Collective periodicity in mean-field models of cooperative behavior

  • Francesca Collet
  • Paolo Dai PraEmail author
  • Marco Formentin


We propose a way to break symmetry in stochastic dynamics by introducing a dissipation term. We show in a specific mean-field model, that if the reversible model undergoes a phase transition of ferromagnetic type, then its dissipative counterpart exhibits periodic orbits in the thermodynamic limit.


Interacting diffusions Random potential Homoclinic bifurcation Mean-field interaction Noise-induced periodicity 

Mathematics Subject Classification

60J60 60K35 37G15 37H20 


  1. 1.
    Dai Pra P., Fischer M., Regoli D.: A Curie–Weiss model with dissipation. J. Stat. Phys. 152, 37–53 (2013)zbMATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    Dawson D.A.: Critical dynamics and fluctuations for a mean-field model of cooperative behavior. J. Stat. Phys. 31, 29–85 (1983)CrossRefGoogle Scholar
  3. 3.
    Elowitz M.B., Leibler S.: A synthetic oscillatory network of transcriptional regulators. Nature 403, 335–338 (2000)CrossRefGoogle Scholar
  4. 4.
    Garcia-Ojalvo J., Elowitz M.B., Strogatz S.H.: Modeling a synthetic multicellular clock: repressilators coupled by quorum sensing. Proc. Natl. Acad. Sci. USA 101, 10955–10960 (2004)zbMATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    Garnier J., Papanicolaou G., Yang T.-W.: Large deviations for a mean field model of systemic risk. SIAM J. Financ. Math. 4, 151–184 (2013)zbMATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    Giacomin, G., Poquet, C.: Noise, interaction, nonlinear dynamics and the origin of rhythmic behaviors. Braz. J. Prob. Stat. (to appear, 2015)Google Scholar
  7. 7.
    Giesecke, K., Spiliopoulos, K., Sowers, R.B., Sirignano, J.A.: Large portfolio asymptotics for loss from default. Math. Finance. doi: 10.1111/mafi.12011 (2012)
  8. 8.
    Lindner B., Garcıa-Ojalvo J., Neiman A., Schimansky-Geier L.: Effects of noise in excitable systems. Phys. Rep. 392, 321–424 (2004)CrossRefGoogle Scholar
  9. 9.
    McKean, H.P.: Stochastic integrals. In: Probability and Mathematical Statistics. Academic Press, New York (1969)Google Scholar
  10. 10.
    Perko, L.: Differential Equations and Dynamical Systems. Springer, New York (2001)Google Scholar
  11. 11.
    Scheutzow M.: Some examples of nonlinear diffusion processes having a time-periodic law. Ann. Probab. 13, 379–384 (1985)zbMATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    Scheutzow M.: Noise can create periodic behavior and stabilize nonlinear diffusions. Stoch. Process. Appl. 20, 323–331 (1985)zbMATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    Schweitzer, F.: Brownian agents and active particles: collective dynamics in the natural and social sciences. Springer Series in Synergetics. Springer, Berlin (2007)Google Scholar
  14. 14.
    Sznitman, A.-S.: Topics in propagation of chaos. In: Ecole d’Eté de Probabilités de Saint-Flour XIX—1989, pp. 165–251. Springer, Berlin (1991)Google Scholar
  15. 15.
    Touboul, J., Hermann, G., Faugeras, O.: Noise-induced behaviors in neural mean field dynamics. arXiv preprint. arXiv:1104.5425 (2011)

Copyright information

© Springer Basel 2015

Authors and Affiliations

  • Francesca Collet
    • 1
  • Paolo Dai Pra
    • 2
    Email author
  • Marco Formentin
    • 3
  1. 1.Dipartimento di MatematicaAlma Mater Studiorum Università di BolognaBolognaItaly
  2. 2.Dipartimento di MatematicaUniversità degli Studi di PadovaPaduaItaly
  3. 3.UTIA, Czech Academy of SciencesPragueCzech Republic

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