Classification and study of a new class of $ \xi^{(as)} $-QSO

TL;DR

This paper introduces a new class of quadratic stochastic operators on 2D simplex, classifies them into 18 non-conjugate classes, and analyzes the limiting behavior of trajectories for some classes.

Contribution

It presents a novel class of $\xi^{(as)}$-QSO, classifies these operators into 18 classes, and studies their trajectory limits, advancing understanding of nonlinear operators in this context.

Findings

Classified $\xi^{(as)}$-QSO into 18 non-conjugate classes.

Analyzed the limiting behavior of trajectories for four classes.

Provided insights into the dynamics of a new class of nonlinear operators.

Abstract

Many systems are presented using theory of nonlinear operators. A quadratic stochastic operator (QSO) is perceived as a nonlinear operator. It has a wide range of applications in various disciplines, such as mathematics, biology, and other sciences. The central problem that surrounds this nonlinear operator lies in the requirement that behavior should be studied. Nonlinear operators, even QSO (i.e., the simplest nonlinear operator), have not been thoroughly investigated. This study aims to present a new class of $\xi^{(as)}$-QSO defined on 2D simplex and to classify it into 18 non-conjugate (isomorphic) classes based on their conjugacy and the remuneration of coordinates. In addition, the limiting points of the behavior of trajectories for four classes defined on 2D simplex are examined.