Nonlinear Dynamics

, Volume 92, Issue 1, pp 3–12 | Cite as

Conflicts among \(\varvec{N}\) armed groups: scenarios from a new descriptive model

  • Sergio Rinaldi
  • Fabio Della Rossa
Original Paper


In this paper, we propose and analyze a new descriptive model of armed conflicts among N groups. The model is composed of \(N^2\) ordinary differential equations, with \(3(N^2+N)\) constant parameters that describe military characteristics and recruitment policies, ranging from pure defensivism to pure fanaticism. The results are only preliminary, but point out interesting (though not very surprising) properties: periodic coexistence is possible, and multiple attractors can exist; governmental groups cannot go extinct if they are highly defensivist, and rebels cannot be eradicated if they are highly fanatic. Shocks due to interventions of short duration of an external army can stabilize/destabilize the system and/or eradicate some group, and the same holds true for small structural changes. Other more subtle questions concerning, for example, the existence of chaotic regimes and the systematic evaluation of the role of strategic factors like power, intelligence, and fanaticism, remain open and require further research.


Mathematical modeling Social systems Conflicts Terrorism Bifurcations Chaos 



The authors are grateful to Gustav Feichtinger and Jonathan Caulkins for their comments and encouragement. The help of three anonymous reviewers has allowed us to significantly improve the quality of the paper.


  1. 1.
    Kress, M.: Modeling armed conflicts. Science 336(6083), 865–869 (2012)CrossRefGoogle Scholar
  2. 2.
    MacKay, N.J.: When Lanchester met Richardson, the outcome was stalemate: a parable for mathematical models of insurgency. J. Oper. Res. Soc. 66(2), 191–201 (2015)CrossRefGoogle Scholar
  3. 3.
    Lanchester, F.W.: Aircraft in warfare: the dawn of the fourth arm. Constable and Company Limited, London (1916)zbMATHGoogle Scholar
  4. 4.
    Richardson, L.F.: Mathematical psychology of war. W. Hunt, Oxford (1919)Google Scholar
  5. 5.
    Richardson, L.F.: Mathematical psychology of war. Nature 135, 830–831 (1935)CrossRefzbMATHGoogle Scholar
  6. 6.
    Schiermeier, Q.: Attempts to predict terrorist attacks hit limits. Nature 517, 419–420 (2015)Google Scholar
  7. 7.
    Lopes, A.M., Machado, J.T., Mata, M.E.: Analysis of global terrorism dynamics by means of entropy and state space portrait. Nonlinear Dyn. 85(3), 1–14 (2016)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2017

Authors and Affiliations

  1. 1.Dipartimento di Elettronica, Informazione e BioingegneriaPolitecnico di MilanoMilanItaly
  2. 2.International Institute for Applied Systems AnalysisLaxenburgAustria

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