On dynamics of quadratic stochastic operators generated by 3-partition on countable state space

• Main Authors: Karim, Siti Nurlaili, Hamzah, Nur Zatul Akmar
• Format: Article
• Language: English
• Published: Ankara University, Faculty of Sciences 2024
• Subjects: QA Mathematics; QA300 Analysis
• Online Access: http://irep.iium.edu.my/117706/7/117706_On%20dynamics%20of%20quadratic%20stochastic%20operators.pdf; http://irep.iium.edu.my/117706/; https://dergipark.org.tr/en/pub/cfsuasmas/issue/88211/1444857

Quadratic stochastic operator (QSO) theory has advanced significantly since the early 1920s and is still growing due to its numerous applications in a variety of fields, particularly mathematics, where QSOs have inspired mathematicians to use and integrate various mathematical knowledge and concepts to better understand their properties and behaviors. Motivated by the relationship between the number of partitions on an infinite state space and the development of the system of equations corresponding to QSOs, this work sought to investigate the dynamics of QSOs formed by three partitions. First, we define and construct the 3-partition QSOs, which result in a system of equations with three variables. We then provide the formulation of the fixed point form and discuss its behavior using Jacobian matrix analysis. Some scenarios of three-partition QSOs with three different parameters are considered to readily investigate the type of fixed point in such systems. It is demonstrated that the operators can have either an attracting or a saddle fixed point but can never be repelling. We show how the saddle fixed point behaves, by identifying a set of points known as the fixed point’s stable manifold.