MATLAB Implementation of Duffing Oscillator with Code Description

The Duffing oscillator is a canonical nonlinear dynamical system frequently used to study chaotic phenomena. This system consists of a forced, damped nonlinear oscillator whose dynamic behavior exhibits rich nonlinear characteristics, including periodic, quasi-periodic, and chaotic motion. Implementing numerical simulation of the Duffing oscillator in MATLAB typically requires using ordinary differential equation (ODE) solvers such as `ode45`. The standard mathematical model of the Duffing oscillator can be expressed as: [ ddot{x} + delta dot{x} + alpha x + beta x^3 = gamma cos(omega t) ] where (x) represents displacement, (delta) is the damping coefficient, (alpha) and (beta) are nonlinear stiffness coefficients, and (gamma) and (omega) represent the amplitude and frequency of external excitation respectively. Implementation approach involves the following steps: Define the Duffing oscillator's differential equation and convert it into a system of first-order equations suitable for MATLAB solving. This conversion typically involves creating a function file that returns the derivatives of state variables. Utilize `ode45` for numerical integration, setting appropriate initial conditions and time range. The solver uses a Runge-Kutta method with adaptive step size control for accurate solution of stiff equations. Observe the system's transition from periodic motion to chaotic states by adjusting parameters such as (gamma) or (omega). This parameter exploration can reveal bifurcation phenomena. Visualize results through phase portraits or time series plots to analyze the system's dynamic characteristics. MATLAB's plotting functions like `plot` and `comet` can effectively demonstrate the oscillator's behavior. This implementation enables investigation of complex behaviors in Duffing oscillators, including bifurcations and chaotic attractors, making it suitable for simulation experiments in nonlinear dynamics and chaos theory research. The code structure typically includes parameter initialization, ODE function definition, solver invocation, and visualization components.