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Complex dynamics of knowledgeable monopoly models with gradient mechanisms

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Abstract

This study explores the dynamics of two monopoly models with knowledgeable players. The first model was initially introduced by Naimzada and Ricchiuti, while the second one is simplified by the famous monopoly model of Puu. We employ several tools based on symbolic computations to analyze the local stability and bifurcations of the two models. To the best of our knowledge, the complete stability conditions of the second model are obtained for the first time. We investigate periodic solutions and their stability. Most importantly, we discover that the topological structure of the parameter space of the second model is much more complex than that of the first one. Specifically, in the first model, the parameter region for the stability of any periodic orbits with a fixed order constitutes a connected set. In the second model, however, the stability regions for the 3-cycle, 4-cycle, and 5-cycle orbits are disconnected sets formed by many disjoint portions. Furthermore, we find that the basins of the two stable equilibria in the second model are disconnected and have complicated topological structures. In addition, the existence of chaos in the sense of Li-Yorke is rigorously proved by finding snapback repellers and 3-cycle orbits.

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Notes

  1. The method of triangular decomposition can be viewed as an extension of the method of Gaussian elimination. The main idea of both methods is to transform a system into a triangular form. However, the triangular decomposition method is available for polynomial systems, while the Gaussian elimination method is just for linear systems. Refer to [37,38,39,40] for more details.

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Acknowledgements

The authors wish to thank Dr. Bo Huang for the beneficial discussions and are grateful to the anonymous referees for their helpful comments. This work has been supported by Philosophy and Social Science Foundation of Guangdong under Grant No. GD21CLJ01, National Natural Science Foundation of China under Grant No. 11601023, and Beijing Natural Science Foundation under Grant No. 1212005.

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Authors and Affiliations

  1. School of Business, Guangzhou College of Technology and Business, Guangzhou, China

    Xiaoliang Li

  2. Sino-French Engineer School, Beihang University, Beijing, China

    Jiacheng Fu & Wei Niu

  3. Beihang Hangzhou Innovation Institute Yuhang, Hangzhou, China

    Wei Niu

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  1. Xiaoliang Li
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Correspondence to Wei Niu.

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Appendices

Appendix A

$$\begin{aligned} SP&= (972K^8 +19440K^7 +127575K^6 +162000K^5 \\&\quad -1552500K^4 -6412500K^3 -5062500K^2 \\&\quad +23437500K+67187500 )( 8503056K^{12} \\&\quad +191318760K^{11}+1523464200K^{10}\\&\quad +3754532250K^9-14134854375K^8 \\&\quad -101982543750K^7 -146939062500K^6 \\&\quad +399469218750K^5 +1522072265625K^4 \\&\quad +261457031250K^3-4576816406250K^2 \\&\quad -1938867187500K +13981445312500),\\ C_4(K,M)&= K^{8} M^{8} +( -126K^{8} -210K^{7}) M^{7} \\&\quad +(6660K^{8} +21300K^{7} +17800K^{6}) M^{6} \\&\quad +( -192024K^{8} -874800K^{7} \\&\quad -1382400K^{6} -731000K^{5}) M^{5} \\&\quad +(3285360K^{8} +18688320K^{7} \\&\quad +41115600K^{6} +39438000K^{5}\\&\quad +13350000K^{4}) M^{4} \\&\quad +( -33957792K^{8} -221940000K^{7}\\&\quad -588016800K^{6} -728172000K^{5} \\&\quad -379740000K^{4} -45500000K^{3}) M^{3} \\&\quad +(206172864K^{8} +1453101120K^{7} \\&\quad +4191652800K^{6} +5433912000K^{5} \\&\quad +2183760000K^{4} -1105200000K^{3} \\&\quad -478000000K^{2}) M^{2} +( -672686208K^{8} \\&\quad -4870886400K^{7} -14246409600K^{6}\\&\quad -16185744000K^{5} +2054160000K^{4} \\&\quad +13262400000K^{3}-7632000000K^{2} \\&\quad -11520000000K) M +906992640K^{8} \\&\quad +6500113920K^{7}+18223833600K^{6} \\&\quad +13351392000K^{5} -25284960000K^{4}\\&\quad -27302400000K^{3} +65376000000K^{2} \\&\quad +30720000000K -102400000000. \end{aligned}$$

Appendix B

Table 5 List of abbreviations
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Table 6 List of symbols in model construction
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Li, X., Fu, J. & Niu, W. Complex dynamics of knowledgeable monopoly models with gradient mechanisms. Nonlinear Dyn 111, 11629–11654 (2023). https://doi.org/10.1007/s11071-023-08414-w

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