Introduction

This study puts core numerical methods to work across three classic dynamical systems and used to be one of my own task during my studies. I implement and compare forward Euler, modified Euler, and fourth-order Runge–Kutta for the smooth IVP (x' = x), empirically confirming first, second, and fourth-order global convergence via log–log error plots. I then analyze the Van der Pol oscillator: for (μ=0.5) distinct initial conditions collapse onto the same limit cycle, while for (μ=5) explicit Euler becomes unstable at coarse steps (e.g., (h=0.08)) and requires smaller steps (e.g., (h=0.01)) to resolve relaxation oscillations an instructive instance of stiffness. Finally, I integrate Newtonian two and three-body systems with RK4, reproducing closed two-body orbits and the celebrated equal-mass figure-eight choreography. Together, these experiments show how scheme order, step size, and problem structure jointly determine accuracy and stability turning abstract error estimates into concrete modeling choices suitable for real-world analysis.