Computing the Duffing Equation with MATLAB Implementation

The Duffing equation serves as a classical model in nonlinear dynamics, widely used for investigating chaotic phenomena. This equation describes nonlinear vibration systems with cubic stiffness, featuring a standard form that includes linear damping terms and external periodic excitation. When parameters enter specific ranges, the system exhibits chaotic behavior characterized by extreme sensitivity to initial conditions.

In MATLAB simulations, numerical methods such as the Runge-Kutta algorithm are typically employed to solve this differential equation. Key chaotic parameters include excitation amplitude, frequency, and damping coefficient - minor variations in these parameters can trigger abrupt transitions from periodic motion to chaotic states. During implementation, careful selection of time steps is crucial to maintain computational accuracy while avoiding excessive computational overhead. MATLAB's ode45 function provides an efficient adaptive step-size implementation of the Runge-Kutta method suitable for this application.

Through analytical tools like phase portraits, time series, and Poincaré sections, researchers can visually observe period-doubling bifurcation pathways and chaotic attractor morphology. This analysis facilitates understanding of complex system dynamics and provides theoretical foundations for vibration control in engineering applications. MATLAB's plotting capabilities enable effective visualization of these dynamic characteristics using functions like plot3 for phase space representation and scatter for Poincaré mapping.