A comparative study of validated Taylor and Chebyshev long time integration of ODEs

Recent advances in the study of dynamical systems have increasingly relied on the rigorous numerical integration of finite-dimensional ordinary differential equations (ODEs). There exist different approaches to this problem, with the Taylor expansion of solutions representing the most traditional one. However, in recent years the Chebyshev expansion has attracted a lot of attention, primarily due to its ability to allow substantially longer integration steps in many cases. This paper focuses on the long time integration of analytical ODEs, and undertakes a comprehensive analysis of these two methods, evaluating their performance and efficacy when applied to a range of representative systems, including the logistic equation, Newton’s equation with different potential functions, the Lorenz system, and the stiff van der Pol equation.