The seminar is free to attend but registration is required – please email the organizers to receive an invitation
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.