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

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.