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Bifurcation Analysis of a Duopoly Game with R&D Spillover, Price Competition and Time Delays

by B. A. Pansera 1, L. Guerrini 2, M. Ferrara 1,3,* and T. Ciano 1,4
Department of Law, Economics and Human Sciences & Decision_lab, ”Mediterranea” University of Reggio Calabria, via dell’Università, 25, 89124 Reggio Calabria, Italy
Department of Management, Polytechnic University of Marche, Piazzale Martelli 8, 60121 Ancona, Italy
Research Affiliate ICRIOS – The Invernizzi Centre for Research in Innovation, Organization, Strategy and Entrepreneurship Bocconi University – Department of Management and Technology, Via Sarfatti, 25, 20100 Milano, Italy
Faculty of Business and Law, University of Portsmouth, Richmond Building, Portland Street, Portsmouth PO1 3DE, UK
Author to whom correspondence should be addressed.
Symmetry 2020, 12(2), 257;
Received: 7 January 2020 / Revised: 2 February 2020 / Accepted: 3 February 2020 / Published: 7 February 2020
(This article belongs to the Special Issue Optimized Machine Learning Algorithms for Modeling Dynamical Systems)
The aim of this study is to analyse a discrete-time two-stage game with R&D competition by considering a continuous-time set-up with fixed delays. The model is represented in the form of delay differential equations. The stability of all the equilibrium points is studied. It is found that the model exhibits very complex dynamical behaviours, and its Nash equilibrium is destabilised via Hopf bifurcations.
Keywords: duopoly; Hopf bifurcation; R& D; time delay duopoly; Hopf bifurcation; R&; D; time delay
MDPI and ACS Style

Pansera, B.A.; Guerrini, L.; Ferrara, M.; Ciano, T. Bifurcation Analysis of a Duopoly Game with R&D Spillover, Price Competition and Time Delays. Symmetry 2020, 12, 257.

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