Computational Approaches for Solving Complex Differential Equations: Implementation of the Direct Numerical Method with Modeling and Analysis of Bifurcation Diagrams and Lyapunov Exponent

In different areas like physics, engineering, biology, for example, to predict the behavior of composite systems one has to solve complex differential equations. This paper reviews the direct numerical method as a computational technique used to solve the difficult Equations above. Therefore, the application of this method will help us identify the behavior of large systems. Some of the methods used in the analysis include; creation of bifurcation diagrams to analyze how the system responds to different parameters, examine of Lyapunov exponent in an effort to understand the initial conditions' sensitiveness. Thus, in this paper uses the mentioned techniques to provide a clear understanding of the naturel of interactions within such systems. In addition to evidencing the value of direct numerical methods for rendering a precise portrayal of the systems, this study also improves our general comprehension of the underlying theory of such systems.