Journal of Nonlinear Science

, Volume 29, Issue 4, pp 1301–1342 | Cite as

Flocking Dynamics of the Inertial Spin Model with a Multiplicative Communication Weight

  • Seung-Yeal Ha
  • Doheon Kim
  • Dohyun Kim
  • Woojoo ShimEmail author


In this paper, we study a flocking dynamics of the deterministic inertial spin (IS) model. The IS model was introduced for the collective dynamics of active particles with an internal angular momentum, or spin. When the generalized moment of inertia becomes negligible compared to spin dissipation (overdamped limit) and mutual communication weight is a function of a relative distance between interacting particles, the deterministic inertial spin model formally reduces to the Cucker–Smale (CS) model with constant speed constraint whose emergent dynamics has been extensively studied in the previous literature. We present several sufficient frameworks leading to the asymptotic mono-cluster flocking, in which spins and relative velocities tend to zero asymptotically. We also provide several numerical simulations for the decoupled and coupled inertial spin models to see the effect of the C–S velocity flocking and compare them with our analytical results.


Cucker–Smale model Flocking Inertial spin model Unit speed constraint The Vicsek model 

Mathematics Subject Classification

82C22 34D05 34C15 


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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematical Sciences and Research Institute of MathematicsSeoul National UniversitySeoulRepublic of Korea
  2. 2.Korea Institute for Advanced StudySeoulRepublic of Korea
  3. 3.Department of Mathematical SciencesSeoul National UniversitySeoulRepublic of Korea
  4. 4.Department of Mathematical SciencesSeoul National UniversitySeoulRepublic of Korea

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