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Journal of Statistical Physics

, Volume 176, Issue 2, pp 478–491 | Cite as

Effects of Local Fields in a Dissipative Curie-Weiss Model: Bautin Bifurcation and Large Self-sustained Oscillations

  • Francesca ColletEmail author
  • Marco Formentin
Article
  • 38 Downloads

Abstract

We modify the spin-flip dynamics of a Curie-Weiss model with dissipative interaction potential [7] by adding a site-dependent i.i.d. random magnetic field. The purpose is to analyze how the addition of the field affects the time-evolution of the observables in the macroscopic limit. Our main result shows that a Bautin bifurcation point exists and that, whenever the field intensity is sufficiently strong and the temperature sufficiently low, a periodic orbit emerges through a global bifurcation in the phase space, giving origin to a large-amplitude rhythmic behavior.

Keywords

Bautin bifurcation Collective noise-induced periodicity Disordered systems Mean-field interaction Non-equilibrium systems Random potential Saddle-node bifurcation of periodic orbits 

Notes

Acknowledgements

The authors are grateful to the anonymous referees whose comments and remarks led to a significant improvement of the paper. They also wish to thank Paolo Dai Pra for fruitful and inspiring discussions. FC was supported by The Netherlands Organisation for Scientific Research (NWO) via TOP-1 grant 613.001.552.

Supplementary material

References

  1. 1.
    Aleandri, M., Minelli, I.G.: Opinion dynamics with Lotka-Volterra type interactions (2018) Preprint, arXiv:1811.05937
  2. 2.
    Andreis, L., Tovazzi, D.: Coexistence of stable limit cycles in a generalized Curie-Weiss model with dissipation. J. Stat. Phys. 173(1), 163–181 (2018)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bertini, L., Giacomin, G., Pakdaman, K.: Dynamical aspects of mean field plane rotators and the Kuramoto model. J. Stat. Phys. 138, 270–290 (2010)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bonilla, L.L., Neu, J.C., Spigler, R.: Nonlinear stability of incoherence and collective synchronization in a population of coupled oscillators. J. Stat. Phys. 67, 313–330 (1992)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Collet, F., Dai Pra, P., Formentin, M.: Collective periodicity in mean-field models of cooperative behavior. NoDEA 22(5), 1461–1482 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Collet, F., Formentin, M., Tovazzi, D.: Rhythmic behavior in a two-population mean-field Ising model. Phys. Rev. E 94(4), 042139 (2016)ADSMathSciNetCrossRefGoogle Scholar
  7. 7.
    Dai Pra, P., Fischer, M., Regoli, D.: A Curie-Weiss model with dissipation. J. Stat. Phys. 152(1), 37–53 (2013)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Ditlevsen, S., Löcherbach, E.: Multi-class oscillating systems of interacting neurons. Stoch. Process. Appl. 127(6), 1840–1869 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Ermentrout, G.B., Terman, D.H.: Mathematical Foundations of Neuroscience, vol. 35. Springer, New York (2010)CrossRefzbMATHGoogle Scholar
  10. 10.
    Ethier, S.N., Kurtz, T.G.: Markov Processes: Characterization and Convergence, 1st edn. Wiley, New York (1986)CrossRefzbMATHGoogle Scholar
  11. 11.
    Fernández, R., Fontes, L.R., Neves, E.J.: Density-profile processes describing biological signaling networks: almost sure convergence to deterministic trajectories. J. Stat. Phys. 136(5), 875–901 (2009)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Giacomin, G., Pakdaman, K., Pellegrin, X., Poquet, C.: Transitions in active rotator systems: invariant hyperbolic manifold approach. SIAM J. Math. Anal. 44(6), 4165–4194 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Giacomin, G., Poquet, C.: Noise, interaction, nonlinear dynamics and the origin of rhythmic behaviors. Braz. J. Prob. Stat. 29, 460–493 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Guckenheimer, J., Holmes, P.: Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, first edn. Springer, New York (1983)CrossRefzbMATHGoogle Scholar
  15. 15.
    Kuznetsov, Y.A.: Elements of Applied Bifurcation Theory, second edn. Springer, New York (1998)zbMATHGoogle Scholar
  16. 16.
    Lindner, B., García-Ojalvo, J., Neiman, A., Schimansky-Geier, L.: Effects of noise in excitable systems. Phys. Rep. 392(6), 321–424 (2004)ADSCrossRefGoogle Scholar
  17. 17.
    Luçon, E., Poquet, C.: Emergence of oscillatory behaviors for excitable systems with noise and mean-field interaction, a slow-fast dynamics approach (2018) Preprint. arXiv:1802.06410
  18. 18.
    Luçon, E., Poquet, C.: Periodicity induced by noise and interaction in the kinetic mean-field FitzHugh-Nagumo model (2018) Preprint, arXiv:1811.00305
  19. 19.
    Odani, K.: Existence of exactly \(N\) periodic solutions for Liénard systems. Funkcial. Ekvac. 39(2), 217–234 (1996)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Perko, L.: Differential Equations and Dynamical Systems, third edn. Springer, New York (2001)CrossRefzbMATHGoogle Scholar
  21. 21.
    Scheutzow, M.: Noise can create periodic behavior and stabilize nonlinear diffusions. Stoch. Process. Appl. 20(2), 323–331 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Scheutzow, M.: Some examples of nonlinear diffusion processes having a time-periodic law. Ann. Probab. 13(2), 379–384 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Shinomoto, S., Kuramoto, Y.: Phase transitions in active rotator systems. Prog. Theor. Phys. 75(5), 1105–1110 (1986)ADSCrossRefGoogle Scholar
  24. 24.
    Sotomayor, J., Mello, L.F., de Carvalho Braga, D.: Bifurcation analysis of the Watt governor system. Comput. Appl. Math. 26(1), 19–44 (2007)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Touboul, J.: The hipster effect: when anticonformists all look the same. Discret. Contin. Dyn. Syst. B (forthcoming)Google Scholar
  26. 26.
    Touboul, J., Hermann, G., Faugeras, O.: Noise-induced behaviors in neural mean field dynamics. SIAM J. Appl. Dyn. Syst. 11(1), 49–81 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Turchin, P.: Complex Population Dynamics: A Theoretical/empirical Synthesis, vol. 35. Princeton University Press, Princeton (2003)zbMATHGoogle Scholar
  28. 28.
    Weidlich, W., Haag, G.: Concepts and Models of a Quantitative Sociology: The Dynamics of Interacting Populations, vol. 14. Springer, New York (2012)zbMATHGoogle Scholar

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Authors and Affiliations

  1. 1.Delft Institute of Applied MathematicsDelft University of TechnologyDelftThe Netherlands
  2. 2.Dipartimento di Matematica “Tullio Levi-Civita”PadovaItaly
  3. 3.Padova Neuroscience CenterPadovaItaly

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