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, if the duopolists adopt the best response reactions or homogeneous adaptive expectations, we establish rigorous conditions for the existence of distinct numbers of positive 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，在非对称情形下关于均衡及其稳定性的解析结果几乎空白。本研究的首要目标正是填补这一空缺。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。我们讨论了多重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。同时探索了Kopel模型中均衡的可能位置。此外，当双寡头采取最优反应策略或同质适应性预期时，我们首次建立了不同数量正均衡存在的严格条件。数值模拟观测似乎支持Kopel模型中混沌现象的存在，但严格证明这一现象颇具挑战性。第二个目标是证明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。