Solving the Duffing Equation

The Duffing equation is a significant class of nonlinear differential equations commonly used to model oscillatory systems with nonlinear restoring forces. This equation has broad applications in engineering and physics, including mechanical vibrations and electrical circuit systems.

In MATLAB, solving the Duffing equation typically employs numerical methods such as the Runge-Kutta algorithm (e.g., using the `ode45` solver), which is well-suited for numerical integration of nonlinear differential equations. By configuring appropriate initial conditions and system parameters (e.g., damping coefficient, nonlinear stiffness), users can obtain the system's temporal response and analyze dynamic characteristics like periodic solutions and chaotic behavior. The implementation involves defining the differential equation in a function file and calling ODE solvers with specified time spans and initial states.

Simulation plots intuitively demonstrate solution behaviors, such as displacement-time relationships and phase portraits (displacement versus velocity), helping researchers understand system stability and nonlinear phenomena. For deeper analysis, techniques like spectral analysis or Poincaré sections can be incorporated to further investigate frequency content and chaotic dynamics through MATLAB's signal processing and plotting capabilities.