This repository serves as a way to quickly find all of my STEM related projects I would like to make public. Each project listed has a link to the repository and either a short description or an abstract to describe what the project is about. Do keep in mind that the papers and provided source code for class projects are NOT the original copies of the papers and source code. These papers were modified in a way so that it is more presentable while maintaining all of the information from the original copy. The source code was reformatted in a way so that it is optimized and structured for readability.
Bifurcation theory plays a fundamental role in understanding the qualitative behavior of dynamical systems. In this paper, we present a comprehensive study of various bifurcations within a carefully constructed system of equations. Our objective is to investigate the dynamic changes and transitions that occur as system parameters are varied. First, we establish a set of equations that capture the essential characteristics of the dynamical system under investigation. These equations are derived based on a thorough analysis of the underlying physical or mathematical phenomenon of interest. We ensure that the system exhibits nonlinear behavior and contains appropriate parameters to facilitate the exploration of bifurcations. Next, we employ analytical techniques to investigate the bifurcation points within the system. By analyzing the stability and eigenvalues of the system, we determine the types of bifurcations that arise, such as saddle-node, transcritical, pitchfork, and Hopf bifurcations. Furthermore, we utilize graphical tools, such as bifurcation diagrams and phase portraits, to visualize and interpret the bifurcation phenomena. In summary, this paper presents a systematic approach to construct a system of equations and investigates the bifurcations within it. Our findings contribute to the broader understanding of bifurcation theory and its applications, providing a valuable framework for exploring and analyzing the dynamic behavior of complex systems.
The double pendulum is a classic example of a chaotic system that exhibits complex and unpredictable behavior. In this paper, we investigate the dynamics of the double pendulum and analyze its chaotic characteristics. We begin by formulating the equations of motion using Lagrangian mechanics, considering the gravitational forces and constraints on the system. Through numerical simulations and computational techniques, we explore the evolution of the double pendulum system under various initial conditions. Our analysis reveals that the double pendulum demonstrates sensitive dependence on initial conditions, commonly referred to as the "butterfly effect." Small variations in the initial conditions lead to significant deviations in the system's subsequent motion, making long-term predictions challenging.
Understanding the intricate interactions within ecosystems is vital for comprehending the delicate balance that sustains life on our planet. This paper is concerned with modeling an ecosystem characterized by the simultaneous occurrence of amensalism, mutualism, and predation, exploring the interplay between these ecological relationships and their impact on species diversity and community stability. The model formulated is shown to admit only positive solutions that are also bounded. We determine the equilibrium points, conduct a comprehensive analysis of their stability and numerical computations of the proposed model are provided. Further, numerical simulations that demonstrated the existence of a Hopf bifurcation about the interior equilibrium point for several parameter values are also provided.
Red Blood Cells are present in the human body and their purpose is to deliver oxygen to the human body while giving carbon dioxide for humans to exhale. The purpose of this paper is to show that Red Blood Cells can maintain an equilibrium state if a certain condition is met. Multiple models have been tested to show that this condition is consistent given the proper assumptions. The first model being analyzed is a system of linear difference equations. The second model being analyzed is a system of linear differential equations. The third model being analyzed is a system of nonlinear difference equations.