On the mathematics of emergence

Cucker, Felipe; Smale, Steve

doi:10.1007/s11537-007-0647-x

Cited by 482 publications

(520 citation statements)

References 15 publications

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“…Flocking patterns have also been shown in models of alignment or orientation averaging in [16,17]. All these models share the objective of pinpointing the minimal effects or interactions leading to certain particular type of pattern or collective motion of the agents.…”

Section: Introductionmentioning

confidence: 93%

“…After several improvements, it has been shown in [16,17,22,21,11] that the asymptotic behavior of the system…”

Section: Asymptotic Speed With Attraction-repulsion Interaction Modelmentioning

confidence: 99%

“…In the Cucker-Smale model, introduced in [16,17], the only mechanism taken into account is the reorientation interaction between agents. Each agent in the swarm tries to mimic other individuals by adjusting/averaging their relative velocity with all the others.…”

Section: The Cucker-smale Modelmentioning

confidence: 99%

“…These IBMs can be considered also as "particle" models in the optic of treating agents or animals as point particles in a physics-based description as in statistical mechanics. Let us also mention that this issue has also received attention from the control engineering viewpoint trying to reproduce these self-organization patterns with artificial robots or devices, see [12,16,36] and the references therein, with the aim of controlling unmanned vehicle operation. Finally, the combination of these minimal bricks in the modeling such as interaction, alignment and orientation with other effects is leading to rich complex behavior and detailed dynamical systems models for particular species, see for instance [5,4,25,2,26,6,46].…”

Section: Introductionmentioning

confidence: 99%

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Collective Behavior of Animals: Swarming and Complex Patterns

Cañizo¹,

Carrillo²,

Rosado³

2010

Arbor

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RESUMEN: En esta nota repasamos algunos modelos basados en individuos para describir el movimiento colectivo de agentes, a lo ABSTRACT:In this short note we review some of the individual based models of the collective motion of agents, called swarming. These models based on ODEs (ordinary differential equations) exhibit a complex rich asymptotic behavior in terms of patterns, that we show numerically. Moreover, we comment on how these particle models are connected to partial differential equations to describe the evolution of densities of individuals in a continuum manner. The mathematical questions behind the stability issues of these PDE (partial differential equations) models are questions of actual interest in mathematical biology research.

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Section: Introductionmentioning

confidence: 93%

“…After several improvements, it has been shown in [16,17,22,21,11] that the asymptotic behavior of the system…”

Section: Asymptotic Speed With Attraction-repulsion Interaction Modelmentioning

confidence: 99%

Section: The Cucker-smale Modelmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Collective Behavior of Animals: Swarming and Complex Patterns

Cañizo¹,

Carrillo²,

Rosado³

2010

Arbor

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“…A possible starting point would be to assume that the fluid medium leads to direct coupling between particle velocities and build this effect into existing models using, for example, the Cucker-Smale velocity matching mechanism. Here particle i is subject to an additional force due to the presence of particle j , given by v j − v i and modulated by a distance-dependent prefactor g(|r i − r j |) [70][71][72]. Because of its simple mathematical form, Cucker-Smale type interactions have been used extensively to study swarming, with coherent morphologies arising depending on the form of g(|r i − r j |).…”

mentioning

confidence: 99%

Swarming in viscous fluids: Three-dimensional patterns in swimmer- and force-induced flows

2016

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We derive a three-dimensional theory of self-propelled particle swarming in a viscous fluid environment. Our model predicts emergent collective behavior that depends critically on fluid opacity, mechanism of self-propulsion, and type of particle-particle interaction. In "clear fluids" swimmers have full knowledge of their surroundings and can adjust their velocities with respect to the lab frame, while in "opaque fluids" they control their velocities only in relation to the local fluid flow. We also show that "social" interactions that affect only a particle's propensity to swim towards or away from neighbors induces a flow field that is qualitatively different from the long-ranged flow fields generated by direct "physical" interactions. The latter can be short-ranged but lead to much longer-ranged fluid-mediated hydrodynamic forces, effectively amplifying the range over which particles interact. These different fluid flows conspire to profoundly affect swarm morphology, kinetically stabilizing or destabilizing swarm configurations that would arise in the absence of fluid. Depending upon the overall interaction potential, the mechanism of swimming ( e.g., pushers or pullers), and the degree of fluid opaqueness, we discover a number of new collective three-dimensional patterns including flocks with prolate or oblate shapes, recirculating pelotonlike structures, and jetlike fluid flows that entrain particles mediating their escape from the center of mill-like structures. Our results reveal how the interplay among general physical elements influence fluid-mediated interactions and the self-organization, mobility, and stability of new three-dimensional swarms and suggest how they might be used to kinetically control their collective behavior. DOI: 10.1103/PhysRevE.93.043112 The collective behavior of self-propelled agents in natural and artificial systems has been extensively studied . Many of the lessons learned from experimental and theoretical work conducted on organisms as diverse as bacteria, ants, locusts, and birds [23][24][25][26][27][28][29][30][31][32][33][34][35][36] have been successfully applied to engineered robotic systems to help frame decentralized control strategies through ad hoc algorithms [37][38][39][40][41][42][43][44]. In most mathematical "swarming" models, particles are assumed to be self-driven by internal mechanisms that impart a characteristic speed. A pairwise short-ranged repulsion and a long-ranged decaying attraction are typically employed as the most realistic choices when modeling aggregating particles [9,11,45]. The interplay between self-propulsion, particle interactions, initial conditions, and number of particles is key in determining the large scale patterns that dynamically arise. In two dimensions, rotating mills and translating flocks are often observed, the latter configuration also arising in three dimensions [9,10,17,19,46,47]. It is possible to classify swarm morphology in terms of interaction strength and length scales, as shown for particles coupled via conserved f...

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