On the mathematics of emergence Authors Felipe Cucker Department of Mathematics City University of Hong Kong Steve Smale Toyota Technological Institute at Chicago Special Feature: The 1st Takagi Lecture

First Online: 28 March 2007 Received: 17 October 2006 Accepted: 24 January 2007 DOI :
10.1007/s11537-007-0647-x

Cite this article as: Cucker, F. & Smale, S. Jpn. J. Math. (2007) 2: 197. doi:10.1007/s11537-007-0647-x
Abstract.
We describe a setting where convergence to consensus in a population of autonomous agents can be formally addressed and prove some general results establishing conditions under which such convergence occurs. Both continuous and discrete time are considered and a number of particular examples, notably the way in which a population of animals move together, are considered as particular instances of our setting.

Keywords and phrases:
emergence
flocking
consensus reaching

Mathematics Subject Classification (2000):
92D50
92D25
91D30
Communicated by: Toshiyuki Kobayashi

This article is based on the 1st Takagi Lectures that the second author delivered at Research Institute for Mathematical Sciences, Kyoto University on November 25 and 26, 2006.

Steve Smale Partially supported by an NSF grant.

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