Shadowing breakdown and large errors in dynamical simulations of physical systems

Simulations play a crucial role in the modern study of physical systems. A major open question for long dynamical simulations of physical processes is the role of discretization and truncation errors in the outcome. A general mechanism is described that can cause extremely small noise inputs to result in errors in simulation statistics that are several orders of magnitude larger. A scaling law for the size of such errors in terms of the noise level and properties of the dynamics is given. This result brings into question trajectory averages that are computed for systems with particular dynamical behaviors, in particular, systems that exhibit fluctuating Lyapunov exponents or unstable dimension variability.