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  • Open Access

Redirecting countermoving swarms through collision

Jason Hindes*, Chinthan B. Prasad, Loy McGuire, and Ira B. Schwartz

  • *Contact author: jason.m.hindes.civ@us.navy.mil
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Phys. Rev. Research 8, 023351 – Published 26 June, 2026

DOI: https://doi.org/10.1103/9tdw-3y1b

Abstract

Multiswarm systems, where two or more swarms of mobile agents occupy the same region of space with different parameters and goals, occur in a variety of biological, engineering, and defense applications. Composites of multiple swarms can produce hybrid spatiotemporal patterns, which compared to single swarming systems, are relatively unexplored. In this work, we develop a framework for studying the collision of countermoving swarms, each with its own preferred, stable velocity before collision. We show that redirection of such swarms after collision occurs when a stable velocity synchronized state of the multiswarm composite exists. Using a rigid-body approximation, we are able to extract how scatter-redirection transitions scale with swarm parameters in a variety of scenarios from reciprocal and nonreciprocal systems to symmetric and antagonistic parameter values. Our results compare well to simulations of both particle modeled agents and wheeled robots.

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References (80)

  1. T. Vicsek and A. Zafeiris, Collective motion, Phys. Rep. 517, 71 (2012).
  2. L. Psimen, Active Matter Within and Around Us: From Self-Propelled Particles to Flocks and Living Forms, 1st ed. (Springer Nature, Cham, 2021).
  3. F. Cucker and S. Smale, Emergent behavior in flocks, IEEE Trans. Autom. Control 52, 852 (2007).
  4. A. Cavagna, A. Cimarelli, I. Giardina, G. Parisi, R. Santagati, F. Stefanini, and M. Viale, Scale-free correlations in starling flocks, Proc. Natl. Acad. Sci. USA 107, 11865 (2010).
  5. H. Levine, W.-J. Rappel, and I. Cohen, Self-organization in systems of self-propelled particles, Phys. Rev. E 63, 017101 (2000).
  6. M. R. D’Orsogna, Y. L. Chuang, A. L. Bertozzi, and L. S. Chayes, Self-propelled particles with soft-core interactions: Patterns, stability, and collapse, Phys. Rev. Lett. 96, 104302 (2006).
  7. J. Hindes, K. Daley, G. Stantchev, and I. B. Schwartz, Swarming network inference with importance clustering of relative interactions, J. Phys.: Complexity 5, 045009 (2024).
  8. R. Olfati-Saber, Flocking for multi-agent dynamic systems: Algorithms and theory, IEEE Trans. Autom. Control 51, 401 (2006).
  9. J. Hindes, V. Edwards, K. S. Kasraie, G. Stantchev, and I. B. Schwartz, Swarm shedding in networks of self-propelled agents, Sci. Rep. 11, 13544 (2021).
  10. M. Melchiorre, L. Salamina, L. S. Scimmi, S. Mauro, and S. Pastorelli, Experiments on the artificial potential field with local attractors for mobile robot navigation, Robotics 12, 81 (2023).
  11. M. Melchiorre, L. S. Scimmi, L. Salamina, S. Mauro, and S. P. Pastorelli, Robot collision avoidance based on artificial potential field with local attractors, in Proceedings of the 19th International Conference on Informatics in Control, Automation and Robotics (SciTePress, Lisbon, Portugal, 2022), pp. 340–350.
  12. K. P. O’Keeffe, H. Hong, and S. H. Strogatz, Oscillators that sync and swarm, Nat. Commun. 8, 1504 (2017).
  13. G. K. Sar, K. O’Keeffe, J. U. Lizárraga, M. A. de Aguiar, C. Bettstetter, and D. Ghosh, Interplay of sync and swarm: Theory and application of swarmalators, Phys. Rep. 1167, 1 (2026).
  14. S. Acharya, G. K. Sar, and D. Ghosh, A two-dimensional swarmalator system with Winfree-type phase coupling, Proc. R. Soc. A 481, 20250315 (2025).
  15. E. Bonabeau, M. Dorigo, and G. Theraulaz, Swarm Intelligence: From Natural to Artificial Systems (Oxford University Press, New York, NY, 1999).
  16. X.-S. Yang, S. Deb, Y.-X. Zhao, S. Fong, and X. He, Swarm intelligence: Past, present, and future, Soft Computing 22, 5923 (2018).
  17. M. N. Ab Wahab, S. Nefti-Meziani, and A. Atyabi, A comprehensive review of swarm optimization algorithms, PLoS One 10, e0122827 (2015).
  18. X. Huang, W. Luo, L. Li, H. Zhang, H. Jiang, and X. Gao, Competitive swarm reinforcement learning improves stability and performance of deep reinforcement learning, Sci. Rep. 15, 43637 (2025).
  19. M. F. Copeland and D. B. Weibel, Bacterial swarming: A model system for studying dynamic self-assembly, Soft Matter 5, 1174 (2009).
  20. A. Be’er and G. Ariel, A statistical physics view of swarming bacteria, Mov. Ecol. 7, 9 (2019).
  21. B. L. Rocha-Gregg and A. Huttenlocher, Swarming motility in host defense, Science 372, 1262 (2021).
  22. I. Giardina, Collective behavior in animal groups: Theoretical models and empirical studies, HFSP J. 2, 205 (2008).
  23. M. Sinhuber, K. van der Vaart, Y. Feng, A. M. Reynolds, and N. T. Ouellette, An equation of state for insect swarms, Sci. Rep. 11, 3773 (2021).
  24. M. Ballerini, N. Cabibbo, R. Candelier, A. Cavagna, E. Cisbani, I. Giardina, V. Lecomte, A. Orlandi, G. Parisi, A. Procaccini, M. Viale, and V. Zdravkovic, Interaction ruling animal collective behavior depends on topological rather than metric distance: Evidence from a field study, Proc. Natl. Acad. Sci. USA 105, 1232 (2008).
  25. G. F. Young, L. Scardovi, A. Cavagna, I. Giardina, and N. E. Leonard, Starling flock networks manage uncertainty in consensus at low cost, PLoS Comput. Biol. 9, e1002894 (2013).
  26. K. Tunstrøm, Y. Katz, C. C. Ioannou, C. Huepe, M. J. Lutz, and I. D. Couzin, Collective states, multistability and transitional behavior in schooling fish, PLoS Comput. Biol. 9, e1002915 (2013).
  27. D. S. Calovi, U. Lopez, S. Ngo, C. Sire, H. Chaté, and G. Theraulaz, Swarming, schooling, milling: Phase diagram of a data-driven fish school model, New J. Phys. 16, 015026 (2014).
  28. X. Huang, F. Arvin, C. West, S. Watson, and B. Lennox, Exploration in extreme environments with swarm robotic system, in Proceedings of the 2019 IEEE International Conference on Mechatronics (ICM) (IEEE, Ilmenau, Germany, 2019), Vol. 1, pp. 193–198.
  29. R. J. Amala Arokia Nathan, I. Kurmi, and O. Bimber, Drone swarm strategy for the detection and tracking of occluded targets in complex environments, Commun. Eng. 2, 55 (2023).
  30. F. Berlinger, M. Gauci, and R. Nagpal, Implicit coordination for 3D underwater collective behaviors in a fish-inspired robot swarm, Sci. Rob. 6, eabd8668 (2021).
  31. L. Gladence, V. M. Anu, A. Anderson, I. Stanley, J. A. Fernando J, and S. Revathy, Swarm intelligence in disaster recovery, in Proceedings of the 2021 5th International Conference on Intelligent Computing and Control Systems (ICICCS) (IEEE, Madurai, India, 2021), pp. 1–8.
  32. V.-L. Heuthe, P. Iyer, G. Gompper, and C. Bechinger, Tunable colloidal swarmalators with hydrodynamic coupling, Nat. Commun. 16, 10984 (2025).
  33. J. Hindes, G. Stantchev, K. S. Kasraie, and I. B. Schwartz, Target-density formation in swarms with stochastic sensing and dynamics, Phys. Rev. Res. 7, 043167 (2025).
  34. E. M. Z. Sánchez, S. Billès, P.-H. Glinel, N. Bredeche, and R. Candelier, Maze-solving with density-driven swarms, arXiv:2509.18359.
  35. S. Ceron, W. Xiao, and D. Rus, Reciprocal and non-reciprocal swarmalators with programmable locomotion and formations for robot swarms, in Proceedings of the 2024 IEEE International Conference on Robotics and Automation (ICRA) (IEEE, Yokohama, Japan, 2024), pp. 12233–12239.
  36. K. Szwaykowska, L. M.-y.-T. Romero, and I. B. Schwartz, Collective motions of heterogeneous swarms, IEEE Trans. Autom. Sci. Eng. 12, 810 (2015).
  37. K. Szwaykowska, L. Mier-y Teran-Romero, and I. B. Schwartz, Patterned dynamics of delay-coupled swarms with random communication graphs, in Proceedings of the 2015 54th IEEE Conference on Decision and Control (CDC) (IEEE, Osaka, Japan, 2015), pp. 6496–6501.
  38. M. S. Islam and I. A. Faruque, Insect visuomotor delay adjustments in group flight support swarm cohesion, Sci. Rep. 13, 6407 (2023).
  39. J. Hindes, V. Edwards, S. Kamimoto, I. Triandaf, and I. B. Schwartz, Unstable modes and bistability in delay-coupled swarms, Phys. Rev. E 101, 042202 (2020).
  40. A. Tiwari, S. Devasia, and J. J. Riley, Low-distortion information propagation with noise suppression in swarm networks, Proc. Natl. Acad. Sci. USA 120, e2219948120 (2023).
  41. M. Moussa and G. Beltrame, On the robustness of consensus-based behaviors for robot swarms, Swarm Intell 14, 205 (2020).
  42. A. Barciś, M. Barciś, and C. Bettstetter, Robots that sync and swarm: A proof of concept in ROS 2, in Proceedings of the 2019 International Symposium on Multi-Robot and Multi-Agent Systems (MRS) (IEEE, New Brunswick, NJ, 2019), pp. 98–104.
  43. E. S. Medeiros and U. Feudel, Local control for the collective dynamics of self-propelled particles, Phys. Rev. E 109, 014312 (2024).
  44. C. Walton, I. Kaminer, Q. Gong, A. H. Clark, and T. Tsatsanifos, Defense against adversarial swarms with parameter uncertainty, Sensors 22, 4773 (2022).
  45. T. Tsatsanifos, A. H. Clark, C. Walton, I. Kaminer, and Q. Gong, Modeling large-scale adversarial swarm engagements using optimal control, in Proceedings of the 2021 60th IEEE Conference on Decision and Control (CDC) (IEEE, Austin, TX, 2021), pp. 1244–1249.
  46. J. Hindes, V. Edwards, M. A. Hsieh, and I. B. Schwartz, Critical transition for colliding swarms, Phys. Rev. E 103, 062602 (2021).
  47. A. Lama, M. di Bernardo, and S. H. L. Klapp, Nonreciprocal field theory for decision-making in multi-agent control systems, Nat. Commun. 16, 8450 (2025).
  48. S. Shahid, Z. Zhen, U. Javaid, and L. Wen, Offense-defense distributed decision making for swarm vs. swarm confrontation while attacking the aircraft carriers, Drones 6, 271 (2022).
  49. D. Armbruster, S. Martin, and A. Thatcher, Elastic and inelastic collisions of swarms, Physica D 344, 45 (2017).
  50. S. Kamimoto, J. Hindes, and I. B. Schwartz, The chaotic milling behaviors of interacting swarms after collision, Chaos 33, 081106 (2023).
  51. C. Kolon and I. Schwartz, The Dynamics of Interacting Swarms, NRL Memo Report No. NRL/MR/6790-18-9782, US Naval Research Laboratory, Washington, DC, 2018.
  52. S. Chatterjee, M. Mangeat, C.-U. Woo, H. Rieger, and J. D. Noh, Flocking of two unfriendly species: The two-species Vicsek model, Phys. Rev. E 107, 024607 (2023).
  53. A. K. Dutta, M. Mangeat, H. Rieger, R. Paul, and S. Chatterjee, Stability of flocking in the reciprocal two-species Vicsek model: Effects of relative population, motility, and noise, Phys. Rev. E 112, 024137 (2025).
  54. J. A. Acebrón, L. L. Bonilla, C. J. Pérez Vicente, F. Ritort, and R. Spigler, The Kuramoto model: A simple paradigm for synchronization phenomena, Rev. Mod. Phys. 77, 137 (2005).
  55. Y. Kuramoto, Chemical oscillations, waves, and turbulence, in Springer Series in Synergetics (Dover Publications, New York, 1984).
  56. Note that each agent will approach its preferred velocity asymptotically when the attraction and repulsion interactions vanish.
  57. W. Ren, Consensus based formation control strategies for multi-vehicle systems, in Proceedings of the 2006 American Control Conference (IEEE, Minneapolis, MN, 2006), p. 6.
  58. W. Ren, Formation keeping and attitude alignment for multiple spacecraft through local interactions, J. Guid. Control. Dyn. 30, 633 (2007).
  59. A. N. Montanari, A. E. D. Barioni, C. Duan, and A. E. Motter, Optimal flock formation induced by agent heterogeneity, Nat. Commun. 16, 9626 (2025).
  60. Y.-L. Chuang, Y. R. Huang, M. R. D’Orsogna, and A. L. Bertozzi, Multi-vehicle flocking: Scalability of cooperative control algorithms using pairwise potentials, in Proceedings of the 2007 IEEE International Conference on Robotics and Automation (IEEE, Rome, Italy, 2007), pp. 2292–2299.
  61. V. Edwards, P. deZonia, M. A. Hsieh, J. Hindes, I. Triandaf, and I. B. Schwartz, Delay induced swarm pattern bifurcations in mixed reality experiments, Chaos 30, 073126 (2020).
  62. Z. You, A. Baskaran, and M. C. Marchetti, Nonreciprocity as a generic route to traveling states, Proc. Natl. Acad. Sci. USA 117, 19767 (2020).
  63. M. A. Tsyganov, J. Brindley, A. V. Holden, and V. N. Biktashev, Quasisoliton interaction of pursuit-evasion waves in a predator-prey system, Phys. Rev. Lett. 91, 218102 (2003).
  64. A. Lama and M. di Bernardo, Shepherding and herdability in complex multiagent systems, Phys. Rev. Res. 6, L032012 (2024).
  65. Yao-li Chuang, M. R. D’Orsogna, D. Marthaler, A. L. Bertozzi, and L. S. Chayes, State transitions and the continuum limit for a 2D interacting, self-propelled particle system, Physica D 232, 33 (2007).
  66. G. Albi, D. Balagué, J. A. Carrillo, and J. von Brecht, Stability analysis of flock and mill rings for second order models in swarming, SIAM J. Appl. Math. 74, 794 (2014).
  67. A. J. Bernoff and C. M. Topaz, A primer of swarm equilibria, SIAM J. Appl. Dyn. Syst. 10, 212 (2011).
  68. E. Rohmer, S. P. N. Singh, and M. Freese, V-REP: A versatile and scalable robot simulation framework, in Proceedings of the 2013 IEEE/RSJ International Conference on Intelligent Robots and Systems (IEEE, Tokyo, Japan, 2013), pp. 1321–1326.
  69. Florian Dörfler and F. Bullo, Synchronization in complex networks of phase oscillators: A survey, Automatica 50, 1539 (2014).
  70. Y. A. Kuznetsov, Elements of Applied Bifurcation Theory (Springer-Verlag, Berlin, 1998), 2nd ed.
  71. D. Manik, D. Witthaut, B. Schäfer, M. Matthiae, A. Sorge, M. Rohden, E. Katifori, and M. Timme, Supply networks: Instabilities without overload, Eur. Phys. J.: Spec. Top. 223, 2527 (2014).
  72. J. Hindes and I. B. Schwartz, Rare slips in fluctuating synchronized oscillator networks, Chaos 28, 071106 (2018).
  73. J. R. Taylor, Classical Mechanics (University Science Books, Sausalito, CA, 2005).
  74. K. Binmore and J. Davies, Calculus: Concepts and Methods (Cambridge University Press, New York, NY, 2002).
  75. One can solve the nonlinear system [Eqs. (8) and (10)] numerically with, e.g., matlab’s fsolve function.
  76. C. Prasad, J. Hindes, R. Suitor, and L. McGuire, Anki-swarm, 2025, https://github.com/chinthanprasad/Anki-Swarm.
  77. A. J. Leverentz, C. M. Topaz, and A. J. Bernoff, Asymptotic dynamics of attractive-repulsive swarms, SIAM J. Appl. Dyn. Syst. 8, 880 (2009).
  78. The integrals over density in this work were numerically approximated by simulating two separate flocking swarms, each with 1000 agents, from random initial conditions with Eq. (1) until t=104. Using each swarm’s steady-state configuration, zB and zR for the blue and red swarms, respectively, the integral operator ∫∫ρB(zB)ρR(zR)dzB2dzR2, can be estimated by 1NBNR∑i∈B,j∈R, similar to a Monte Carlo integration approximation.
  79. J. Hindes, I. B. Schwartz, and M. Tyloo, Stability of Kuramoto networks subject to large and small fluctuations from heterogeneous and spatially correlated noise, Chaos 33, 113129 (2023).
  80. J. Hindes, P. Jacquod, and I. B. Schwartz, Network desynchronization by non-Gaussian fluctuations, Phys. Rev. E 100, 052314 (2019).

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Phys. Rev. Research 8, 023351– Published 26 June, 2026
  • Received 11 March 2026
  • Accepted 5 June 2026
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