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Stephen Smale, a wellknown differential geometry, differential topology, dynamical systems guy, co-authored the Cucker-Smale model of FLOCKING:
a dynamical system where point particles move in ways influenced by their neighbors, as you can see in a school of fish, flock of birds, cloud of gnats.
Flocks can be frightened by predators, they can divide, merge back together, and change shape, all under a kind of leaderless collective uncontrol, seemingly without a guiding plan.
Here is a new paper about modeling flock behavior:
http://arxiv.org/abs/1102.5575
A new model for self-organized dynamics and its flocking behavior
Sebastien Motsch, Eitan Tadmor
(Submitted on 28 Feb 2011)
We introduce a new model for self-organized dynamics which, we argue, addresses several drawbacks of the celebrated Cucker-Smale (C-S) model. The new model does not only take into account the distance between agents, but instead, the influence between agents is scaled in term of their relative distance. Consequently, the new model does not involve any explicit dependence on the number of agents; only their geometry in phase space is taken into account. Our new model lacks, however, the symmetry property of the original C-S model, which was the key for the various recent studies of C-S flocking behavior. To this end, we introduce here a new framework to analyze the phenomenon of flocking in the presence of non-symmetric influence matrix. With this aim in mind, we develop a new concept of active sets. We then present a unified framework for studying the flocking behavior for rather general classes of dynamical systems, including the proposed new model and strongly asymmetric models with "leaders". The methodology presented in this paper, based on the notion of active sets, carries over from the particle to kinetic and hydrodynamic descriptions. In particular, we discuss the hydrodynamic formulation of our new model, and prove its unconditional flocking for slowly decaying influence functions.
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a dynamical system where point particles move in ways influenced by their neighbors, as you can see in a school of fish, flock of birds, cloud of gnats.
Flocks can be frightened by predators, they can divide, merge back together, and change shape, all under a kind of leaderless collective uncontrol, seemingly without a guiding plan.
Here is a new paper about modeling flock behavior:
http://arxiv.org/abs/1102.5575
A new model for self-organized dynamics and its flocking behavior
Sebastien Motsch, Eitan Tadmor
(Submitted on 28 Feb 2011)
We introduce a new model for self-organized dynamics which, we argue, addresses several drawbacks of the celebrated Cucker-Smale (C-S) model. The new model does not only take into account the distance between agents, but instead, the influence between agents is scaled in term of their relative distance. Consequently, the new model does not involve any explicit dependence on the number of agents; only their geometry in phase space is taken into account. Our new model lacks, however, the symmetry property of the original C-S model, which was the key for the various recent studies of C-S flocking behavior. To this end, we introduce here a new framework to analyze the phenomenon of flocking in the presence of non-symmetric influence matrix. With this aim in mind, we develop a new concept of active sets. We then present a unified framework for studying the flocking behavior for rather general classes of dynamical systems, including the proposed new model and strongly asymmetric models with "leaders". The methodology presented in this paper, based on the notion of active sets, carries over from the particle to kinetic and hydrodynamic descriptions. In particular, we discuss the hydrodynamic formulation of our new model, and prove its unconditional flocking for slowly decaying influence functions.
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