Abstract:  The butterfly effect is a well-known phenomenon in fluid dynamics. A small perturbation to a chaotic dynamical system, such as turbulent flows or the Earth's atmosphere, can lead to large differences at a later time. 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 in aerodynamic 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. If the weather is ergodic, then it seems unlikely that the butterfly in Brazil can affect the long-time statistics of weather, also known as the climate of Texas. Indeed, for many scale-resolved aerodynamic simulations to be meaningful, we must believe that their statistics are not super sensitive to perturbations such as numerics and modeling approximations. The concept and theory of shadowing in dynamical systems support such claims for stability.
     The speaker, having dedicated almost a decade of research into the theory and computation of shadowing, has recently found the approach insufficient. Even systems that satisfy most idealized assumptions in shadowing theory can be arbitrarily sensitive to parameter perturbations. This question thus resurfaces: can a butterfly in Brazil change the climate of Texas? In this talk, we will illustrate why the theory of shadowing cannot give a negative answer to 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, a sufficiently intelligent butterfly could wield robust control over the climate.
     While still seeking an answer to the question in the title for realistic flow physics, we emphasize its importance to aerodynamics and control. If the answer is yes, we must question whether we can trust any numerical or laboratory simulation in predicting the statistics of chaotic aerodynamics, even if numerics and modeling can be made arbitrarily precise. A positive answer also reveals an opportunity to control the statistics of large and powerful chaotic flows with tiny and even imperceptible control actions.

BIO:  Qiqi Wang  (pronounces as Chi-chi Wong) got his BS in mathematics from the University of Science and Technology at Hefei, China. He then got his Ph.D. in Computational and Mathematical Engineering at Stanford with a minor in Aeronautics and Astronautics. Since 2009, Qiqi Wang has been in ACDL, first as an assistant professor, then an associate professor, and then a tenured associate professor. Qiqi Wang co-founded FlexCompute, offering a fast and easy to use Computational Fluid Dynamics service.