MATLAB Implementation of Duffing Bifurcation Diagram with Dynamics Analysis

The Duffing equation serves as a classic model for studying nonlinear dynamical systems, where bifurcation diagrams and Lyapunov exponent analysis are crucial for understanding system stability. Implementing this in MATLAB requires integrating numerical computation methods with eigenvalue calculation techniques. Bifurcation Diagram Generation Workflow Bifurcation diagrams reveal system state transitions by scanning specific parameters (e.g., driving force amplitude) while keeping other parameters constant. Key implementation steps include: - Performing long-time numerical integration for each parameter value using MATLAB's ode45 solver (or other ODE solvers) with proper tolerance settings - Discarding transient responses and recording steady-state solutions after system stabilization - Extracting system states via Poincaré section method or direct sampling approach - Projecting state points corresponding to different parameter values onto a 2D plane to form the bifurcation diagram Lyapunov Exponent Calculation Essentials The largest Lyapunov exponent characterizes system sensitivity to initial conditions, with computational considerations: - Simultaneous integration of the original equations and linearized variational equations using coupled ODE functions - Implementing Gram-Schmidt orthogonalization at regular intervals to prevent vector direction collapse - Obtaining stable exponent values through long-term averaging of divergence rates Practical Implementation Recommendations: - Set appropriate parameter scanning steps with denser sampling near bifurcation points using adaptive step algorithms - Ensure sufficient integration time for steady-state convergence while balancing computational efficiency - Utilize MATLAB's event detection feature (odeset Event property) for precise Poincaré section intersection capture - For chaotic regions, validate results by combining phase portraits and power spectrum analysis through FFT computations This implementation framework applies to various Duffing system variants (hard/soft spring types) and can be extended to other nonlinear oscillators by modifying the nonlinear stiffness terms in the governing equations. Core MATLAB functions involved include ode45 for integration, quiver for phase space visualization, and lyap for stability analysis in linearized systems.