Numerical methods for Chaotic ODE

This paper explores backward error analysis for numerical solutions of ordinary differential equations, particularly focusing on chaotic systems. Three approaches are examined: residual assessment, the method of modified equations, and shadowing. We investigate how these methods explain the success of numerical simulations in capturing the behavior of chaotic systems, even when facing issues like spurious chaos introduced by numerical methods or suppression of chaos by numerical methods. Finally, we point out an open problem, namely to explain why the statistics of long orbits are usually correct, even though we do not have a theoretical guarantee why this should be so.