Welcome to my repository of explorations in Ordinary Differential Equations (ODEs), featuring a series of intriguing challenges and their resolutions. These explorations delve into various dynamical systems and applied ODE topics from the coursework MATH350.
An investigation into the family of differential equations ( x' = ax - x^3 - b ), exploring bifurcations in the parameter space and their impacts on equilibrium points. The exploration extends to the analysis of periodic solutions and bifurcation behavior under varying parameters.
- Detailed Analysis: Exploration 1.6 Solution
A dive into the complex behaviors of chemical systems, exemplified by the chlorine dioxide-iodine-malonic acid reaction. This exploration focuses on finding stability regions and stable limit cycles, applying the Poincaré-Bendixson theorem to simplified nonlinear systems.
- Comprehensive Exploration: Exploration 10.7 Solution
A series of challenges showcasing Euler, Improved Euler, and Runge-Kutta methods in action. This includes detailed visual demonstrations and an examination of error ratios, highlighting the nuances of each method and their applicability to different ODEs.
- In-depth Exploration: Exploration 7.5 Solution
An exploration into the sensitive dependence on initial conditions in ODEs, using the equation ( \frac{dy}{dt} = e^t \sin(y) ) This includes an analysis of Euler and Runge-Kutta methods' performances under varying initial conditions.
- Fascinating Findings: Exploration 7.6 Solution
This challenge involves analyzing a 3-parameter linear system represented by the differential equation ( X' = \begin{pmatrix} a & b \ c & 0 \end{pmatrix} ). The exploration involves exploring different behaviors in the solution space as parameters vary, such as identifying regions in the parameter space corresponding to various system behaviors like saddles, centers, and spiral sinks.
- Detailed Insights: Exploration 4.3 Solution
Each challenge in this repository not only demonstrates my proficiency in tackling complex ODE problems but also showcases the beauty and intricacy of dynamical systems. Whether you're a grad school committee member, a potential employer, or simply an enthusiast in mathematics, these explorations offer a deep dive into the fascinating world of ODEs, all underscored by rigorous analytical and numerical methods.