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

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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 frst 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 efciency 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 signifcant 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. Te CAFC-NEVC mandate will inevitably intensify the integration and competition in the auto industry and increase the complexity of automakers' production decisions. Te advantage of new energy vehicles compared with traditional cars is that new energy vehicles are more environmentally friendly. It has higher fuel efciency, 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. Terefore, 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.
Te 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 efect 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]. Te research of Vooren et al. shows that the CAFC-NEVC mandate "labels" energysaving 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 efect of R&D patent growth and commercialization is signifcant [5]. Te research of Melton et al. believes that the CAFC-NEVC mandate efectively guides and stimulates the R&D and innovation activities of new energy vehicle companies and provides more models, production, and sales [6]. Te research of Syke and Axsen indicates that the mandatory CAFC-NEVC mandate efectively avoids the free-riding phenomenon in the spillover of R&D innovation [7]. Te research of Whistance et al. believes that the CAFC-NEVC mandate also plays an active role in promoting R&D and innovation in the felds of energy supply and infrastructure [8]. Te 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 decisionmaking, proft, 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 diferent 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 proft. 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]. Tis production decisionmaking model means that interests can be shared according to the shareholding ratio, but decision-making remains relatively independent (for example, fnancial investment rather than equity investment in reality), which is essentially diferent 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]. Terefore, 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]. Tese 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]. Te above studies deal with the irregular fuctuations 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 frst determines the output can obtain a larger auto market share and obtain higher profts, thus showing a frst-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. Te results of the 2 Discrete Dynamics in Nature and Society 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. Te 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 diferent 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 frst 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][27][28][29][30][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. Te possible innovations are as follows: frst, based on the principle of limited cooperation, beneft 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 fuctuations in points and fuctuations in demand for TEVs and NEVs, the strategy for adjusting R&D and production plans is given.
Te remainder of the paper is organized as follows: in Section 1, 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, timedelayed feedback control is used to stabilize the chaotic behaviors of the system. We conclude the paper in Section 5 with discussions on possible extensions.

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. Terefore, 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 competition π ι (θ ι , θ φ ) � ϕ − θ ι − α φ θ φ , ι, φ � 1, 2; ι ≠ φ, ϕ represents the upper limit of the market size, the market price, α j (0 ≤ α j ≤ 1) represents the substitution rate of j's products for i's, measure the cross-infuence of the change in demand of automobile manufacturer j on the change in demand of automobile manufacturer i. Te lower the α j value, the lower the replacement rate. High substitution rates often lead to more intense market competition [32]. If α j � 0 and α j � 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. Te latter is a complete substitute for the former, but the reverse is not true, that is, α 1 � 1, α 2 � α < 1. Te production cost functions of TEVs and NEVs are respectively C i (q i (t)) � c i q i (t) and a marginal cost c i is a positive number, and the cost of NEVs is higher than that of TEVs, that is, c 2 > c 1 [33].
Due to the CAFC-NEVC mandate, automakers will incur additional costs and benefts. 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 is e 0 L/100 km, the actual value of e 0 (1 − B 1 ) is L/100 km, and the normal number B i represents the technological level of the car companies' innovative products. Te government's standard value is (e 0 − △e) L/100 km, then the credits of the average fuel consumption of the enterprise are the product of the diference between the average fuel consumption of the enterprise and the actual value and the calculation quantity of the TEVs of the enterprise, that is, (B 1 e 0 − △e)q 1 (t); When B 1 e 0 > △e, a positive fuel consumption credits will be formed, which can be used to carry over to subsequent years but cannot be traded; When B 1 e 0 ≤ △e, negative fuel consumption credits are formed, and the carryover positive fuel consumption credits can only be used to compensate and return to zero. Terefore, the total credit demand of the TEV enterprises is is the credits coefcient of the TEVs. Te NEV manufacturers will earn positive NEV credits from the production of NEVs based on the cruising distance. Te NEV credits coefcient is β 2 (For example, NEV credits β 2 � 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; Terefore, each NEV will get higher credits. Te 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 fuctuation of credit prices is relatively small to meet the risk constraints of automobile manufacturers. With the intensifcation of credit price volatility, it will break through the upper limit of risk tolerance for auto manufacturers, and the market will become Discrete Dynamics in Nature and Society 3 unstable and chaotic. Te main factors afecting the production and competition strategies of both parties under the CAFC-NEVC mandate are shown in Figure 1.

Model Construction.
For the number of cars produced by each enterprise in a certain period, the auto enterprise's capital accumulation K i (t) determines its potential to produce cars in that period, and it is represented by the Te 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. K i (t − 1) represents the capital stock of the enterprise i in the period t − 1, and x i (t) 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 K i (t − 1) fows into the next economic period after the remaining θK i (t − 1), where θ (0 < θ < 1) is the depreciation rate. Terefore, the relationship between the capital stock of enterprise i in the two adjacent periods is Based on the above assumptions about the functional relationship of related variables, the profts of TEV manufacturer 1 and NEV manufacturer 2 in the t period are calculated as follows: Substituting the specifc expressions of each variable into equations (1) and (2), and obtaining the partial derivative of π i (x 1 (t), x 2 (t)) with respect to x i (t), the marginal profts of TEV enterprises and NEV enterprises are obtained as follows: 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 proft. In other words, if the observed proft margin of the enterprises i 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. Terefore, the dynamic adjustment mechanism of an enterprise i's R\&D investment can be expressed in the following form: where α i (x i (t)) is a positive adjustment function, which represents the adjustment range of the enterprise i's R\&D investment based on marginal proft. Tis paper still considers the linear adjustment function [15,18,19,23,24,28]: , the coefcient α i > 0 represents the speed at which the frm adjusts its R\&D investment strategy according to the marginal proft signal. Terefore, the dynamic equation (5) can be rewritten as follows: Using simultaneous equations (1)-(6), we can get a fourdimensional discrete dynamic model as follows: In order to be consistent in expression, this article replaces K i (t − 1) with I i (t), and obviously replaces I i (t + 1) with K i (t). Ten, the dynamic system (7) can be rewritten into the following standard form: 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 proft.

Equilibrium Point Stability Analysis
In the discrete dynamical system (8), let x i (t + 1) � x i (t) and I i (t + 1) � I i (t), i � 1, 2, we can obtain the following equation:  Discrete Dynamics in Nature and Society By solving the equations (9), we can obtain the four equilibrium points of the discrete dynamic system (8) as follows: where It is easy to know that E 0 , E 1 and E 2 are boundary equilibrium points, and E * is the only interior point equilibrium. Considering the real economic signifcance 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, B 1 , B 2 and θ are both positive parameters, when E 1 , E 2 , and E * are both greater than zero, the parameters must meet the following conditions: In the analysis later in this paper, the non-negativity conditions (12a)-(12d) are assumed to be valid.

Stability of the Boundary Equilibrium Point.
In order to discuss the stability of each equilibrium point (x 1 , x 2 , I 1 , I 2 ) of the discrete dynamic system (8), we calculate and obtain the Jacobian matrix corresponding to the discrete dynamic system (8) as follows: where 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 (x 1 , x 2 , I 1 , I 2 ) is asymptotically stable.

Proposition 1. Te boundary equilibrium point E 0 is the unstable equilibrium point.
Proof. Te specifcs of the Jacobian matrix (13) at the boundary equilibrium point E 0 are as follows: 6 Discrete Dynamics in Nature and Society By calculation, four eigenvalues of the Jacobian matrix (14) can be obtained, and the specifc expression is as follows: From the adjustment coefcient α i > 0 and the nonnegative condition (12a) of the equilibrium point, the latter two eigenvalues meet |λ 3,4 | > 1 are established. Terefore, according to the stability determination condition, the equilibrium point of the discrete dynamic system, E 0 is the unstable boundary equilibrium. □ Proposition . Te boundary equilibrium points E 1 and E 2 are both unstable equilibrium points.
Proof. Te specifc of the Jacobian matrix (13) at the boundary equilibrium point E 1 is as follows: where In the same way, four eigenvalues of the Jacobian matrix (15) can be obtained by calculation, and the specifc expression is as follows: By presupposing the parameters B 1 and α 2 > 0, then using the inequality (12d), we can conclude that λ 2 > 1 is established. Terefore, according to the stability determination condition of the equilibrium point of the discrete dynamic system, it can be known that E 1 is an unstable boundary equilibrium. A similar approach shows that E 2 is unstable too.

Stability of the Interior Point
Equilibrium. Next, we mainly discuss the stability of the interior equilibrium point E * . Te specifc of the Jacobian matrix (13) at the boundary equilibrium point E * is as follows: First, let the characteristic polynomial of the matrix J(E * ) be P(λ), and the specifc expression is P(λ) � λ 4 + p 1 λ 3 + p 2 λ 2 + p 3 λ + p 4 . Ten, through numerical calculation, we can obtain the coefcients of the polynomial P(λ) in the following form: Discrete Dynamics in Nature and Society According to the Schur-Cohn stability criterion, if the characteristic polynomial P(λ), that is, all the eigenvalues of the Jacobian matrix J(E * ) lie in the unit circle on the complex plane, the coefcients of the polynomial P(λ) need to meet the following conditions at the same time: From the hypothesis of the positive and negative system parameters and the inequality (12c)-(12d), we know that P(1) > 0 and 1 ± p 4 > 0 are obviously valid in a stable condition. Terefore, according to Jury's criterion, the sufcient 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 − p 1 + p 2 − p 3 + p 4 > 0 and |M ± 3 | > 0, that is, are established, and all eigenvalues of J(E * ) satisfy |λ i | < 1(i � 1, 2, 3, 4). Terefore, when condition (20) is satisfed, 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.

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, .131, and α � 0.58. Figure 2 shows the complex dynamic behavior of equilibrium solution oscillation of a discrete dynamical system (8) for diferent R\&D input adjustment speed α 1 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 α 1 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 3 shows the bifurcation diagram of the discrete dynamic system (8) as the adjustment speed α 1 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 α 1 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 fnally enters chaos. Figure 3 not only shows the diferent 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.
In order to highlight the infuence of the market price P 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 α 1 at diferent 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 α 1 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 afected 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 efect of environmental protection. Figure 5 shows the phase diagram of the discrete dynamic system (8) under diferent 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 twodimensional phase diagram corresponding to the value of α 1 under diferent motion states of the orbit in Figure 2. By Figure 5, it can be found that when the adjustment speed α 1 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.
Select the adjustment coefcient α 1 � 16.32 of the R&D investment in the chaotic state of the discrete dynamic system (8) in Figure 2 Figure 6. With the increase in the number of iterations, the same state variable is gradually separated under the infuence 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.

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 infuence on the stability of the system. Te 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 confdence in government regulation, which will have a great impact on the stable and rapid development of the economy. Terefore, this paper uses the delay feedback control method [26][27][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 moment t, and the amount of change is proportional to x 1 (t) − x 1 (t + 1), that is, K(x 1 (t) − x 1 (t + 1)), where K represents the intensity of regulation. By adjusting the value of K, the instability in the economic system can be eliminated. Te management implications of this method are as follows: When market chaos Discrete Dynamics in Nature and Society  occurs, it is necessary to invest according to the diference 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:    Discrete Dynamics in Nature and Society , 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). Te Jacobian matrix corresponding to the discrete dynamic system (21) is obtained in the following form: It can be seen from Figure 3 that when the R\&D investment adjustment coefcient α 1 � 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: Suppose the characteristic polynomial of matrix (23) is P(λ) � λ 4 + u 1 λ 3 + u 2 λ 2 + u 3 λ + u 4 , and when its coefcients satisfy the stability condition of Schur-Cohn criterion the following conditions, that is,  Figure 6: Te discrete dynamic system (8) sensitive dependence on the initial value after instability.
Te characteristic roots of matrix (23) are located in the unit circle on the complex plane. Tus, the interior point equilibrium of the controlled discrete dynamic system (21) is stable under the value of this set of parameters. Te 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. Discrete Dynamics in Nature and Society Next, numerical simulation is used to study the efectiveness 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 efectively controlled. Figure 8 shows the evolution of the controlled discrete dynamic system (21) from unstable to stable under diferent gain intensities. Trough 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.

Conclusion and Discussion
Tis 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 specifc market structures, the impact of R&D investment on output has attracted considerable attention. Te diference 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 infuence of R\&D investment and the CAFC-NEVC mandate on the output and market stability of automobiles is considered. Specifcally, with the increase or decrease of R&D investment and the market price P 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.
Te study found that if the adjustment level of R&D investment decreases with the market price P 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 efciently and quickly and make the market quickly recover a stable and orderly state; the control efect will be better if the gain strength is larger, which provided a scientifc basis for decision makers to efectively solve the market instability.
Finally, this study also has some limitations, namely, that enterprises producing the same goods can expand the study of diferentiated products-substitutes or complementary products-further. In this case, the dynamic Stackelberg competition and the infuence of the CAFC-NEVC mandate can also be considered in detail.

Data Availability
Te numerical simulations data used to support the fndings of this study were supplied by LiuWei Zhao under license and so cannot be made freely available. Requests for access to these data should be made to LiuWei Zhao, e-mail address: 136901672@qq.com.

Conflicts of Interest
Te authors declare that there are no conficts of interest.