Investigation of Iterative Time Step Value Effects on Chaotic Results Using a Nonlinear Reaction–Diffusion System

The Euler method is widely used for numerically solving differential equations due to its simplicity, but improper choice of time step size or neglect of system behavior can lead to significant errors that compromise simulation validity. Using the Gray–Scott reaction–diffusion model, the computational limitations of the Euler method are examined across distinct behavioral regimes and near regime transitions. Results show that the threshold for accuracy depends strongly on system dynamics, with increased sensitivity near transition regions. In particular, chaotic regimes require a drastically smaller iteration time step to maintain accuracy. These findings emphasize that while the Euler method can be sufficient in many cases, caution is needed to ensure reliable results, either through adjustment of iteration steps or the use of more accurate numerical schemes.