. A numerical study is conducted in Section 3 to demonstrate the analytical results and discuss insights. In Section 5, time-delayed feedback control is used to stabilize the chaotic behaviors of the system. We conclude the paper in Section 5 with discussions on possible extensions.


## 2. Problem Description and Model Construction

### 2.1. Problem Description

In this study, the product is the Cournot competition in the automobile market between the alternative TEV manufacturer 1 and the NEV manufacturer 2. Therefore, the market prices of their respective automobile products are jointly determined by their respective outputs, that is, through the inverse demand function. In order to facilitate research, this paper adopts the inverse linear demand function commonly used in marketing and operations to study production competitionπιθι,θφ=ϕ−θι−αφθφ,ι,φ=1,2;ι≠φ, ϕ represents the upper limit of the market size, the market price, αj (0≤αj≤1) represents the substitution rate of j’s products for i’s, measure the cross-influence of the change in demand of automobile manufacturer j on the change in demand of automobile manufacturer i. The lower the αj value, the lower the replacement rate. High substitution rates often lead to more intense market competition [32]. If αj=0 and αj=1 correspond to complete independence and complete substitution, respectively. Due to high production costs, low cruising distance, an inconvenient power supply, and other reasons, consumers generally believe that NEVs are inferior to TEVs. The latter is a complete substitute for the former, but the reverse is not true, that is, α1=1, α2=α<1. The production cost functions of TEVs and NEVs are respectively Ciqit=ciqit and a marginal cost ci is a positive number, and the cost of NEVs is higher than that of TEVs, that is, c2>c1 [33].Due to the CAFC-NEVC mandate, automakers will incur additional costs and benefits. In addition to reducing fuel consumption, TEV manufacturers are also required to produce a certain percentage of NEVs. Assuming that the initial average fuel consumption of TEV manufacturers ise0 L/100 km, the actual value of e01−B1 is L/100 km, and the normal number Bi represents the technological level of the car companies’ innovative products. The government’s standard value is e0−△e L/100 km, then the credits of the average fuel consumption of the enterprise are the product of the difference between the average fuel consumption of the enterprise and the actual value and the calculation quantity of the TEVs of the enterprise, that is, B1e0−△eq1t; When B1e0>△e, a positive fuel consumption credits will be formed, which can be used to carry over to subsequent years but cannot be traded; When B1e0≤△e, negative fuel consumption credits are formed, and the carryover positive fuel consumption credits can only be used to compensate and return to zero. Therefore, the total credit demand of the TEV enterprises is γ+△e−B1e0+q1t, and γ+△e−B1e0+q1t is the credits coefficient of the TEVs. The NEV manufacturers will earn positive NEV credits from the production of NEVs based on the cruising distance. The NEV credits coefficient is β2 (For example, NEV credits β2=0.012R+0.8, R is the driving range of NEVs), the higher the cruising distance, the higher the credits earned by NEVs. According to the CAFC-NEVC mandate, the higher the credit factor of the NEV, the longer the mileage; Therefore, each NEV will get higher credits. The credit of both parties will be traded in the credits market, and the market price of the credits is P.In a stable credit market, the fluctuation of credit prices is relatively small to meet the risk constraints of automobile manufacturers. With the intensification of credit price volatility, it will break through the upper limit of risk tolerance for auto manufacturers, and the market will become unstable and chaotic. The main factors affecting the production and competition strategies of both parties under the CAFC-NEVC mandate are shown in Figure1.Figure 1
Conceptual framework of the CAFC-NEVC mandate.


### 2.2. Model Construction

For the number of cars produced by each enterprise in a certain period, the auto enterprise’s capital accumulationKit determines its potential to produce cars in that period, and it is represented by the linear form qit=BiKit, i=1,2. The action strategy of each car enterprise is to select R\&D investment in various periods and meet the requirements: the two car enterprises formulate corresponding R\&D investment strategies on a discrete time axis. Kit−1 represents the capital stock of the enterprise i in the period t−1, and xit represents the R\&D investment of the enterprise in a single period in the t period. Due to the certain depreciation of capital, the capital stock Kit−1 flows into the next economic period after the remaining θKit−1, where θ (0<θ<1) is the depreciation rate. Therefore, the relationship between the capital stock of enterprise i in the two adjacent periods is Kit=θKit−1+xit (i=1,2), and thus: qit=BiθKit−1+xit (i=1,2).Based on the above assumptions about the functional relationship of related variables, the profits of TEV manufacturer 1 and NEV manufacturer 2 in thet period are calculated as follows:(1)π1x1t,x2t=q1tϕ−q1t−αq2t−c1t−x1t−β1q1tP,(2)π2x1t,x2t=q2tϕ−q2t−q1t−c2t−x2t−β2q2tP.Substituting the specific expressions of each variable into equations (1) and (2), and obtaining the partial derivative of πix1t,x2t with respect to xit, the marginal profits of TEV enterprises and NEV enterprises are obtained as follows:(3)φ1t=∂π1x1t,x2t∂x1t=ϕB1−2B12θK1t−1+x1t−αB2B1θK2t−1+x2t−1−c1B1−β1B1P,(4)φ2t=∂π2x1t,x2t∂x2t=ϕB2−2B22θK2t−1+x2t−B1B2θK1t−1+x1t−c2B2−1−β2B2P.Based on the assumption that participants have bounded rationality, TEV enterprises and NEV enterprises will adjust their R\&D investment strategies in the next period according to their local knowledge of marginal profit. In other words, if the observed profit margin of the enterprisesi in the current period t is positive, then they will increase their R\&D investment in the period t+1; otherwise, they will reduce their R\&D investment. Therefore, the dynamic adjustment mechanism of an enterprise i’s R\&D investment can be expressed in the following form:(5)xit+1=xit+αixitφiti=1,2,where αixit is a positive adjustment function, which represents the adjustment range of the enterprise i's R\&D investment based on marginal profit. This paper still considers the linear adjustment function [15, 18, 19, 23, 24, 28]: αixit=αixit, the coefficient αi>0 represents the speed at which the firm adjusts its R\&D investment strategy according to the marginal profit signal. Therefore, the dynamic equation (5) can be rewritten as follows:(6)xit+1=xit+αixitφiti=1,2.Using simultaneous equations (1)–(6), we can get a four-dimensional discrete dynamic model as follows:(7)x1t+1=x1t+α1x1tϕB1−2B12θK1t−1+x1t−αB2B1θK2t−1+x2t−1−c1B1−β1B1P,x2t+1=x2t+α2x2tϕB2−2B22θK2t−1+x2t−B1B2θK1t−1+x1t−c2B2−1−β2B2P,K1t=θK1t−1+x1t,K2t=θK2t−1+x2t.In order to be consistent in expression, this article replacesKit−1 with Iit, and obviously replaces Iit+1 with Kit. Then, the dynamic system (7) can be rewritten into the following standard form:(8)x1t+1=x1t+α1x1tϕB1−2B12θI1t+x1t−αB2B1θI2t+x2t−1−c1B1−β1B1P,x2t+1=x2t+α2x2tϕB2−2B22θI2t+x2t−B1B2θI1t+x1t−c2B2−1−β2B2P,I1t+1=θI1t+x1t,I2t+1=θI2t+x2t.Discrete dynamic system (8) expresses the assumption that both the market inverse demand function and the production cost function are in linear form, constructing a competition model of R&D investment strategy in each period of which game participants with boundedly rational expectations are adjusted according to the marginal profit.


## 2.1. Problem Description

In this study, the product is the Cournot competition in the automobile market between the alternative TEV manufacturer 1 and the NEV manufacturer 2. Therefore, the market prices of their respective automobile products are jointly determined by their respective outputs, that is, through the inverse demand function. In order to facilitate research, this paper adopts the inverse linear demand function commonly used in marketing and operations to study production competitionπιθι,θφ=ϕ−θι−αφθφ,ι,φ=1,2;ι≠φ, ϕ represents the upper limit of the market size, the market price, αj (0≤αj≤1) represents the substitution rate of j’s products for i’s, measure the cross-influence of the change in demand of automobile manufacturer j on the change in demand of automobile manufacturer i. The lower the αj value, the lower the replacement rate. High substitution rates often lead to more intense market competition [32]. If αj=0 and αj=1 correspond to complete independence and complete substitution, respectively. Due to high production costs, low cruising distance, an inconvenient power supply, and other reasons, consumers generally believe that NEVs are inferior to TEVs. The latter is a complete substitute for the former, but the reverse is not true, that is, α1=1, α2=α<1. The production cost functions of TEVs and NEVs are respectively Ciqit=ciqit and a marginal cost ci is a positive number, and the cost of NEVs is higher than that of TEVs, that is, c2>c1 [33].Due to the CAFC-NEVC mandate, automakers will incur additional costs and benefits. In addition to reducing fuel consumption, TEV manufacturers are also required to produce a certain percentage of NEVs. Assuming that the initial average fuel consumption of TEV manufacturers ise0 L/100 km, the actual value of e01−B1 is L/100 km, and the normal number Bi represents the technological level of the car companies’ innovative products. The government’s standard value is e0−△e L/100 km, then the credits of the average fuel consumption of the enterprise are the product of the difference between the average fuel consumption of the enterprise and the actual value and the calculation quantity of the TEVs of the enterprise, that is, B1e0−△eq1t; When B1e0>△e, a positive fuel consumption credits will be formed, which can be used to carry over to subsequent years but cannot be traded; When B1e0≤△e, negative fuel consumption credits are formed, and the carryover positive fuel consumption credits can only be used to compensate and return to zero. Therefore, the total credit demand of the TEV enterprises is γ+△e−B1e0+q1t, and γ+△e−B1e0+q1t is the credits coefficient of the TEVs. The NEV manufacturers will earn positive NEV credits from the production of NEVs based on the cruising distance. The NEV credits coefficient is β2 (For example, NEV credits β2=0.012R+0.8, R is the driving range of NEVs), the higher the cruising distance, the higher the credits earned by NEVs. According to the CAFC-NEVC mandate, the higher the credit factor of the NEV, the longer the mileage; Therefore, each NEV will get higher credits. The credit of both parties will be traded in the credits market, and the market price of the credits is P.In a stable credit market, the fluctuation of credit prices is relatively small to meet the risk constraints of automobile manufacturers. With the intensification of credit price volatility, it will break through the upper limit of risk tolerance for auto manufacturers, and the market will become unstable and chaotic. The main factors affecting the production and competition strategies of both parties under the CAFC-NEVC mandate are shown in Figure1.Figure 1
Conceptual framework of the CAFC-NEVC mandate.


## 2.2. Model Construction

For the number of cars produced by each enterprise in a certain period, the auto enterprise’s capital accumulationKit determines its potential to produce cars in that period, and it is represented by the linear form qit=BiKit, i=1,2. The action strategy of each car enterprise is to select R\&D investment in various periods and meet the requirements: the two car enterprises formulate corresponding R\&D investment strategies on a discrete time axis. Kit−1 represents the capital stock of the enterprise i in the period t−1, and xit represents the R\&D investment of the enterprise in a single period in the t period. Due to the certain depreciation of capital, the capital stock Kit−1 flows into the next economic period after the remaining θKit−1, where θ (0<θ<1) is the depreciation rate. Therefore, the relationship between the capital stock of enterprise i in the two adjacent periods is Kit=θKit−1+xit (i=1,2), and thus: qit=BiθKit−1+xit (i=1,2).Based on the above assumptions about the functional relationship of related variables, the profits of TEV manufacturer 1 and NEV manufacturer 2 in thet period are calculated as follows:(1)π1x1t,x2t=q1tϕ−q1t−αq2t−c1t−x1t−β1q1tP,(2)π2x1t,x2t=q2tϕ−q2t−q1t−c2t−x2t−β2q2tP.Substituting the specific expressions of each variable into equations (1) and (2), and obtaining the partial derivative of πix1t,x2t with respect to xit, the marginal profits of TEV enterprises and NEV enterprises are obtained as follows:(3)φ1t=∂π1x1t,x2t∂x1t=ϕB1−2B12θK1t−1+x1t−αB2B1θK2t−1+x2t−1−c1B1−β1B1P,(4)φ2t=∂π2x1t,x2t∂x2t=ϕB2−2B22θK2t−1+x2t−B1B2θK1t−1+x1t−c2B2−1−β2B2P.Based on the assumption that participants have bounded rationality, TEV enterprises and NEV enterprises will adjust their R\&D investment strategies in the next period according to their local knowledge of marginal profit. In other words, if the observed profit margin of the enterprisesi in the current period t is positive, then they will increase their R\&D investment in the period t+1; otherwise, they will reduce their R\&D investment. Therefore, the dynamic adjustment mechanism of an enterprise i’s R\&D investment can be expressed in the following form:(5)xit+1=xit+αixitφiti=1,2,where αixit is a positive adjustment function, which represents the adjustment range of the enterprise i's R\&D investment based on marginal profit. This paper still considers the linear adjustment function [15, 18, 19, 23, 24, 28]: αixit=αixit, the coefficient αi>0 represents the speed at which the firm adjusts its R\&D investment strategy according to the marginal profit signal. Therefore, the dynamic equation (5) can be rewritten as follows:(6)xit+1=xit+αixitφiti=1,2.Using simultaneous equations (1)–(6), we can get a four-dimensional discrete dynamic model as follows:(7)x1t+1=x1t+α1x1tϕB1−2B12θK1t−1+x1t−αB2B1θK2t−1+x2t−1−c1B1−β1B1P,x2t+1=x2t+α2x2tϕB2−2B22θK2t−1+x2t−B1B2θK1t−1+x1t−c2B2−1−β2B2P,K1t=θK1t−1+x1t,K2t=θK2t−1+x2t.In order to be consistent in expression, this article replacesKit−1 with Iit, and obviously replaces Iit+1 with Kit. Then, the dynamic system (7) can be rewritten into the following standard form:(8)x1t+1=x1t+α1x1tϕB1−2B12θI1t+x1t−αB2B1θI2t+x2t−1−c1B1−β1B1P,x2t+1=x2t+α2x2tϕB2−2B22θI2t+x2t−B1B2θI1t+x1t−c2B2−1−β2B2P,I1t+1=θI1t+x1t,I2t+1=θI2t+x2t.Discrete dynamic system (8) expresses the assumption that both the market inverse demand function and the production cost function are in linear form, constructing a competition model of R&D investment strategy in each period of which game participants with boundedly rational expectations are adjusted according to the marginal profit.


## 3. Equilibrium Point Stability Analysis

In the discrete dynamical system (8), let xit+1=xit and Iit+1=Iit, i=1,2, we can obtain the following equation:(9)x1tϕB1−2B12θI1t+x1t−αB2B1θI2t+x2t−1−c1B1−β1B1P=0,x2tϕB2−2B22θI2t+x2t−B1B2θI1t+x1t−c2B2−1−β2B2P=0,1−θI1t=x1t,1−θI2t=x2t.By solving the equations (9), we can obtain the four equilibrium points of the discrete dynamic system (8) as follows:(10)E0=0,0,0,0,E1=1−θB1ϕ−c1−Pβ1−12B12,0,B1ϕ−c1−Pβ1−12B12,0,E2=0,1−θB2ϕ−c2−Pβ2−12B22,0,B2ϕ−c2−Pβ2−12B22,E∗=x1∗,x2∗,I1∗,I2∗,where x1∗=1−θB1B2ϕ−c1α−2+P2β1−αβ2−α−2B2/4−αB12B2,(11)x2∗=1−θB2+B1B2ϕ+c1−2c2+Pβ1−2β2−24−αB12B2,I1∗=B1B2ϕ−c1α−2+P2β1−αβ2−α−2B24−αB12B2,I2∗=B2+B1B2ϕ+c1−2c2+Pβ1−2β2−24−αB12B2.It is easy to know thatE0, E1 and E2 are boundary equilibrium points, and E∗ is the only interior point equilibrium. Considering the real economic significance of the equilibrium point of the discrete dynamical system, we only discuss the situation where the equilibrium point is nonnegative in this article. Since b, B1, B2 and θ are both positive parameters, when E1, E2, and E∗ are both greater than zero, the parameters must meet the following conditions:(12a)B1ϕ−c1−Pβ1−1>0,(12b)B2ϕ−c2−Pβ2−1>0,(12c)B1B2ϕ−c1α−2+P2β1−αβ2−α−2B2>0,(12d)B2+B1B2ϕ+c1−2c2+Pβ1−2β2−2>0.In the analysis later in this paper, the non-negativity conditions (12a)–(12d) are assumed to be valid.


### 3.1. Stability of the Boundary Equilibrium Point

In order to discuss the stability of each equilibrium pointx1,x2,I1,I2 of the discrete dynamic system (8), we calculate and obtain the Jacobian matrix corresponding to the discrete dynamic system (8) as follows:(13)J=1+α1M−α1B1B2x1α−2α1θB12x1−θB1B2x1αα1−α2x2B1B21+α2N−α2θB1B2x2−2α2θB22x210θ0010θ,where M=B1ϕ−αx2+θI2B2−c1−Pβ1−1−22x1+θI1B12 and N=B2ϕ−x1+θI1B1−c2−Pβ2−1−22x2+θI2B22.According to the Schur–Cohn stability criterion [34], when all the characteristic roots of the characteristic polynomial corresponding to the Jacobian matrix (13) are in the unit circle on the complex plane; that is, when the modulus of any characteristic root is less than 1, the equilibrium point x1,x2,I1,I2 is asymptotically stable.Proposition 1.
The boundary equilibrium pointE0 is the unstable equilibrium point.Proof.
The specifics of the Jacobian matrix (13) at the boundary equilibrium point E0 are as follows:(14)JE0=1+α1B1ϕ−c1−Pβ1−100001+α2B2ϕ−c2−Pβ2−10010θ0010θ.
By calculation, four eigenvalues of the Jacobian matrix (14) can be obtained, and the specific expression is as follows: λ1=λ2=θ, λ3=1+α1B1ϕ−c1−Pβ1−1, λ4=1+α2B2ϕ−c2−Pβ2−1.
From the adjustment coefficientαi>0 and the nonnegative condition (12a) of the equilibrium point, the latter two eigenvalues meet λ3,4>1 are established. Therefore, according to the stability determination condition, the equilibrium point of the discrete dynamic system, E0 is the unstable boundary equilibrium.Proposition 2.
The boundary equilibrium pointsE1 and E2 are both unstable equilibrium points.Proof.
The specific of the Jacobian matrix (13) at the boundary equilibrium point E1 is as follows:(15)JE1=1+α1θ−1H1α1B2θ−1H12B1θθ−1α1H1θθ−1α1B2H12B101+α2H22B10010θ0010θ,where H1=aB1−B1c1−1 and H2=B1B2a+c1−2c2−2B1+B2.
In the same way, four eigenvalues of the Jacobian matrix (15) can be obtained by calculation, and the specific expression is as follows: λ1=θ, λ2=1+α2B2+B1−2+B2a+c1−2c2+Pβ1−2Pβ2/2B1,(16)λ3,4=121+θ+α11−θ1−B1ϕ−c1−Pβ1±−4θ+1+θ+α11−θ1−B1ϕ−c1−Pβ12.
By presupposing the parametersB1 and α2>0, then using the inequality (12d), we can conclude that λ2>1 is established. Therefore, according to the stability determination condition of the equilibrium point of the discrete dynamic system, it can be known that E1 is an unstable boundary equilibrium. A similar approach shows that E2 is unstable too.


### 3.2. Stability of the Interior Point Equilibrium

Next, we mainly discuss the stability of the interior equilibrium pointE∗. The specific of the Jacobian matrix (13) at the boundary equilibrium point E∗ is as follows:(17)JE∗=1−2α1bB12x1∗−α1bB1B2x1∗−2θα1bB12x1∗−θα1bB1B2x1∗−α2bB1B2x2∗1−2α2bB22x2∗−θα2bB1B2x2∗−2θα2bB22x2∗10θ0010θ.First, let the characteristic polynomial of the matrixJE∗ be Pλ, and the specific expression is Pλ=λ4+p1λ3+p2λ2+p3λ+p4. Then, through numerical calculation, we can obtain the coefficients of the polynomial Pλ in the following form:(18)p1=2B1B2α−4−2α1+α2−θ4−α−2α1+α2−α2θ−1B2X+α1θ−1B12B2Y−α−α2θ−1B224−αB1B2,p2=1α−4B1B22α2α1θ−1−θ−1θ−1B22+α1θ−1B1221+α2+θ−α2θ+α2θ−1B2XB2Y−α+B1B2α1−α1α2θ−12+4θ+θ2−4α1α2θ−1−θ−1θ−1+1−α2+4θ+θ2−α2θ2+α2θ−1B2ϕ−21+θ−αα11−θ+4α1θ−1−21+θc1−2Pβ1+41+θ−4+αα1θ−1c2−4Pα1β1−2Pθβ1−Pαα1θβ2+4Pα1β1θ+4Pβ2+4Pα1β2+Pαα1β2+4Pθβ2−4Pα1θβ2,p3=2θα21−θB22+B1B2α2B21−θX−2α1−2α2−4−α1+θ−α1B121−θB2Y−α/4−αB1B2,p4=θ2,where X=ϕ+c1−2c2+Pβ1−2Pβ2 and Y=−2ϕ+αϕ+2c1−αc2+2Pβ1−αPβ2.According to the Schur-Cohn stability criterion, if the characteristic polynomialPλ, that is, all the eigenvalues of the Jacobian matrix JE∗ lie in the unit circle on the complex plane, the coefficients of the polynomial Pλ need to meet the following conditions at the same time:(19)P1=1+p1+p2+p3+p4=α1α2θ−12B2+B1−2+B2X2B2+B1−α+B2Yα−4B1B2>0,P−1=1−p1+p2−p3+p4=1−4+αB1B22B22α2−1+θα1−1+θ−21+θ+α1B12−1+θ4+2α2+4θ−2α2θ+α2B2−1+θX−α+B2Y+B1B2−4α1−1+θ−21+θα2−1+θ−21+θ+α−α1α2−1+θ2+41+θ2+α2B2−1+θ−4ϕ−αα1ϕ−4θϕ+αα1ϕθ+4−1+α1−1+θ−θc1+−4+αα1−1+θ+81+θc2−4Pβ1−4Pα1β1−4Pθβ1+4Pθα1β1+8Pβ2+4Pα1β2+Pαα1β2+8Pθβ2−4Pθα1β2−Pαα1θβ2>01±p4=1±θ2>0.From the hypothesis of the positive and negative system parameters and the inequality (12c)-(12d), we know that P1>0 and 1±p4>0 are obviously valid in a stable condition. Therefore, according to Jury’s criterion, the sufficient condition for the interior equilibrium points E∗ of the discrete dynamical system (8) to be asymptotically locally stable is if it meets the following conditions: P−1=1−p1+p2−p3+p4>0 and M3±>0, that is,(20)p1+p3<1+p2+p4p2−p2p4+p4−p43−p1p3+p12p4<1−p42+p2p4−p2p42+p1p3p4−p32⇔p1+p3<1+p2+p43+3p4>p21−p41−p42−p21−p4+p1−p3p3−p1p4>0p4<1,are established, and all eigenvalues of JE∗ satisfy λi<1i=1,2,3,4. Therefore, when condition (20) is satisfied, the internal equilibrium point E∗ is locally stable. From the above analysis of the equilibrium point, it can be seen that the stability of the equilibrium point is dependent on the values of the system parameters.


## 3.1. Stability of the Boundary Equilibrium Point

In order to discuss the stability of each equilibrium pointx1,x2,I1,I2 of the discrete dynamic system (8), we calculate and obtain the Jacobian matrix corresponding to the discrete dynamic system (8) as follows:(13)J=1+α1M−α1B1B2x1α−2α1θB12x1−θB1B2x1αα1−α2x2B1B21+α2N−α2θB1B2x2−2α2θB22x210θ0010θ,where M=B1ϕ−αx2+θI2B2−c1−Pβ1−1−22x1+θI1B12 and N=B2ϕ−x1+θI1B1−c2−Pβ2−1−22x2+θI2B22.According to the Schur–Cohn stability criterion [34], when all the characteristic roots of the characteristic polynomial corresponding to the Jacobian matrix (13) are in the unit circle on the complex plane; that is, when the modulus of any characteristic root is less than 1, the equilibrium point x1,x2,I1,I2 is asymptotically stable.Proposition 1.
The boundary equilibrium pointE0 is the unstable equilibrium point.Proof.
The specifics of the Jacobian matrix (13) at the boundary equilibrium point E0 are as follows:(14)JE0=1+α1B1ϕ−c1−Pβ1−100001+α2B2ϕ−c2−Pβ2−10010θ0010θ.
By calculation, four eigenvalues of the Jacobian matrix (14) can be obtained, and the specific expression is as follows: λ1=λ2=θ, λ3=1+α1B1ϕ−c1−Pβ1−1, λ4=1+α2B2ϕ−c2−Pβ2−1.
From the adjustment coefficientαi>0 and the nonnegative condition (12a) of the equilibrium point, the latter two eigenvalues meet λ3,4>1 are established. Therefore, according to the stability determination condition, the equilibrium point of the discrete dynamic system, E0 is the unstable boundary equilibrium.Proposition 2.
The boundary equilibrium pointsE1 and E2 are both unstable equilibrium points.Proof.
The specific of the Jacobian matrix (13) at the boundary equilibrium point E1 is as follows:(15)JE1=1+α1θ−1H1α1B2θ−1H12B1θθ−1α1H1θθ−1α1B2H12B101+α2H22B10010θ0010θ,where H1=aB1−B1c1−1 and H2=B1B2a+c1−2c2−2B1+B2.
In the same way, four eigenvalues of the Jacobian matrix (15) can be obtained by calculation, and the specific expression is as follows: λ1=θ, λ2=1+α2B2+B1−2+B2a+c1−2c2+Pβ1−2Pβ2/2B1,(16)λ3,4=121+θ+α11−θ1−B1ϕ−c1−Pβ1±−4θ+1+θ+α11−θ1−B1ϕ−c1−Pβ12.
By presupposing the parametersB1 and α2>0, then using the inequality (12d), we can conclude that λ2>1 is established. Therefore, according to the stability determination condition of the equilibrium point of the discrete dynamic system, it can be known that E1 is an unstable boundary equilibrium. A similar approach shows that E2 is unstable too.


## 3.2. Stability of the Interior Point Equilibrium

Next, we mainly discuss the stability of the interior equilibrium pointE∗. The specific of the Jacobian matrix (13) at the boundary equilibrium point E∗ is as follows:(17)JE∗=1−2α1bB12x1∗−α1bB1B2x1∗−2θα1bB12x1∗−θα1bB1B2x1∗−α2bB1B2x2∗1−2α2bB22x2∗−θα2bB1B2x2∗−2θα2bB22x2∗10θ0010θ.First, let the characteristic polynomial of the matrixJE∗ be Pλ, and the specific expression is Pλ=λ4+p1λ3+p2λ2+p3λ+p4. Then, through numerical calculation, we can obtain the coefficients of the polynomial Pλ in the following form:(18)p1=2B1B2α−4−2α1+α2−θ4−α−2α1+α2−α2θ−1B2X+α1θ−1B12B2Y−α−α2θ−1B224−αB1B2,p2=1α−4B1B22α2α1θ−1−θ−1θ−1B22+α1θ−1B1221+α2+θ−α2θ+α2θ−1B2XB2Y−α+B1B2α1−α1α2θ−12+4θ+θ2−4α1α2θ−1−θ−1θ−1+1−α2+4θ+θ2−α2θ2+α2θ−1B2ϕ−21+θ−αα11−θ+4α1θ−1−21+θc1−2Pβ1+41+θ−4+αα1θ−1c2−4Pα1β1−2Pθβ1−Pαα1θβ2+4Pα1β1θ+4Pβ2+4Pα1β2+Pαα1β2+4Pθβ2−4Pα1θβ2,p3=2θα21−θB22+B1B2α2B21−θX−2α1−2α2−4−α1+θ−α1B121−θB2Y−α/4−αB1B2,p4=θ2,where X=ϕ+c1−2c2+Pβ1−2Pβ2 and Y=−2ϕ+αϕ+2c1−αc2+2Pβ1−αPβ2.According to the Schur-Cohn stability criterion, if the characteristic polynomialPλ, that is, all the eigenvalues of the Jacobian matrix JE∗ lie in the unit circle on the complex plane, the coefficients of the polynomial Pλ need to meet the following conditions at the same time:(19)P1=1+p1+p2+p3+p4=α1α2θ−12B2+B1−2+B2X2B2+B1−α+B2Yα−4B1B2>0,P−1=1−p1+p2−p3+p4=1−4+αB1B22B22α2−1+θα1−1+θ−21+θ+α1B12−1+θ4+2α2+4θ−2α2θ+α2B2−1+θX−α+B2Y+B1B2−4α1−1+θ−21+θα2−1+θ−21+θ+α−α1α2−1+θ2+41+θ2+α2B2−1+θ−4ϕ−αα1ϕ−4θϕ+αα1ϕθ+4−1+α1−1+θ−θc1+−4+αα1−1+θ+81+θc2−4Pβ1−4Pα1β1−4Pθβ1+4Pθα1β1+8Pβ2+4Pα1β2+Pαα1β2+8Pθβ2−4Pθα1β2−Pαα1θβ2>01±p4=1±θ2>0.From the hypothesis of the positive and negative system parameters and the inequality (12c)-(12d), we know that P1>0 and 1±p4>0 are obviously valid in a stable condition. Therefore, according to Jury’s criterion, the sufficient condition for the interior equilibrium points E∗ of the discrete dynamical system (8) to be asymptotically locally stable is if it meets the following conditions: P−1=1−p1+p2−p3+p4>0 and M3±>0, that is,(20)p1+p3<1+p2+p4p2−p2p4+p4−p43−p1p3+p12p4<1−p42+p2p4−p2p42+p1p3p4−p32⇔p1+p3<1+p2+p43+3p4>p21−p41−p42−p21−p4+p1−p3p3−p1p4>0p4<1,are established, and all eigenvalues of JE∗ satisfy λi<1i=1,2,3,4. Therefore, when condition (20) is satisfied, the internal equilibrium point E∗ is locally stable. From the above analysis of the equilibrium point, it can be seen that the stability of the equilibrium point is dependent on the values of the system parameters.


## 4. Numerical Simulations of the Model

In order to analyze the dynamic evolution behavior of the discrete dynamic system (8) more intuitively, we will use numerical simulation to visually describe the dynamic evolution process of the discrete dynamic system (8) as the model parameters change. We also focus on analyzing the impact of the market price P of credit transactions and the adjustment speed of R&D investment on the dynamics of the system. Let ϕ=4, c1=0.3, c2=0.4, θ=0.61, B1=0.53, B2=0.715, β1=4.3, β2=5.1, α2=1, P=0.131, and α=0.58.Figure2 shows the complex dynamic behavior of equilibrium solution oscillation of a discrete dynamical system (8) for different R\&D input adjustment speed α1 and credits transaction market price P. By comparing, Figures 2(a) and 1(a), it can be found that the complex attractors appear in the imbalance of Nash equilibrium as the adjustment speed α1 of R&D investment increases. Comparing Figures 2(b) and 1(c), it can be found that the Nash equilibrium amplitude of the discrete dynamic system (8) will change accordingly with the market price P of the credits transaction changes.Figure 2
The trajectory of time at differentα1 and P. (a)α1=16, P=0.125. (b)α1=17.4, P=0.125(c)α1=17.4, P=0.131.
(a)(b)(c)Figure3 shows the bifurcation diagram of the discrete dynamic system (8) as the adjustment speed α1 of R\&D investment changes when the market price P of credits transactions are P=0.125 and P=0.131, respectively. It is easy to see that in Figures 3(a) and 2(b), with the adjustment speed α1 of R\&D investment continues to increase, the discrete dynamic system (8) evolves from stable equilibrium to inverse period multiplication and inverse Neimark-Sacher bifurcation, and finally enters chaos. Figure 3 not only shows the different paths of chaos in the discrete dynamic system (8), but also shows that the greater the market price P of the credits transaction, the earlier the discrete dynamic system (8) enters the chaotic state. In addition, it can also be found that with the increase in the market price P of the credits transaction, the stable area of the discrete dynamic system (8) is reduced, which can explain why the behavior of the discrete dynamic system (8) becomes more unstable.Figure 3
The bifurcation diagram of the discrete dynamic system (8) with the adjustment speed α1 of R&D investment under the market price P of the credits transaction.
(a)(b)In order to highlight the influence of the market priceP of the credits transaction on the stability of the orbital evolution of the discrete dynamical system (8). Figure 4 shows the bifurcation diagram of the discrete dynamic system (8) with the market price P of the credits transaction under the adjustment speed α1 at different R&D investments. By observing Figure 4, it can be found that the discrete dynamic system (8) has inverse period-doubling bifurcation and inverse Neimark-Sacher bifurcation, respectively. As the adjustment speed α1 of R\&D investment increases, the discrete dynamic system (8) enters chaos more quickly. It can be explained that for the economic market of the credit transaction, the increase in the price of credit transactions market will increase the inventory of credits on the market. From an environmental perspective, the emergence of this phenomenon is conducive to promoting environmental protection, indicating that enterprises have invested in R\&D, energy conservation, and emission reduction. However, when the price of credits continues to rise, it will increase the inventory of credits in the market. After the supply of credit in the market reached a certain level, it caused an oversupply on the credit market, which affected the stability of the market. From an economic point of view, this phenomenon conforms to the law of market development. However, from the perspective of the harmonious development of environmental protection, this phenomenon is an undesirable endless loop that fails to achieve the desired effect of environmental protection.Figure 4
The bifurcation diagram of the discrete dynamic system (8) with the market price P of the credits transaction under the adjustment speed α1 at different R&D investment.
(a)(b)Figure5 shows the phase diagram of the discrete dynamic system (8) under different adjustment speeds of R\&D investment. Figure 5 gives a more detailed description of the evolution of the discrete dynamic system (8), which is a two-dimensional phase diagram corresponding to the value of α1 under different motion states of the orbit in Figure 2. By Figure 5, it can be found that when the adjustment speed α1 of R&D investment increases, the Nash equilibrium of the discrete dynamic system (8) will lose its stability, which will increase the number of complex strange attractors.Figure 5
The phase diagram of the discrete dynamic system (8) under different adjustment speeds of R&D investment increases.Select the adjustment coefficientα1=16.32 of the R&D investment in the chaotic state of the discrete dynamic system (8) in Figure 2(b), and the initial values are x10,x20,I10,I20=0.32,0.45,0.75,0.86, and 0.32001,0.45,0.75,0.86, respectively. The evolution graph of the orbit of the state variable of the discrete dynamic system (8) over time is shown in Figure 6. With the increase in the number of iterations, the same state variable is gradually separated under the influence of the two initial value conditions and evolves according to their respective orbits, which means that the discrete dynamic system (8) has sensitive dependence on the initial conditions.Figure 6
The discrete dynamic system (8) sensitive dependence on the initial value after instability.
(a)(b)


## 5. Using the Time-Delay Feedback Method to Control System Chaos

According to the numerical simulation results of the evolution process of the discrete dynamic system (8), the adjustment speed of R&D investment and the market price changes of credit transactions have a great influence on the stability of the system. The changes in the speed of adjustment of R&D investment and the market price of credit transactions have caused the monopoly market to bifurcate and cause unstable economic growth. If the economy falls into chaos, it should be controlled immediately. Otherwise, it will only get worse and cause serious consequences. It will not only worsen the market economy and fall into disorderly competition, but also make investors have bad expectations for the future and shake their confidence in government regulation, which will have a great impact on the stable and rapid development of the economy. Therefore, this paper uses the delay feedback control method [26–28] to control the chaotic behavior of the discrete dynamic system (8) so that it can return to a stable development state.If the chaotic state of the market is changed by changing the enterprise’s R&D investment, the enterprise needs to change the investment in R&D at every momentt, and the amount of change is proportional to x1t−x1t+1, that is, Kx1t−x1t+1, where K represents the intensity of regulation. By adjusting the value of K, the instability in the economic system can be eliminated. The management implications of this method are as follows: When market chaos occurs, it is necessary to invest according to the difference between the R&D investment in this period and a certain time before, which can realize the long-term stability of the system. By simplifying the discrete dynamical system (8) after adding the control items, we can obtain the controlled discrete dynamical system in the following form:(21)x1t+1=x1t+α11+K∗x1tϕB1−2B12θI1t+x1t−αB2B1θI2t+x2t−1−c1B1−β1B1P,x2t+1=x2t+α2x2tϕB2−2B22θI2t+x2t−B1B2θI1t+x1t−c2B2−1−β2B2P,I1t+1=θI1t+x1t,I2t+1=θI2t+x2t.Next, analyze the stability of the controlled discrete dynamic system (21). It is easy to know that the controlled discrete dynamic system (21) has the same equilibrium point as the discrete dynamic system (8). The Jacobian matrix corresponding to the discrete dynamic system (21) is obtained in the following form:(22)J=1+α1M1+K−α1B1B2x1α1+K−2α1θB12x11+K−θB1B2x1αα11+K−α2x2B1B21+α2N−α2θB1B2x2−2α2θB22x210θ0010θ.It can be seen from Figure3 that when the R\&D investment adjustment coefficient α1=16.32 and the credits transaction price P=0.18, the discrete dynamic system (8) shows a chaotic state. Under the original parameter values, the expression of the Jacobian matrix (22) at interior point equilibrium E∗ can be calculated as follows:(23)J=1−3.471731+K−1.972571+K−1.735861+K−1.183541+K−0.330.25−0.165−0.45100.6100100.61.Suppose the characteristic polynomial of matrix (23) is Pλ=λ4+u1λ3+u2λ2+u3λ+u4, and when its coefficients satisfy the stability condition of Schur–Cohn criterion the following conditions, that is,(24)1+u2+u4>u1+u3,1±u4>0,1−u42+u2u4−u2u42+u1u3u4−u32>u2−u2u4+u4−u43−u1u3+u12u4.The characteristic roots of matrix (23) are located in the unit circle on the complex plane. Thus, the interior point equilibrium of the controlled discrete dynamic system (21) is stable under the value of this set of parameters. The chaotic of a discrete dynamical system (8) can also be controlled to the expected stable orbit, and the value range of feedback gain intensity can be solved. From the stability conditions, we get that all the eigenvalues of the matrix (23) will lie inside the unit disc by providing K>0.0856.Next, numerical simulation is used to study the effectiveness of controlling the chaotic phenomenon of the discrete dynamic system (8). In Figure 7, the feedback control term is added to the iterative equation of the controlled discrete dynamic system (21). When the feedback gains strength K>0.0856, the unstable behavior of the discrete dynamic system (8) can be effectively controlled. Figure 8 shows the evolution of the controlled discrete dynamic system (21) from unstable to stable under different gain intensities. Through comparison, it can be found that the time required for the feedback to gain strength K=0.15 to reach a steady state is shorter than K=0.05, that is, the feedback gains strength value, and the more chaotic behavior can quickly control the stable orbit.Figure 7
Bifurcation diagram of a discrete dynamical system (21) with the change of feedback gain intensity.Figure 8
Evolution paths of discrete dynamical systems (21) under different gain intensities.


## 6. Conclusion and Discussion

This paper mainly studies the impact of the R&D investment and CAFC-NEVC mandate of the TEV enterprises and the NEV enterprises on automobile output and market stability. As mentioned above, for specific market structures, the impact of R&D investment on output has attracted considerable attention. The difference between this paper and the earlier papers is that when the market structure itself is sensitive to the level of R&D investment, that is, when the market structure is endogenous, the influence of R\&D investment and the CAFC-NEVC mandate on the output and market stability of automobiles is considered. Specifically, with the increase or decrease of R&D investment and the market priceP of the credit transaction, the stability of the oligopoly market is analyzed. Finally, the delay feedback control method is used to control the unstable behavior of the dynamic system.The study found that if the adjustment level of R&D investment decreases with the market priceP of the credit transaction, then the Nash equilibrium of the Cournot game is stable only in a certain range; if it exceeds a certain threshold, the game system will lose its original stability. We also found that the equilibrium is stable only when the adjustment level of R&D investment drops. In addition, the time-delay feedback control method could control the unstable behavior of the dynamic system efficiently and quickly and make the market quickly recover a stable and orderly state; the control effect will be better if the gain strength is larger, which provided a scientific basis for decision makers to effectively solve the market instability.Finally, this study also has some limitations, namely, that enterprises producing the same goods can expand the study of differentiated products—substitutes or complementary products— further. In this case, the dynamic Stackelberg competition and the influence of the CAFC-NEVC mandate can also be considered in detail.



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