# Complex Dynamics of a Model with R&D Competition

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The Model

## 3. Stability and Existence of Bifurcations

**Lemma**

**1.**

- (1)
- If $\pm i\omega $$(\omega >0)$ is a pair of purely imaginary roots of the characteristic equation, then ${\omega}^{2}$ is a positive root of the above quartic polynomial equation.
- (2)
- There exists at least one positive solution to (7) provided that $n\ne 1$ and $\left(\right)open="|"\; close="|">1+n$

**Proof.**

**Proposition**

**1.**

**Proof.**

**Proposition**

**2.**

**Proof.**

**Remark**

**1.**

**Theorem**

**1.**

- (1)
- If Equation (7) has no positive root, then the equilibrium ${E}_{*}$ of system (2) is locally asymptotically stable for $\tau \ge 0.$
- (2)
- If Equation (7) has a unique positive root ${\omega}_{*},$ then there exists a ${\tau}_{*}0,$ where ${\tau}_{*}=min\{{\tau}_{*}^{j},j\in \mathbb{Z}\},$ such that the equilibrium ${E}_{*}$ of system (2) is locally asymptotically stable when $\tau \in [0,{\tau}_{*}).$ As τ increases, the system dynamic may switch from stable to unstable, a Hopf bifurcation occurs, and then back to stable, and so on, according to $sign\left[G({\omega}_{*},{\theta}_{*})\right].$
- (3)
- If Equation (7) has at least two positive roots, then there may exist many stability switches, with the occurrence of a Hopf bifurcation at each switch.

## 4. Concluding Remarks

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Kamien, M.I.; Muller, E.; Zang, I. Research joint ventures and R&D cartels. Am. Econ. Rev.
**1992**, 82, 1293–1306. [Google Scholar] - d’Aspremont, C.; Jacquemin, A. Cooperative and noncooperative R&D in duopoly with spillovers. Am. Econ. Rev.
**1988**, 78, 1133–1137. [Google Scholar] - Qiu, L.D. On the dynamic efficiency of Bertrand equilibria. J. Econ. Theory
**1997**, 75, 213–229. [Google Scholar] [CrossRef] [Green Version] - Amir, R. Modelling imperfectly appropriable R&D via spillovers. Int. J. Ind. Organ.
**2000**, 18, 1013–1032. [Google Scholar] - Shibata, T. Market structure and R&D investment spillovers. Econ. Model.
**2014**, 43, 321–329. [Google Scholar] - Matsumura, T.; Matsushima, N.; Cato, S. Competitiveness and R&D competition revisited. Econ. Model.
**2013**, 31, 541–547. [Google Scholar] - Cavalli, F.; Naimzada, A. Complex dynamics and multi- stability with increasing rationality in market games. Chaos Solitons Fractals
**2016**, 93, 151–161. [Google Scholar] [CrossRef] - Dubiel-Teleszynski, T. Nonlinear dynamics in a heterogeneous duopoly game with adjusting players and diseconomies of scale. Commun. Nonlinear Sci. Numer. Simul.
**2011**, 16, 296–308. [Google Scholar] [CrossRef] - Peng, Y.; Lu, Q.; Xiao, Y. A dynamic Stackelberg duopoly model with different strategies. Chaos Solitons Fractals
**2016**, 85, 128–134. [Google Scholar] [CrossRef] - Ding, Z.W.; Zhu, X.F.; Jiang, S.M. Dynamical Cournot game with bounded rationality and time delay for marginal profit. Math. Comput. Simul.
**2014**, 100, 1–12. [Google Scholar] [CrossRef] - Bischi, G.I.; Lamantia, F. A dynamic model of oligopoly with R&D externalities along networks. Part I. Math. Comput. Simul.
**2012**, 84, 51–65. [Google Scholar] - Bischi, G.I.; Lamantia, F. A dynamic model of oligopoly with R&D externalities along networks. Part II. Math. Comput. Simul.
**2012**, 84, 66–82. [Google Scholar] - Smith, H. An Introduction to Delay Differential Equations with Applications to the Life Sciences; Springer: Berlin/Heidelberg, Germany, 2010. [Google Scholar]
- Zhang, Y.; Zhou, W.; Chu, T.; Chu, Y.; Yu, J. Complex dynamics analysis for a two-stage Cournot duopoly game of semi-collusion in production. Nonlinear Dyn.
**2018**, 91, 819–835. [Google Scholar] [CrossRef] - Cao, Y.; Colucci, R.; Guerrini, L. On the stability analysis of a delayed two-stage Cournot model with R&D spillovers. Math. Comput. Simul.
**2021**. [Google Scholar] [CrossRef] - Chen, S.S.; Shi, J.P.; Wei, J.J. Time delay-induced instabilities and Hopf bifurcations in general reaction-diffusion systems. J. Nonlinear Sci.
**2013**, 23, 1–38. [Google Scholar] [CrossRef]

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**MDPI and ACS Style**

Ferrara, M.; Ciano, T.; Gangemi, M.; Guerrini, L.
Complex Dynamics of a Model with R&D Competition. *Symmetry* **2021**, *13*, 2262.
https://doi.org/10.3390/sym13122262

**AMA Style**

Ferrara M, Ciano T, Gangemi M, Guerrini L.
Complex Dynamics of a Model with R&D Competition. *Symmetry*. 2021; 13(12):2262.
https://doi.org/10.3390/sym13122262

**Chicago/Turabian Style**

Ferrara, Massimiliano, Tiziana Ciano, Mariangela Gangemi, and Luca Guerrini.
2021. "Complex Dynamics of a Model with R&D Competition" *Symmetry* 13, no. 12: 2262.
https://doi.org/10.3390/sym13122262