Lefschetz Center for Dynamical Systems Seminar
Abstract: Collins and Stewart pointed out that the characteristic gaits of quadrupeds --- walk, trot, pace, bound, etc. --- can be described by spatio-temporal symmetries of periodic functions where the spatial symmetries are leg permutations and the temporal symmetries are phase shifts. For example, when a horse paces it moves both left legs in unison and then a half-period later it moves both right legs, and so on. This form of motion is determined by invariance with respect to two symmetries: (1) Interchange front and back legs, and (2) swap left and right legs with a half-period phase shift.
Biologists postulate the existence of a central pattern generator (CPG) in the neural system which sends periodic signals to the legs. CPGs can be thought of as electrical circuits that produce periodic signals and can be modelled by coupled systems (or cells) of differential equations. Symmetries of the coupled system are just permutations of the cells that respect the coupling.
In this lecture we discuss animal gaits and describe how to determine the kinds of spatio-temporal symmetries that can exist robustly in systems with symmetry. We then show how to construct coupled cell systems that naturally produce periodic solutions having the symmetries of quadrupedal gaits. This construction leads to a number of predictions about animal gaits. We discuss these predictions and some of the supporting evidence.
This research is joint with Luciano Buono, Ian Stewart, and Jim Collins.
Center for Fluid Mechanics Seminar
Abstract: The aim of this seminar is to present some new results on simulation of bubble-induced modifications in two-phase flows. Using two-way coupling approach, presence of bubbles is modelled by point forces moving in the fluid flow, two bubbly flows are investigated :
Tuesday, September 14, 1999, Center for Fluid Mechanics Seminar Continued... - Bubble convection :
Injection of bubbles at low void fraction in an initially quiescent liquid induces a fluctuating anisotropic flow. Increasing level of fluctuations is observed up to a critical void fraction injection rate. Beyond this critical value a Rayleigh-Taylor type instability develops. The striking phenomenon is that no selection wavelength seems to exist.
- Modifications of a plane mixing layer :
A spatially evolving mixing layer is simulated and modifications of coherent structures dynamic is investigated. The role of entrapment phenomenon of bubbles in the vortical structures is demonstrated to be a major key to understand induced modifications.
Comparison with experiments shows that the qualitative trends are well represented by this model.
Stochastic Systems Seminar
Brown Analysis Seminar
Applied Mathematics Colloquium
Abstract: The Euler equations for the motion of an ideal fluid are a system of degenerate , non-elliptic PDE. We discuss how asymptotic methods of geometric optics can be used to obtain a sufficient condition for instability. We then consider the augmented system of PDE that govern ideal MHD. In this case the geometric approach gives an instability criterion in terms of the growth rate of the evolution operator of a system of local, hyperbolic PDE. This condition is applied to obtain sufficient conditions for instability of rotating MHD configurations.
This is joint work with Misha Vishik.
PDE Seminar
<--- 1999 Index