Prof. Qiqi Wang, Massachusetts Institute of Technology (MIT), USA
Title: The climatic butterfly effect — do numerical simulations capture the statistics of chaotic systems?
The butterfly effect is a well-known phenomenon in fluid dynamics. A small perturbation can lead to large differences later in a chaotic dynamical system, such as turbulent flows. Lorenz famously posed the question, does the flap of a butterfly’s wings in Brazil set off a tornado in Texas? The answer is now widely accepted to be yes. This result has significant consequences to simulations that resolve chaotic dynamics, e.g., in fluid flows.
Whereas a tiny perturbation can change the state of a chaotic system, it is unclear whether it can change the long-time statistics. Statistics of many turbulent flows are known to be stable, insensitive to initial conditions. Ergodic theory provided a foundation for such stability. Indeed, for many unsteady flow simulations to be meaningful, we must believe that their statistics are not super sensitive to perturbations such as numerics. Many researchers rationalize his belief with the concept and theory of shadowing in dynamical systems.
Having dedicated a decade of research into the theory of shadowing, the speaker has found problems in this theory. Even systems that satisfy the most idealized assumptions in the shadowing theory can be arbitrarily sensitive to parameter perturbations. This question thus resurfaces: do numerical simulations capture the statistics of chaotic systems? In this talk, we will illustrate why the theory of shadowing cannot answer this question. We will then construct a simple mathematical model in which arbitrarily small perturbations can significantly influence the statistics of a stable, ergodic system. In Lorenz’s analogy, an intelligent butterfly could control the climate. Engineers must construct numerical simulations more meticulously to predict the statistics of chaotic fluid flows.
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