On the probability of finding a nonphysical solution through shadowing

TL;DR

This paper demonstrates that shadowing solutions in chaotic systems can often be nonphysical, challenging assumptions about the stability of statistical properties under small perturbations and impacting climate control and turbulence modeling.

Contribution

It provides a theoretical proof that shadowing solutions can be almost surely nonphysical, revealing limitations in current understanding of chaos and numerical simulations.

Findings

Shadowing solutions can be nonphysical with high probability.

Small perturbations may significantly alter statistical behavior.

Implications for climate control and turbulence simulation accuracy.

Abstract

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.