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Collision dynamics in dissipative systems

Presented by: 
Y Nishiura Tohoku University
Thursday 15th March 2012 - 12:00 to 12:50
Spatially localized dissipative structures are ubiquitous such as vortex, chemical blob, discharge patterns, granular patterns, and binary convective motion. When they are moving, it is unavoidable to observe various types of collisions. One of the main questions for the collision dynamics is that how we can describe the large deformation of each localized object at collision and predict its output. The strong collision usually causes topological changes such as merging into one body or splitting into several parts as well as annihilation. It is in general quite difficult to trace the details of the deformation unless it is a very weak interaction. We need a change in our way of thinking to solve this issue. So far we may stick too much to the deformation of each localized pattern and become shrouded in mystery. We try to characterize the hidden mechanism behind the deformation process instead. It may be instructive to think about the following metaphor: the droplet falling down the landscape with many valleys and ridges. The motion of droplets on such a rugged landscape is rather complicated; two droplets merge or split at the saddle points and they may sin k into the underground, i.e., annihilation. On the other hand, the profile of the landscape remains unchanged and in fact it controls the behaviors of droplets. It may be worth to describe the landscape itself rather than complex deformation, namely to find where is a ridge or a valley, and how they are combined to form a whole landscape. Such a change of viewpoint has been proposed recently claiming that the network of unstable patterns relevant to the collision process constitutes the backbone structure of the deformation process, namely the deformation is guided by the connecting orbits among the nodes of the network. Each node is typically an unstable ordered pattern such as steady state or time-periodic solution. This view point is quite useful not only for the problems mentioned above but also for more generalized collision problems, especially, the dynamics in heterogeneous media is one of the interesting applications, since the encounter with heterogeneity can be regarded as a collision. Similarly questions of adaptability to external environments in biological systems fall in the above framework when they are reformulated in an appropriate way. In summary a highly unstable and transient state arising in collision problems is an organizing center which produces various outputs in order to adjust the emerging environments.
University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons