Exponential stabilization of fixed and random time impulsive delay differential system with applications.

Area of Science:

• Dynamical Systems and Control Theory
• Nonlinear Science
• Mathematical Biology

Background:

• Functional differential equations and chaotic delayed equations exhibit complex dynamics.
• Impulsive effects, both fixed and random, significantly influence system stability and synchronization.
• Understanding these effects is crucial for modeling real-world phenomena like biological systems and resonators.

Purpose of the Study:

• To analyze the p-th moment global exponential stability of functional differential equations and scalar chaotic delayed equations under random impulsive perturbations.
• To investigate the p-th moment global exponential synchronization for these systems.
• To explore the impact of fixed and random time impulses on system behavior and validate the theoretical results with practical models.

Main Methods:

• Lyapunov function and Razumikhin technique were employed to prove the main theoretical results.
• The developed methods were applied to analyze the Mackey Glass blood cell production model.
• The Ikeda bistable resonator model was used to demonstrate the effectiveness of the proposed impulse control strategies.

Main Results:

• The study establishes criteria for achieving p-th moment global exponential stability and synchronization in the presence of random impulses.
• The application to the Mackey Glass and Ikeda models showcases the practical implications of the theoretical findings.
• Graphical representations effectively illustrate the influence of fixed and random impulses on system dynamics.

Conclusions:

• The research provides a robust framework for analyzing and controlling complex delayed systems with random impulsive effects.
• The findings contribute to a deeper understanding of stability and synchronization in chaotic systems.
• The study highlights the significance of impulse control in managing the behavior of biological and physical models.