Japanese Journal of Mathematics

, Volume 2, Issue 1, pp 197–227

On the mathematics of emergence


    • Department of MathematicsCity University of Hong Kong
  • Steve Smale
    • Toyota Technological Institute at Chicago
Special Feature: The 1st Takagi Lecture

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


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:

emergenceflockingconsensus reaching

Mathematics Subject Classification (2000):

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© The Mathematical Society of Japan and Springer 2007