Superposition of linear and quadratic operators arising in statistical mechanics

A significant portion of scientific and applied research worldwide focuses on the study of nonlinear dynamic systems. Understanding the behavior of such systems based on their initial conditions is crucial for predicting future developments. The theory of dynamical systems plays a vital role in deepening our comprehension of intricate phenomena across various domains, including biology, physics, economics, and healthcare. The current research addresses the superposition of a linear operator and a Volterra quadratic stochastic operator arising in statistical mechanics, which is established on a two-dimensional simplex space. The paper determines the stationary states of the combined transformation and classified their nature. Furthermore, it is demonstrated that nearly every trajectory under this transformation tends to approach the third corner of the two-dimensional simplex.