Nonlinear dynamic in a remanufacturing duopoly game: spectral entropy analysis and chaos control

Rami Amira; Mohammed Salah Abdelouahab; Nouressadat Touafek; Mouataz Billah Mesmouli; Hasan Nihal Zaidi; Taher S. Hassan; Rami Amira; Mohammed Salah Abdelouahab; Nouressadat Touafek; Mouataz Billah Mesmouli; Hasan Nihal Zaidi; Taher S. Hassan

doi:10.3934/math.2024374

AIMS Mathematics

2024, Volume 9, Issue 3: 7711-7727. doi: 10.3934/math.2024374

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Research article

Nonlinear dynamic in a remanufacturing duopoly game: spectral entropy analysis and chaos control

1.
Laboratory of Mathematics, Informatics and Systems (LAMIS), Echahid Cheikh Larbi Tebessi University, Tebessa, 12002, Algeria
2.
Laboratory of Mathematics and Their Interactions, Abdelhafid Boussouf University Center of Mila, Mila 43000, Algeria
3.
LMAM Laboratory, Faculty of Exact Sciences and Informatics, University of Jijel, 18000 Jijel, Algeria
4.
Department of Mathematics, College of Science, University of Ha'il, Ha'il 2440, Saudi Arabia
5.
Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt

Received: 29 December 2023 Revised: 07 February 2024 Accepted: 18 February 2024 Published: 23 February 2024
MSC : 39A23, 39A28, 39A30, 39A33, 39A60

In this study, our focus is on stabilizing a competitive game involving an original equipment manufacturer (OEM) and a third-party remanufacturer (TPR). To assess the presence of chaos within the dynamics of this game, we employ various analytical tools, including spectral entropy (SE), bifurcation diagrams, and Lyapunov exponents. The unpredictable nature of chaotic dynamics significantly influences the market and has negative implications for the strategic decisions of both firms. Our approach to counteracting this chaotic behaviour and stabilizing the system revolves around the implementation of the Ott, Grebogi, and Yorke (OGY) method. Crucially, our analysis highlights that the marginal costs ($ c_n $ and $ c_r $) incurred by the OEM and TPR emerge as pivotal factors in achieving stabilization within the game. To provide a tangible demonstration of the effectiveness of our proposed stabilization strategy in the context of this competitive environment, we conducted numerical simulations.
- remanufacturing,
- duopoly game,
- spectral entropy,
- chaos control,
- OGY method
Citation: Rami Amira, Mohammed Salah Abdelouahab, Nouressadat Touafek, Mouataz Billah Mesmouli, Hasan Nihal Zaidi, Taher S. Hassan. Nonlinear dynamic in a remanufacturing duopoly game: spectral entropy analysis and chaos control[J]. AIMS Mathematics, 2024, 9(3): 7711-7727. doi: 10.3934/math.2024374

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Abstract

In this study, our focus is on stabilizing a competitive game involving an original equipment manufacturer (OEM) and a third-party remanufacturer (TPR). To assess the presence of chaos within the dynamics of this game, we employ various analytical tools, including spectral entropy (SE), bifurcation diagrams, and Lyapunov exponents. The unpredictable nature of chaotic dynamics significantly influences the market and has negative implications for the strategic decisions of both firms. Our approach to counteracting this chaotic behaviour and stabilizing the system revolves around the implementation of the Ott, Grebogi, and Yorke (OGY) method. Crucially, our analysis highlights that the marginal costs ($ c_n $ and $ c_r $) incurred by the OEM and TPR emerge as pivotal factors in achieving stabilization within the game. To provide a tangible demonstration of the effectiveness of our proposed stabilization strategy in the context of this competitive environment, we conducted numerical simulations.

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