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Pursuit on regular surfaces with application to consensus problems

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Abstract

In this work we develop geometric and numerical methods to analyze the behavior of a system of organisms or particles under various types of pursuit on a regular surface. We consider different types of pursuit similar to some mechanisms proposed by several authors, although treated in a different way. Using different concepts from diverse areas of mathematics, global and time- invariant relationships between the involved particles in the system are characterized. In order to consider different dynamical behaviors depending on individual positions and velocities we applied our geometric pursuit framework defined on regular surfaces to a specific consensus problem.

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Data availability statement

All data generated or analyzed during this study are included in this published article [and its supplementary information files].

Notes

  1. Locally, system (3) defines a vector field G on TM and its flow is called the geodesic flow on TM where \(t\rightarrow \varphi (t,p,v)\) is the unique trajectory of G which satisfies the initial condition \(\varphi (0,p,v)=(p,v)\).

  2. The diameter D(S) of a surface S is, by definition \(D(S)=\underset{p,q\in S}{\sup } \rho (p,q), \) where \(\rho \) is the intrinsic distance function defined in S.

  3. Couzin et al. identified four dynamical behaviors, one of which is not exhibited in the simulations of our model.

  4. We remind that the model developed by Couzin et al. is in order to investigate the spatial dynamics of animal groups, and they labeled this behavior as swarm in analogy of the motion of insects such as mosquitoes.

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Acknowledgements

This work was supported by CONACYT Project CB2016-286437

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  1. CIMAT, Guanajuato, Gto, 36000, Mexico

    Francisco J. Solis & Carlos Yebra

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  1. Francisco J. Solis
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  2. Carlos Yebra
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Correspondence to Francisco J. Solis.

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This work was supported by CONACyT, Mexico project CB2016-286437.

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Solis, F.J., Yebra, C. Pursuit on regular surfaces with application to consensus problems. Nonlinear Dyn 105, 3423–3438 (2021). https://doi.org/10.1007/s11071-021-06800-w

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