MATLAB Implementation of Poincaré Section for Duffing Equation

This project presents a MATLAB implementation for generating Poincaré sections of the Duffing equation. The Duffing equation represents a class of second-order nonlinear ordinary differential equations containing cubic nonlinearity terms, commonly used to model vibrational systems in physics and engineering. The Poincaré section method serves as a fundamental technique for analyzing dynamical systems by projecting three-dimensional phase space trajectories onto a two-dimensional plane, enabling clearer observation of system behavior patterns and characteristics. In this implementation, we employ MATLAB's numerical ODE solver ode45 to compute solutions for the Duffing equation. The algorithm involves defining specific cutting conditions for the Poincaré section and systematically recording system state variables (position and velocity) each time the trajectory intersects the predefined plane. Key implementation aspects include: - Parameter configuration for the Duffing oscillator (nonlinear stiffness, damping, and forcing terms) - Event detection programming using ODE options to capture precise intersection points - State-space data collection at periodic intervals corresponding to external forcing phases The collected intersection data enables generation of phase portraits and trajectory visualizations, facilitating analysis of periodic orbits, chaotic behavior, and bifurcation patterns. This code-based approach provides practical insights into nonlinear dynamics through numerical experimentation and visualization techniques.