Complex dynamics of knowledgeable monopoly models with gradient mechanisms

TL;DR

This paper analyzes the complex dynamics of two knowledgeable monopoly models, revealing intricate stability regions, bifurcations, and chaos, with the second model exhibiting more complex topological structures than the first.

Contribution

It provides the first complete stability conditions for the second monopoly model and uncovers its complex parameter space and chaotic behavior.

Findings

Complete stability conditions for the second model are obtained.

The second model's parameter space has disconnected stability regions.

Chaos in the models is rigorously proved using snapback repellers and cycles.

Abstract

In this paper, we explore 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 from a famous monopoly introduced by 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 also investigate periodic solutions as well as 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 orbit 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 also 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 in the two models, respectively.