Impact of tolerances in the RK45 method on bifurcation analysis

Bifurcation analysis is a critical tool for understanding the behavior of nonlinear dynamical systems. In this study, we explore the impact of tolerance settings in the RK45 method on the accuracy of bifurcation analysis. While tighter tolerances can improve the accuracy of numerical solutions, they also significantly increase the computational cost, which is particularly important in bifurcation analysis that requires numerous iterations to explore parameter spaces. Using the Duffing equation as a case study, we investigate how variations in tolerance settings affect the observed bifurcation phenomena. We conduct numerical experiments to compare the performance of the RK45 method with that of the 4th-order Runge-Kutta method (RK4), focusing on the accuracy of the solution trajectories and the computational efficiency. Our results show that default tolerance settings in the RK45 method may fail to capture certain bifurcation phenomena, potentially leading to misleading conclusions. Interestingly, the bifurcation diagrams obtained by varying rtol revealed a rich variety of bifurcations, representing a novel discovery not previously reported. To address this, we identify a range of tolerance values, approximately 1×10-4 to 1×10-5, that balance accuracy with computational efficiency. These findings offer practical guidelines for selecting appropriate tolerance settings in numerical integration, ensuring accurate bifurcation analysis while minimizing computational resources.