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On a Volterra Cubic Stochastic Operator.

U U Jamilov1, A Yu Khamraev2, M Ladra3

  • 1Institute of Mathematics, Academy of Sciences of Uzbekistan, Tashkent, Uzbekistan, 100170. jamilovu@yandex.ru.

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Summary

This study analyzes Volterra cubic stochastic operators, detailing their fixed points and invariant sets. Researchers demonstrate these operators possess the ergodic property, crucial for understanding system dynamics.

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Cubic stochastic operatorKolmogorov systemQuadratic stochastic operatorVolterra and non-Volterra operator

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Area of Science:

  • Stochastic analysis
  • Dynamical systems theory

Background:

  • Volterra cubic stochastic operators are fundamental in modeling complex systems.
  • Understanding their asymptotic behavior is key to predicting long-term system states.

Purpose of the Study:

  • To characterize the fixed points and invariant sets of Volterra cubic stochastic operators.
  • To investigate the asymptotic behavior and ergodic properties of these operators.

Main Methods:

  • Fixed point analysis
  • Invariant set determination
  • Lyapunov function construction
  • Asymptotic behavior analysis

Main Results:

  • A comprehensive description of the fixed point set was established.
  • Invariant sets were identified for the operators.
  • Lyapunov functions were successfully constructed to analyze stability.
  • The ergodic property of the operators was proven.

Conclusions:

  • The study provides a thorough understanding of Volterra cubic stochastic operators.
  • The identified ergodic property is significant for applications in probability and dynamical systems.
  • The methods developed offer a framework for analyzing similar stochastic operators.