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

Year 2024, Volume: 73 Issue: 4, 1114 - 1133

Siti Nurlaili Karim , Nur Zatul Akmar Hamzah

https://doi.org/10.31801/cfsuasmas.1444857

Abstract

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.

Keywords

Quadratic stochastic operator, dynamics, system of equations, measurable partitions

Ethical Statement

This work does not involve live subjects (human or animal). The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Project Number

Fundamental Research Grant Scheme (FRGS) grant from the Malaysian Ministry of Education, project code FRGS/1/2021/ STG06/UIAM/02/1, project ID FRGS21-219-0828

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