Article · Research in Mathematics · July 15, 2025
On the family of non-constrained Volterra cubic operators
Uygun Umurovich Jamilov, A. Yu. Khamrayev, Asliddin Z. Norov, Davron Aslonqulovich Juraev, Irfan Nurhidayat, Mohd Asif Shah
Abstract
This paper investigates a new class of non-constrained Volterra cubic stochastic operators (VCSOs) defined on a two-dimensional simplex. Unlike traditional approaches that impose structural constraints, we analyze the dynamic behavior of these operators in their general form. We establish the existence and classification of fixed points and identify invariant sets associated with the operator. Through rigorous analytical methods, including Jacobian analysis and Lyapunov functions, we demonstrate that all trajectories originating from points within the simplex converge to fixed points, confirming the operator’s regularity. Several illustrative examples and counterexamples are provided to validate the theoretical results. The findings contribute to the broader understanding of nonlinear stochastic dynamics and offer a theoretical foundation for modeling complex systems in biology, economics and engineering.
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