MATLAB Code Implementation for Duffing Bifurcation Diagram and Lyapunov Exponent Calculation

The Duffing bifurcation diagram serves as a powerful visualization method for characterizing nonlinear dynamical systems. To compute Lyapunov exponents for continuous Duffing equations, specialized numerical programs are essential. This implementation typically utilizes Runge-Kutta integration methods (such as ode45 in MATLAB) to solve the Duffing differential equation system, incorporating algorithms like the Wolf method or Benettin's approach for Lyapunov exponent calculation. The code can simulate complex nonlinear behaviors in various physical systems including electronics, mechanical engineering, and chemical systems. Furthermore, the program integrates seamlessly with numerical computation tools like MATLAB and Python, enabling comprehensive analysis of system dynamics through parameter variation studies and phase space reconstruction. The implementation includes key functions for system discretization, Jacobian matrix calculation, and orthogonalization procedures to ensure numerical stability. Therefore, possessing a program capable of computing Lyapunov exponents for continuous Duffing equations proves invaluable for nonlinear dynamics research and engineering applications.