The butterfly effect is a well-known phenomenon. A small perturbation to a chaotic dynamical system, such as turbulent flows or the Earth’s ocean and atmosphere, can lead to significant differences at a later time. The answer to the famous question, can the flap of a butterfly’s wings in Brazil set off a tornado in Texas, is now accepted to be yes. This phenomenon has significant consequences on modern computation in applications ranging from aerospace engineering to climate research.
Whereas a small perturbation can change the state of a chaotic system, it is unclear whether it can change the statistics. Ergodic theory shows that time-averaged statistics can be insensitive to initial conditions. If the weather is ergodic, it seems unlikely that a butterfly can change the weather’s statistics, also known as the climate. Indeed, for computed statistics about a chaotic system to be valid, the statistics can not be super sensitive to numerics and modeling approximations. Claims for such stability are supported by the theory of shadowing in dynamical systems.
Having dedicated a decade of research into shadowing in dynamical systems, the speaker has recently found the theory insufficient. In this talk, we show that even systems that satisfy the most idealized assumptions can be arbitrarily sensitive to tiny perturbations. This result raises questions on the fidelity of modern computation in predicting the statistics of chaotic systems. The result also reveals the possibility of controlling the long-time behavior of chaotic systems with little effort. These new mathematical results may significantly impact applications ranging from rocket design to climate change mitigation.