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Exploring nonlinear chaotic systems with applications in stochastic processes.
H G Abdelwahed1,2, Islam M Elbaz3,4, M A Sohaly5
1Department of Physics, College of Science and Humanities, Prince Sattam bin Abdulaziz University, Al-Kharj, 11942, Saudi Arabia.
Scientific Reports
|December 23, 2024
Summary
This study analyzes the stability of random systems using Lyapunov functions. It establishes criteria for various stability types and applies them to HIV/AIDS dynamics and financial models.
Area of Science:
- Stochastic Analysis
- Dynamical Systems Theory
- Mathematical Biology
- Financial Mathematics
Background:
- Stochastic differential equations (SDEs) are crucial for modeling systems with inherent randomness.
- Understanding the stability of equilibrium points in these systems is vital for predicting long-term behavior.
- Existing methods often struggle with systems exhibiting random coefficients and external noise like Brownian motion.
Purpose of the Study:
- To develop a comprehensive stability theory for stochastic models with random variable coefficients.
- To establish criteria for asymptotic mean-square stability, stability in probability, and stochastic global exponential stability.
- To apply these stability concepts to real-world problems in epidemiology and finance.
Main Methods:
- Construction of generalized Lyapunov functions tailored for stochastic systems.
- Derivation of necessary and sufficient conditions for different types of stability.
- Analysis of specific models including HIV/AIDS persistence and financial market dynamics.
- Numerical simulations and stability region analysis to validate theoretical results.
Main Results:
- Identified distinct stability conditions based on Lyapunov function properties.
- Established criteria for asymptotic mean-square stability, stability in probability, and stochastic global exponential stability.
- Demonstrated the application to HIV/AIDS models, confirming stochastic global exponential stability for the endemic equilibrium when the basic reproductive number exceeds one.
- Derived sufficient conditions for stability in stochastic market and Ornstein-Uhlenbeck models.
Conclusions:
- The developed stability theory provides a robust framework for analyzing complex stochastic systems.
- The findings offer critical insights into disease persistence dynamics and financial market stability.
- Numerical examples and simulations validate the theoretical advancements and their practical applicability.
Keywords:
Financial market modelsGlobal Mean-square stability (GMSS)HIV/AIDS stochastic modelPersistenceStochastic and random systemsMore Related Videos
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