Numerical analysis and chaos control: a study of lorentz systems with visual basic for application implementation

Abstract

The study focuses on the numerical analysis and chaos control of Lorenz systems, leveraging Visual Basic for Application in Microsoft Excel for modeling and visualization. Chaotic systems, including the Lorenz attractor, represent a fundamental concept in nonlinear dynamics and chaos theory, characterized by sensitivity to initial conditions, nonlinearity, and fractal dimensionality. These properties make such systems valuable for analyzing complex processes in physics, biology, engineering, and economics. The research extends traditional exploration of the Lorenz attractor by introducing numerical methods such as the four-point Adams method with adaptive step selection. Classical parameter sets and non-classical modifications are examined. Additionally, a modified Lorenz system incorporating a supplementary term is analyzed, demonstrating distinct dynamic behaviors and trajectories. This work highlights the applicability of the developed Visual Basic for Application-based tools for solving nonlinear differential equations and visualizing complex attractors. The integration of Adams and Krylov methods enhances computational efficiency and precision. The outcomes align with previous studies and suggest that the software can address a wide range of applied mathematical and engineering challenges, including chaos management in dynamic systems. The findings underline the potential of the Lorenz attractor as a testbed for chaos control methods and numerical analysis techniques, with broader implications for scientific and practical applications across various disciplines.