The Complexities in the R&D Competition Model with Spillover Effects in the Supply Chain

Tis study aims to investigate the research and development (R&D) competition within the supply chain, focusing on two aspects: R&D competition at the manufacturing level and competition in pricing strategies. Tis paper establishes a dynamic game model of R&D competition, comprising two manufacturers and two retailers, with both manufacturers exhibiting bounded rationality. Te key fndings are as follows: (1) an increase in the adjustment speed positively afects the chaotic nature of the R&D competition system, leading to a state of disorder. Tis chaotic state has adverse implications for manufacturing proftability. (2) Te spillover efect exhibits a positive relationship with the level of chaos in the R&D competition system. A greater spillover efect contributes to a more turbulent environment, which subsequently impacts the proftability of manufacturers. (3) R&D cost parameters exert a positive infuence on the stability of the R&D competition system. When the system reaches a state of equilibrium, an escalation in the R&D cost parameters poses a threat to manufacturer proftability. (4) Retailer costs play a detrimental role in the stability of the R&D competition system. As retailer costs increase, there is a decline in R&D levels, thereby diminishing manufacturer proftability. (5) To mitigate the chaotic state, we propose the implementation of the time-delayed feedback control (TDFC) method, which refects a more stable state in the R&D competition system.


Introduction
In the era of the knowledge economy and digitalization, businesses are increasingly emphasizing their technological innovation endeavors to adapt to external environmental changes and gain a competitive advantage.Numerous occurrences of cooperative innovation have manifested within various supply chains [1], as evidenced by partnerships between renowned automotive manufacturers GM and Benz with intermediaries from diverse nations.Likewise, in the context of China's community-based vegetable procurement, collaborative endeavors between vegetable suppliers and retailers have led to signifcant business model innovations.Technological innovation plays a vital role in driving economic growth at both national and regional levels, as well as boosting company profts [2].Consequently, competition among frms in research and development (R&D) has become more intense.R&D activities are instrumental in enhancing frms' core competitiveness, reducing operational costs, and cultivating unique aspects of their business that provide an edge over competitors.However, engaging in R&D activities also presents decisionmaking challenges for frms [3], while the complexity associated with managing R&D activities leads to numerous management hurdles [4].Moreover, the competitive nature of R&D strategies contributes to intricate behaviors within the entire system.In the real world, the spillover efect has emerged as a signifcant phenomenon in R&D activities, making it impossible for manufacturers to exclude competitors solely through their own R&D eforts.
As a consequence, the collaboration and R&D activities within the supply chain have led to an escalation of competitive dynamics [2].In the event of competition failing, it precipitates detrimental implications for all involved entities within the supply chain.Te losses incurred by enterprises such as Shi Hui Tuan and Ding Dong Maicai in 2021 substantially diminished the profts of vegetable suppliers and retailers, and in some cases, even resulted in their outright closure.Hence, given this scenario, it becomes imperative to address the ensuing questions concerning channel strategies: How do channels engage in competitive R&D?What are the ramifcations of such competitive endeavors?How can the competitive behavior of channel partners be regulated?
Te innovation of this article is comprised of the following facets: First, the present study presents an extension of the R&D competition model within the domain of the supply chain.By amalgamating R&D theory with the tenets of chaos theory, we systematically analyze the equilibrium points of the R&D competition system under the conditions of bounded rationality among participants, subsequently investigating the multifaceted dynamics that emerge in various scenarios.Second, we employ the time-delayed feedback control (TDFC) method as a strategic approach.By employing TDFC, our research endeavors to profciently regulate chaotic phenomena.Our fndings have revealed that TDFC not only diminishes the incidence of bifurcation and chaotic phenomena but also ensures the preservation of the original level of innovation, thereby safeguarding the efcacy of market mechanisms.

Literature Review
During the literature review, we have classifed the papers into two distinct categories: the R&D stream that primarily focuses on investigating the spillover efects of research and development and the chaos theory stream that primarily focuses on the topics of bifurcation and chaos control.
Te primary focus of the frst stream lies in the realm of R&D within the supply chain.In this domain, corporations fnd themselves equipped with a competitive advantage through their ventures in R&D.Te R&D activities of manufacturers exhibit a dual beneft, positively impacting both end-consumers and retailers [5].Indeed, it is imperative to recognize that these R&D endeavors can give rise to the manifestation of diverse competitive strategies among enterprises [1].In their insightful exploration of R&D collaboration within a supply chain, the authors of [6] assert that frms should diligently evaluate the spillover efect prior to embarking on cooperative R&D initiatives.Moreover, their research seeks to unravel the intricacies that underscore the occurrence of R&D-induced complexity, elucidating that frms tend to garner comparatively reduced profts within volatile environments.Signifcantly, the signifcance of R&D activities stems from their notable capacity to engender and propagate spillover efects throughout the supply chain.It is worthwhile to acknowledge that such activities can engender noticeable cost reductions for participating frms, a phenomenon often attributed to technological breaches or the sharing of erudite insights among researchers [7].Market inquiry not only posits that R&D competition may yield greater fnancial gains vis-à-vis collaboration, as posited by [2], but also presents the inquiry into cooperative behaviors within a competitive R&D framework involving three prominent oligopolistic entities, as scrutinized by [8].Nevertheless, it is incumbent upon us to underscore that R&D endeavors can yield intricate phenomena within the supply chain, as amply demonstrated by [9] investigation into the complexities that permeate a duopoly Stackelberg model of R&D competition.Te contribution of R&D to the value creation process by precipitating conspicuous cost reductions across the entire supply chain remains an irrefutable fact.In this vein, an inquiry by [10] into the chaos that often pervades R&D initiatives within monopolistic frms and [11] exploration of the sophisticated dynamics at play within high-tech manufacturers substantiate this assertion.
Te second stream of research focuses on the domain of chaos theory, which has garnered signifcant attention from scholars in recent years.Complexity is a pervasive phenomenon observed in economic and supply chain systems alike.Within chaotic systems, the sensitivity to initial conditions intensifes, leading to managerial challenges in decision-making processes [12].Consequently, systems can display transitions between chaotic and stable states, underscoring the criticality of efective chaos control strategies.Notably, a growing number of researchers have adopted the framework of bounded rationality to investigate economic models, as exemplifed by studies conducted by [13,14].Te authors of [15] have analyzed the dynamic behavior of discretizing a continuous-time Leslie prey-predator model.Te authors of [16] have examined complex behavior in investment patterns within spatial public goods games.Tese inquiries have shed light on the impact of bounded rationality on decision-making challenges within various market systems such as supply chains and platform ecosystems.Time delays and the infuence of bounded rationality have been identifed as key factors contributing to the intricate dynamics exhibited by these systems, including phenomena such as bifurcation and chaos.Consequently, research endeavors have proposed feedback control methods as viable means to address these complex behaviors.Notable studies by the authors of [17][18][19], the authors of [19] have thoroughly examined the application and efectiveness of feedback control methods.Additionally, time-delayed feedback control (TDFC) has emerged as a prominent technique used to stabilize unstable periodic orbits within non-linear dynamical systems, as demonstrated by the work of [20].Furthermore, the authors of [21], have proposed the incorporation of upper or lower bounds to alleviate chaos in dynamical systems, ofering an alternative approach to chaos control.
In summary, despite the existing research on supply chain innovation and chaos dynamics in academia, there are certain defciencies in their integration.Given the widespread phenomena and highlighted signifcance of collaborative R&D within the supply chain, related studies hold crucial signifcance.Te bounded rationality exhibited by supply chain participants necessitates exploring the occurrences of excessive or insufcient innovation, resulting in heightened market volatility.

Complexity
Te remaining sections of this paper are structured as follows: Section 3 presents the R&D competition model, while Section 4 explores the dynamic analysis of the game, specifcally highlighting the stable regions within the R&D competition game.Section 5 utilizes numerical simulations to illustrate the bifurcation diagrams, maximum Lyapunov exponents, and strange attractors.We implement feedback control on systems that have already entered chaotic states.Te conclusions and managerial recommendations are provided in Section 6.

Model
Tis investigation establishes a comprehensive model of supply chain R&D competition, as graphically displayed in Figure 1.In this context, the supply chain confguration encompasses two manufacturers and two retailers, where the manufacturers strategically formulate R&D policies to effectively compete for market share by engaging the retailers as intermediaries.Te structural framework of this discourse aligns closely with the pioneering contribution by [1].Te systemic dynamics of the supply chain R&D competition model follow a sequential game sequence, commencing with the manufacturers' deliberation on their respective R&D levels.Subsequently, the manufacturers judiciously determine their wholesale prices, followed by the retailers' formulation of optimal retail prices.Lastly, consumers exercise discernment in selecting their consumption tendencies.
Following the study by the authors of [22], retailers are confronted with a counter-demand function that can be represented as follows: where a denotes the magnitude of the market and q i (i � 1, 2) represents the quantity supplied by retailer i.
In the context of manufacturers' R&D activities, spillover efects are present.To simplify the analysis, we assume equal spillover efects between the two frms.Consequently, the cost functions for the two retailers can be defned as follows: Here, α signifes the extent of spillover efects, while c i (i � 1, 2) corresponds to the cost incurred by retailer i, and x i (i � 1, 2) denotes the R&D level of manufacturer i.It is worth noting that the R&D endeavors of the manufacturer yield cost reduction efects on both retailers.For computational ease, we make the assumption that the marginal cost of the manufacturer is negligible.
Te proft functions of the manufacturer can be defned as follows: where w i (i � 1, 2) represents the price set by manufacturer i and β i (i � 1, 2) denotes the parameter for R&D cost.We refer to the cost function as outlined in [23], setting it to be quadratic, indicating the phenomenon of diminishing returns in research and development benefts.Te proft function for retailers is given by By substituting equations (1) and (2) into equation ( 4), the retailer's function can be derived as follows: Taking the partial derivatives of π ri (i � 1, 2) in equation (5) with respect to q i (i � 1, 2), we obtain the necessary frstorder conditions: Solving these equations leads to the following solutions: Substituting equation ( 7) back into equation ( 5), the resulting expressions for the manufacturer's proft functions are as follows: Finally, diferentiating π mi (i � 1, 2) in equation ( 8) with respect to w i (i � 1, 2), we obtain the frst-order necessary conditions: Te solutions for w i (i � 1, 2) are given by Substituting equation (10) back into equation (8), and diferentiating π mi with respect to x i (i � 1, 2), we obtain Taking into account the practical constraints faced by manufacturers, namely, information and resource limitations, it becomes evident that arriving at entirely rational decisions is unattainable.Terefore, in line with the research conducted by [24], we adopt the assumption that each manufacturer operates under bounded rationality.Tis implies that manufacturers rely on previous-stage information when making R&D decisions.As a result, a dynamic system can be derived as follows: 4 Complexity In equation (12), c i (i � 1, 2) represents the adjustment speed of manufacturer i.

Equilibrium
Upon reaching a state of equilibrium in the R&D competition system, it becomes necessary to satisfy the following conditions: , where It is important to note that E1, E2, and E3 represent the corner solutions, bearing less signifcance in the context of R&D competition.Hence, our sole focus centers on the equilibrium solution E4, while analyzing the equilibrium conditions between the two manufacturers.
We can derive the equilibrium E4 in the R&D competition system.Te Jacobian matrix can be expressed as follows: Te trace of the Jacobian matrix is calculated as follows: Te determinant is determined as follows: Consequently, the characteristic polynomial of the Jacobian matrix can be expressed as follows: By evaluating the expression 2 ) > 0, we can conclude that equation (17) has two real roots [24].
According to the Jury rule [24,25], the stability condition for the R&D competition system states that Te condition (1) can be expressed as follows: In order to satisfy this condition, it follows that Te condition ( 2) is given by We can conclude that when When Det ≥ 0, the condition (3) can be represented as follows: We can conclude that 6 Complexity When Det < 0, We can conclude that when Det < 0, Figure 2 illustrates the distinct areas of stability: Area I represents a stable region, while Area II is characterized as unstable.In Area II, both conditions (1) and (2) are satisfed.Area III, on the other hand, falls within the realm of instability, meeting condition (2) but failing to satisfy conditions (1) and (3).Similarly, Area IV is unstable but complies with conditions (2) and (1), albeit falling short in satisfying condition (1) completely.By examining Figure 2, we discern that a low adjustment speed of manufacturers leads to a stable state within the R&D competition system.However, when the adjustment speed surpasses a certain threshold, the R&D competition system undergoes bifurcations or even transitions into chaotic states.Tese circumstances, in turn, pose challenges in managing the R&D level of manufacturers, thus increasing the complexities of their management.

Simulation
From the previous discussion, it is apparent that the R&D competition system may exhibit nonlinear behavior and potentially undergo bifurcation or even enter a chaotic state.To elucidate the impact of various factors in the system, numerical simulation methods are employed.In this section, our primary focus is on examining the infuence of decisionmaking behavior and R&D characteristics on dynamic behavior.Decision-making behavior comprises the adjustment speed of manufacturers and the marginal cost of retailers, which can also serve as a representation of the market conditions faced by manufacturers.R&D characteristics primarily encompass the infuence of spillover efects and R&D costs.Tere are four factors that impact the R&D competition system: adjustment speed, spillover efect, R&D cost, and retailer marginal cost.
We shall utilize a parameter set based on actual industry conditions.Tese parameters shall be incorporated into the R&D competition system (12) and subsequently subjected to computational simulation via MATLAB software.In the event of nonlinear behavior observed in the simulation outcomes, bifurcation and chaos phenomena manifest in the bifurcation diagram.Tis occurrence is accompanied by fuctuations in the maximum Lyapunov exponent, signifying an unstable phenomenon.Consequently, the predictability of corporate behavior diminishes, imposing heightened challenges for enterprise managers and resulting in consequential losses due to nonlinear behavior.

Adjustment Speed Efects.
In this subsection, we illustrate the efects of manufacturer's adjustment speed on stability.We utilize a fxed parameter set a with initial R&D levels set at x 1 (1) � 0.1 and x 2 (1) � 0.2. Figure 3 demonstrates that, holding other parameters constant, an increase in the adjustment speed c 1 has a destabilizing efect.
Trough numerical evidence of the complex dynamics in R&D competition system (12), we observe the bifurcation diagram of x 1 and x 2 concerning c 1 .Figure 3 reveals that the R&D competition system (12) remains stable when Complexity c 1 < 1.342.In this stable state, the R&D level of manufacturer 1 exceeds that of manufacturer 2, possibly due to cost pressures leading to higher R&D level for manufacturer 1.When c 1 ranges from 1.342 to 1.649, the R&D competition system (12) undergoes a period-2 bifurcation.Subsequently, as c 1 > 1.649, the R&D competition system (12) experiences a period-4 bifurcation.Ultimately, when c 1 > 1.734, a state of chaos emerges.It is notable that within the stable regime, variations in the adjustment speed utilized by manufacturers do not signifcantly impact the changes in R&D level.However, when the adjustment speed surpasses an optimal threshold, the R&D competition system exhibits nonlinear behaviors, thereby rendering predictability of R&D actions arduous and exacerbating managerial complexities within manufacturers.
Te largest Lyapunov exponent (LE) exhibits bifurcations and chaotic dynamics, whereby positive LE values indicate the occurrence of chaotic behavior.Te LE can be understood as the exponential rate at which the infnitesimally small separation between two nearby initial states in the evolving phase space expands over time [26].In Figure 4, the LE remains negative when c 1 < 1.342, thus indicating the stability of the R&D competition system (12) due to the low adjustment speed of x 1 .When c 1 approximates 1.342 and 1.649, the LE attains a zero value, suggesting the onset of bifurcation within the R&D competition system (12).Beyond c 1 ≈ 1.734, the LE primarily assumes positive values, signifying the entrance into a chaotic regime for the R&D competition system (12).Figure 5 visually presents the manifestation of strange attractors in the R&D competition system (12) at c 1 � 1.830.Additionally, Figure 5 demonstrates the partial synchronization of x 1 and x 2 during the state of chaos within the R&D competition system (12).
Figure 6 depicts the aggregate profts of Manufacturer 1 over 100 iterations based on the adjustment speed.As the adjustment speed increases, Manufacturer 1 eventually enters a chaotic state.It is evident that the adjustment speed has no impact on proft variations when the R&D competition system is in a stable state.However, following the occurrence of bifurcation, Manufacturer 1 experiences a signifcant decline in profts.Tis observation implies that while the adjustment speed does not afect the R&D level to a certain extent, the nonlinear behavior results in proft losses throughout the entire supply chain.Moreover, it introduces greater management challenges for manufacturers within the supply chain.

Retailer Marginal Cost Efect.
In this section, we elucidate the infuence of retailer marginal cost c 1 on the stability of R&D competition system (12).Te simulation in this section aligns closely with the previously presented data.We constrain the parameters to the following values: a � 10, 4}, and initialize the R&D level with x 1 (1) � 0.1 and x 2 (1) � 0.2.With the purpose of providing a fully representation of the impact of market capability on dynamic behavior, we judiciously adjust the prior adjustment speed, accentuating the increase in retailer 2's adjustment speed.Consequently, the R&D competition system's precariousness rises, which enhances the manifestation of chaotic phenomena, while preserving its evolutionary tendencies intact.Figure 7 substantiates the realization of stability in R&D competition system (12) in response to infated retailer marginal costs.
Within Figure 7, we peruse the comprehensive bifurcation diagram depicting x 1 and x 2 as nuanced by retailer 1's marginal cost c 1 .Ascertained therein is the R&D competition system's manifestation of chaotic behavior for c 1 values less than 0.605, transmuting into a bifurcation regime within the range of 0.605 < c 1 < 2.251.Profoundly, when c 1 exceeds the threshold of 2.251, the R&D competition system (12) stabilizes.It is pertinent to note that, during this phase of stability, retailer 1's marginal cost escalation precipitates 8 Complexity the decrement of x 1 and the concomitant increment of x 2 .
Evidently, greater retailer marginal costs instigate R&D competition system stability, implying that a stronger market advantage paradoxically engenders a heightened propensity towards chaotic dynamics.Figure 8 presents the Lyapunov exponent (LE) in relation to retailer marginal cost.For c 1 values less than 0.605, the LE exhibits negative values, indicating the R&D competition system (12) displays chaotic behavior.At approximately c 1 ≈ 0.680 and c 1 ≈ 0.960, the LE assumes a value of zero, signaling a transition to bifurcation in R&D competition system (12).Beyond c 1 ≈ 0.960, the LE reverts to negative values, suggesting the R&D competition system (12) attains stability.Figure 9 showcases the strange attractors of R&D competition system (12) at c 1 � 0.345.
Figure 10 portrays the aggregate proft of Manufacturer 1 over 100 iterations in relation to the impact of Retailer 1's marginal cost.It is evident that as Retailer 1's marginal cost escalates, Manufacturer 1's aggregate proft diminishes.While reducing marginal cost may seem economically advantageous, it proves deleterious to system stability.As previously expounded, the market infuence of R&D manufacturers directly afects their R&D levels.Greater market infuence corresponds to higher levels of R&D, resulting in maximal profts.However, it is crucial to recognize that such a scenario also engenders a heightened inclination towards system instability.Under these circumstances, manufacturers must strive to strike a balance among their R&D decisions, proftability, and market stability.

Spillover Efects. Tis analysis examines the infuence of spillover efects on the proftability and stability of R&D competition systems. Utilizing the fxed parameter set
7}, we consider the initial R&D levels as x 1 (1) � 0.1 and x 2 (1) � 0.2.We adjust the speed of adaptation to yield an uninterrupted visualization of the occurrence of chaotic phenomena.

Complexity
Figure 11 presents the bifurcation diagram of x 1 and x 2 with respect to the spillover efects, providing insight into the stability of the R&D competition system (12).Stability is achieved when α < 0.437, during which the Nash equilibrium point E4 remains locally stable.Specifcally, for α0.437 < α < 0.802, the R&D competition system (12) undergoes a transition into a 2-period bifurcation, while for α > 0.802, it exhibits chaotic behavior.Notably, due to their lower R&D costs, Manufacturer 1 gains a competitive advantage in innovation, surpassing Manufacturer 2 in terms of R&D levels.However, as spillover efects increase, Manufacturer 1 experiences a decline in R&D level, while Manufacturer 2 sees an improvement.Figure 11 illustrates that, while holding other parameters constant, an increase in spillover efects leads to instability in the R&D competition system (12).Noteworthy is the supply chain system's bifurcation and resulting instability when spillover efects are high, culminating in intermittent chaos within the R&D competition system [27].
Figure 12 showcases the Lyapunov exponent (LE) as it pertains to α. Notably, the LE denoted values below zero when α < 0.437, shedding light on the inherent stability of the R&D competition system (12) amidst low spillover effects.A threshold is observed at α ≈ 0.437, where the LE converges to zero, implying the emergence of a two-period bifurcation within the R&D competition system (12).Upon exceeding α > 0.802, the LE predominantly assumes positive values, signifying the R&D competition system's entry into a chaotic domain.Figure 8 efectively captures the intermittent chaos traits distinctly manifested in the R&D competition system.In parallel, Figure 13 visually delineates the strange attractors of the R&D competition system (12) at α � 0.910.Moreover, Figure 13 underscores the coalescence between x 1 and x 2 under chaotic circumstances within R&D competition system (12).
Figure 14 visually represents the aggregate proft of Manufacturer 1 over 100 iterations, elucidating its dependency on the spillover efect.Notably, a positive correlation is observed between Manufacturer 1's aggregate proft and the magnitude of spillover.Contrary to alternative scholarly fndings, an increase in spillover efects among distinct manufacturers yields a collective rise in their profts.However, as the spillover efect intensifes, the R&D competition system experiences bifurcations, impeding the rate of proft escalation.Simultaneously, the occurrence of these bifurcations leads to an augmented burden of managerial costs.Consequently, prudent supply chain managers are compelled to impose limitations on the intensity of spillover efects.

R&D Cost
Efects.Tis section presents a comprehensive analysis of the infuence of R&D costs on the stability of R&D competition system (12).Te specifed parameter set is held constant, while the initial R&D levels are set as x 1 (1) � 0.1 and x 2 (1) � 0.2.Consistent with the preceding discourse, adaptations are made to the adjustment speed, while maintaining the constancy of the remaining coefcients.10 Complexity Figure 15 visually portrays the R&D competition system's response to varying R&D costs, revealing the attainment of stability as these costs escalate.
In Figure 15, the bifurcation diagram of x 1 and x 2 is presented in relation to Manufacturer 1's R&D cost.It is evident that the R&D competition system (12) manifests a state of chaos when β 1 < 0.295, transitions into a state of bifurcation for 0.446 > β 1 > 0.295, and ultimately achieves stability when c 1 > 0.295.Under stable area, an increase in Manufacturer 1's R&D cost leads to a decrease in x 1 and an increase in x 2 .Notably, this observation refects the occurrence of intense market competition and potential chaos when R&D costs are low, ultimately giving way to a more stable system when R&D costs are relatively high.Moreover, it underscores the signifcance of enhancing R&D capabilities for manufacturers striving to maintain a competitive edge.
Figure 16 depicts the Lyapunov exponent (LE) in relation to β 1 : for β 1 < 0.295, the LE exceeds zero, indicating chaotic behavior in R&D competition system (12).At approximately β 1 � 0.446, the LE reaches zero, signifying the onset of bifurcation in the system.As β 1 surpasses the threshold of 0.446, the LE becomes negative, indicating R&D competition system (12) attains stability.Figure 17 illustrates the emergence of strange attractors in R&D competition system (12) when β 1 � 0.293.
Figure 18 depicts the progressive variation of manufacturer 1's aggregate proft over 100 iterations in response to changes in its R&D cost.A discernible pattern emerges,  revealing a negative relationship between Manufacturer 1's R&D cost and its aggregate proft.Specifcally, as the R&D cost increases, the aggregate proft experiences a consistent decline.Tis fnding underscores the intricate trade-of between proftability and stability within the context of R&D investment.Notably, when the R&D cost is low, the R&D competition system achieves a relatively higher proft level.However, this regime also coincides with a greater propensity for bifurcation and chaotic phenomena.Conversely, when the R&D cost is set at a higher value, the R&D competition system gravitates towards stability but at the expense of diminished proftability.Tus, it becomes imperative for manufacturers to diligently strike a delicate equilibrium between sustaining proftability and ensuring system stability.

Time-Delayed Feedback Control.
Based on the preceding discourse, it is discernible that under specifc circumstances, the R&D competition system manifests phenomena characterized by bifurcation and chaos.Tese phenomena yield a deleterious impact on the proftability of the R&D competition system, thereby posing formidable challenges to the management of the supply chain.To efectively govern the chaos and curtail fnancial losses with the manufacturers, the adoption of control mechanisms emerges as a customary choice.Within the context of Section 5.1, the R&D competition system (12) transitions into a state of chaos.To ameliorate this condition, we employ the Time-Delay Feedback Control (TDFC) method, as expounded by [27], as a means of mitigating the chaotic state prevalent in R&D competition system.Mathematically, the TDFC mechanism can be characterized by the equation   12 Complexity )), where the parameter k assumes the role of the chaos control parameter.As a consequence, the R&D competition system (12) undergoes a transformation.Notably, the ensuing R&D TDFC system can be succinctly presented as follows:

Complexity 13
So we get the R&D TDFC system as follows: In accordance with Section 4.1, the parameter set a was established as fxed, while the initial R&D levels assumed values of x 1 (1) � 0.1 and x 2 (1) � 0.2.As an outcome of this confguration, the R&D competition system demonstrated a state of chaos.Figure 19 efectively portrays the bifurcation diagram and the corresponding Lyapunov exponent (LE) in relation to the parameter k.A detailed scrutiny unveils that the R&D competition TDFC system devolves into a state of chaos when k assumes values below the threshold of 0.056, exhibits bifurcation within the parameter range of 0.056 < k < 0.366, and ultimately achieves stability when k surpasses the value of 0.366.Employing the TDFC method proves instrumental in attenuating intricate occurrences, namely bifurcation and chaos, without imposing any perceivable detriment upon the equilibrium state and its correlated innovation quotient.
In Figure 20, the absence of the TDFC mechanism elicits a chaotic state with x 1 .Continuing along this line of inquiry, subsequent explorations are undertaken for two distinct scenarios, where k is set at 0.2 and 0.5, respectively.Te acquisition of empirical evidence conclusively afrms that upon setting k equal to 0.2, the R&D competition TDFC system undergoes a two-period bifurcation, whereas an unequivocally stable state is attained as k acquires the value of 0.5.

Conclusions and Discussion
6.1.Conclusions.Tis study explores the stability of the R&D competition model within the context of the supply chain.It has been observed that four infuential factors, namely adjustment speed, spillover efect, R&D cost efect, and retailer marginal cost, signifcantly impact the stability of the R&D system.Trough simulated simulation of these factors, it has been ascertained that an escalation in adjustment speed and spillover efects precipitates a state of bifurcation, leading ultimately to the onset of a chaotic state in the R&D competition system.Te adjustment speed does not exert any infuence on the equilibrium R&D level; however, the occurrence of complex phenomena, such as bifurcation, yields a decline in manufacturer profts.Additionally, the spillover efect heightens the R&D level, resulting in augmented profts.Nevertheless, when this efect surpasses a certain threshold, R&D competition system instability ensues, thereby impeding proft growth rates.Conversely, elevations in R&D cost and retailer marginal cost promote the R&D competition system stability; nonetheless, they concomitantly give rise to diminished manufacturer profts.To combat the issue of chaotic dynamics, the TDFC mechanism has been introduced, which efectively mitigates chaos.

Teoretical Contributions.
Our study makes theoretical contributions in the following aspects: (1) We investigate the stability of R&D competition within the context of the supply chain, extending upon prior works such as [23,28], by integrating R&D competition, chaos theory, and the supply chain.(2) We enhance our understanding of the stability of the R&D competition system in the supply chain, shedding light on the origins of R&D competition system instability and providing valuable insights for further advancement of R&D practices.(3) We provide a foundation for managerial decision-making by examining critical factors infuencing R&D behavior, including adjustment speed, spillover efects, R&D costs, and retailer marginal costs.(4) Recognizing the trade-of between proft and stability, we propose the TDFC mechanism as an efective tool to alleviate chaos, aiming to enrich R&D research with the integration of chaos theory.

Managerial Implications.
Our research yields managerial implications for supply chain decision-making: First, we fnd that the R&D competition system becomes chaotic and loses control when the retailer's marginal cost reaches a sufciently low level.Tis indicates that supply chain innovation must necessarily take into account the retailer costs.When retailer costs are excessively reduced, such as in the case of team buying, it can lead to complex behaviors.At this time, the business model represented by team buying, which excessively relies on the cost-driven operating model, should be moderately controlled.Retailers with cost advantages may introduce complexities to the R&D competition system.Terefore, managers should carefully balance manufacturer proftability and stability.Second, although chaos may emerge in simulations, we conclude that the chaotic state can be efectively managed through the implementation of the TDFC method.Managers should consider the adoption of similar control mechanisms to mitigate chaotic behaviors and enhance system stability.

Limitations and Further
Research.Tis paper acknowledges several limitations.First, our R&D competition model assumes symmetry in the spillover efects among frms.However, in the real world, diferent frms may exhibit diverse spillover efects with each other, leading to potentially more interesting conclusions.Second, our assumption of collaboration between manufacturers and retailers overlooks power dynamics and possible competition between them.It is essential to pay closer attention to this aspect, as it is likely to yield more intriguing fndings.

Figure 20 :
Figure 20: Te time series diference between the R&D competition system (12) and under TDFC method.