Solving Delay Differential Equations Using dde23

To solve delay differential equations using dde23, first define all parameters and initial conditions for the differential equation system. The implementation typically involves creating a function handle that specifies the derivative expressions, where delays are explicitly handled through lag variables. For time history analysis, use dde23's output structure to plot solution trajectories over time, which helps visualize system dynamics and stability properties. For bifurcation analysis, develop a program that systematically varies key parameters while monitoring solution behavior. This requires embedding dde23 within parameter sweeps and tracking qualitative changes in solutions using appropriate detection algorithms. Key implementation aspects include proper handling of history functions for delayed states and configuring solver options for numerical stability. When plotting time series curves, consider using MATLAB's plotting functions like plot() or comet() for dynamic visualization, while bifurcation diagrams can be generated using scatter plots with parameter values on the x-axis and solution metrics on the y-axis. The dde23 solver automatically manages discontinuities in derivative history, but users should verify solution consistency through sensitivity analyses. Overall, solving delay differential equations with dde23 presents a complex yet fascinating computational challenge that requires careful parameter tuning and numerical experimentation to obtain meaningful results.