Bifurcations of Random Differential Equations with Bounded Noise

  • Ale Jan Homburg
  • Todd R. Young
  • Masoumeh Gharaei
Part of the Modeling and Simulation in Science, Engineering and Technology book series (MSSET)


We review recent results from the theory of random differential equations with bounded noise. Assuming the noise to be “sufficiently robust in its effects” we discuss the feature that any stationary measure of the system is supported on a “Minimal Forward Invariant” (MFI) set. We review basic properties of the MFI sets, including their relationship to attractors in systems where the noise is small. In the main part of the paper we discuss how MFI sets can undergo discontinuous changes that we have called hard bifurcations. We characterize such bifurcations for systems in one and two dimensions and we give an example of the effects of bounded noise in the context of a Hopf–Andronov bifurcation.


Bounded noises Random differential equations Stationary measures Stochastic bifurcations Hopf–Andronov bifurcation Hard bifurcations 


  1. 1.
    Araújo, V.: Ann. Inst. Henri Poincaré, Analyse non linéaire 17, 307 (2000)Google Scholar
  2. 2.
    Arnold, L.: Random Dynamical Systems. Springer, Berlin (1998)CrossRefMATHGoogle Scholar
  3. 3.
    Arnold, L.: IUTAM Symposium on Nonlinearity and Stochastic Structural Dynamics, Madras, 1999. Solid Mech. Appl., vol. 85, pp. 15. Kluwer Academic, Dordrecht (2001)Google Scholar
  4. 4.
    Arnold, L., Bleckert, G., Schenk-Hoppé, K.R.: In: Crauel, H., Gundlach, M. (eds.) Stochastic Dynamics (Bremen, 1997), pp. 71. Springer, Berlin (1999)Google Scholar
  5. 5.
    Arnold, L., Kliemann, W.: On unique ergodicity for degenerate diffusions. Stochastics 21, 41 (1987)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Arnold, L., Sri Namachchivaya, N., Schenk-Hoppé, K.R.: Int. J. Bifur. Chaos Appl. Sci. Eng. 6, 1947 (1996)Google Scholar
  7. 7.
    Arnold, V.I., Afraimovich, V.S., Il’yashenko, Yu.S., Shil’nikov, L.P.: Bifurcation Theory and Catastrophe Theory. Springer, Berlin (1999)MATHGoogle Scholar
  8. 8.
    Aubin, J.P., Cellina, A.: Differential Inclusions. Springer, Berlin (1984)CrossRefMATHGoogle Scholar
  9. 9.
    Bakhtin, Y., Hurth, T.: Nonlinearity 25, 2937 (2012) (unpublished)Google Scholar
  10. 10.
    Bashkirtseva, I., Ryashko, L., Schurz, H.: Chaos Solit. Fract. 39, 72 (2009)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Bena, I.: Int. J. Modern Phys. B 20, 2825 (2006)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Botts, R.T., Homburg, A.J., Young, T.R.: Discrete Contin. Dyn. Syst. Ser. A 32, 2997 (2012)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Bovier, A., Eckhoff, M., Gayrard, V., Klein, M.: J. Eur. Math. Soc. 6, 399 (2004)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Bovier, A., Gayrard, V., Klein, M.: J. Eur. Math. Soc. 7, 69 (2005)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Colombo, G., Pra, P.D., Křivan, V., Vrkoč, I.: Math. Control Signals Syst. 16, 95 (2003)CrossRefMATHGoogle Scholar
  16. 16.
    Colonius, F., Gayer, T., Kliemann, W.: SIAM J. Appl. Dyn. Syst. 7, 79 (2007)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Colonius, F., Homburg, A.J., Kliemann, W.: J. Differ. Equat. Appl. 16, 127 (2010)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Colonius, F., Kliemann, W.: In: Crauel, H., Gundlach, M. (eds.) Stochastic Dynamics. Springer, Berlin (1999)Google Scholar
  19. 19.
    Colonius, F., Kliemann, W.: The Dynamics of Control. Birkhauser Boston, Boston (2000)CrossRefGoogle Scholar
  20. 20.
    Crauel, H., Imkeller, P., Steinkamp, M.: In: Crauel, H., Gundlach, M. (eds.) Stochastic Dynamics. Springer, Berlin (1999)Google Scholar
  21. 21.
    d’Onofrio, A., Gandolfi, A., Gattoni, S.: Phys. A Stat. Mech. Appl. 91, 6484 (2012)Google Scholar
  22. 22.
    Doob, J.L.: Stochastic Processes. Wiley, New York (1953)MATHGoogle Scholar
  23. 23.
    Froyland, G., Stancevic, O.: ArXiv:1106.1954v2 [math.DS] (2011) (unpublished)Google Scholar
  24. 24.
    Gayer, T.: J. Differ. Equat. 201, 177 (2004)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Ghil, M., Chekroun, M.D., Simonnet, E.: Phys. D 237, 2111 (2008)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Homburg, A.J., Young, T.R.: Regular Chaotic Dynam. 11, 247 (2006)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Homburg, A.J., Young, T.R.: Topol. Methods Nonlin. Anal. 35, 77 (2010)MathSciNetMATHGoogle Scholar
  28. 28.
    Horsthemke, W., Lefever, R.: Noise-Induced Transitions: Theory and Applications in Physics, Chemistry, and Biology. Springer Series in Synergetics, vol. 15. Springer, Berlin (1984)Google Scholar
  29. 29.
    Johnson, R.: In: Crauel, H., Gundlach, M. (eds.) Stochastic Dynamics. Springer, Berlin (1999)Google Scholar
  30. 30.
    Kliemann, W.: Ann. Probab. 15, 690 (1987)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Kunita, H.: Stochastic Flows and Stochastic Differential Equations. Cambridge University Press, Cambridge (1990)MATHGoogle Scholar
  32. 32.
    Kuznetsov, Yu.A.: Elements of Applied Bifurcation Theory. Springer, Berlin (2004)CrossRefMATHGoogle Scholar
  33. 33.
    Lamb, J.S.W., Rasmussen, M., Rodrigues, C.S.: ArXiv:1105.5018v1 [math.DS] (2011) (unpublished)Google Scholar
  34. 34.
    Mallick, K., Marcq, P.: Eur. Phys. J. B 36 119 (2003)CrossRefGoogle Scholar
  35. 35.
    Marsden, J.E., McCracken, M.: The Hopf Bifurcation and Its Applications. Springer, Berlin (1976)CrossRefMATHGoogle Scholar
  36. 36.
    Nadzieja, T.: Czechoslovak Math. J. 40, 195 (1990)MathSciNetGoogle Scholar
  37. 37.
    Ridolfi, L., D’Odorico, P., Laio, F.: Noise-Induced Phenomena in the Environmental Sciences. Cambridge University Press, Cambridge (2011)CrossRefMATHGoogle Scholar
  38. 38.
    Rodrigues, C.S., Grebogi, C., de Moura, A.P.S.: Phys. Rev. E 82, 046217 (2010)MathSciNetCrossRefGoogle Scholar
  39. 39.
    Schütte, C., Huisinga, W., Meyn, S.: Ann. Appl. Probab. 14, 419 (2004)MathSciNetCrossRefMATHGoogle Scholar
  40. 40.
    Tateno, T.: Phys. Rev. E 65, 021901 (2002)MathSciNetCrossRefGoogle Scholar
  41. 41.
    Wieczorek, S.: Phys. Rev. E 79, 036209 (2009)MathSciNetCrossRefGoogle Scholar
  42. 42.
    Zeeman, E.C.: Nonlinearity 1, 115 (1988)MathSciNetCrossRefMATHGoogle Scholar
  43. 43.
    Zeeman, E.C.: Bull. Lond. Math. Soc. 20, 545 (1988)MathSciNetCrossRefMATHGoogle Scholar
  44. 44.
    Zmarrou, H., Homburg, A.J.: Ergod. Theor. Dyn. Syst. 27, 1651 (2007)MathSciNetCrossRefMATHGoogle Scholar
  45. 45.
    Zmarrou, H., Homburg, A.J.: Discrete Contin. Dyn. Syst. Ser. B 10, 719 (2008)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Ale Jan Homburg
    • 1
    • 2
  • Todd R. Young
    • 3
  • Masoumeh Gharaei
    • 1
  1. 1.KdV Institute for MathematicsUniversity of AmsterdamAmsterdamThe Netherlands
  2. 2.Department of MathematicsVU University AmsterdamHV AmsterdamThe Netherlands
  3. 3.Department of MathematicsOhio UniversityAthensUSA

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