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Stability switching and its directions in cournot duopoly game with three delays


The authors appreciate the constructive suggestions and comments by anonymous referees which have greatly helped to improve the paper. The first author highly acknowledges the financial supports from the Japan Society for the Promotion of Science (Grant-in-Aid for Scientific Research (C) 20K01566). The usual disclaimers apply
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  • A three-delay duopoly is considered where the firms have identical implementation delays with different information delays. The equilibrium is locally asymptotically stable without delays however this stability is lost with increasing values of the delays. The stability properties of the equilibrium depend on the common implementation delay of the firms and on the sum of the two information delays. The stability switching curves are first analytically characterized and illustrated, and then the direction of the stability switching is determined at each point of the curves. The possibility of multiple pure imaginary eigenvalues is also discussed when the directions of the stability switches cannot be determined. Simulation examples illustrate the theoretical results.

    Mathematics Subject Classification: 34A34, 34C23, 34D20, 34K18.


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  • Figure 1.  Graphs of $ F(\omega ) $ in black, $ G_{1}(\omega ) $ in red and $ G_{2}(\omega ) $ in blue

    Figure 2.  Stability switching curves

    Figure 3.  Dynamics at point $ a $ for various values of $ \tau _{1} $ and $ \tau _{2} $

    Figure 4.  Dynamics at point $ b $ for various values of $ \tau _{1} $ and $ \tau _{2} $

    Figure 5.  Dynamics at point $ c $ for various values of $ \tau _{2} $ and $ \tau _{3} $

    Figure 6.  Graphs of $ F(\omega ) $ in black, $ G_{1}(\omega ) $ in red and $ G_{2}(\omega ) $ in blue

    Figure 7.  Stability switching curves

    Figure 8.  Nonlinear adjustment coefficient functions

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