  This paper proves that shadowing solutions can be almost surely nonphysical.
This finding invalidates the argument that small perturbations in a chaotic
system can only have a small impact on its statistical behavior. This
theoretical finding has implications for many applications in which chaotic
mechanics plays an important role. It suggests, for example, that we can
control the climate through subtle perturbations. It also suggests that
numerical simulations of chaotic dynamics, such as turbulent flows, may fail to
predict the true long-term or statistical behavior.
