Since Kopel's duopoly model was proposed about three decades ago, there are almost no analytical results on the equilibria and their stability in the asymmetric case. The first objective of our study is to fill this gap. This paper analyzes the asymmetric duopoly model of Kopel analytically by using several tools based on symbolic computations. We discuss the possibility of the existence of multiple positive equilibria and establish necessary and sufficient conditions for a given number of positive equilibria to exist. The possible positions of the equilibria in Kopel's model are also explored. Furthermore, in the asymmetric model of Kopel, if the duopolists adopt the best response reactions or homogeneous adaptive expectations, we establish rigorous conditions for the local stability of equilibria for the first time. The occurrence of chaos in Kopel's model seems to be supported by observations through numerical simulations, which, however, is challenging to prove rigorously. The second objective is to prove the existence of snapback repellers in Kopel's map, which implies the existence of chaos in the sense of Li-Yorke according to Marotto's theorem.

翻译：自约三十年前Kopel双寡头模型提出以来，关于非对称情形下均衡点及其稳定性的解析结果几乎空白。本研究的第一目标旨在填补这一空白。本文利用基于符号计算的多种工具，对Kopel的非对称双寡头模型进行解析分析。我们讨论了多个正均衡存在的可能性，并建立了给定数量正均衡存在的充要条件。同时，还探究了Kopel模型中均衡点的可能位置。此外，在Kopel的非对称模型中，当双寡头企业采取最优反应策略或均匀自适应预期时，我们首次建立了均衡点局部稳定性的严格条件。数值模拟观测似乎支持Kopel模型中混沌现象的出现，但对其进行严格证明颇具挑战性。第二目标则是证明Kopel映射中存在snapback repeller，根据Marotto定理，这意味着Li-Yorke意义下的混沌存在性。