Noisy heteroclinic networks

  • Yuri BakhtinEmail author


We consider a white noise perturbation of dynamics in the neighborhood of a heteroclinic network. We show that under the logarithmic time rescaling the diffusion converges in distribution in a special topology to a piecewise constant process that jumps between saddle points along the heteroclinic orbits of the network. We also obtain precise asymptotics for the exit measure for a domain containing the starting point of the diffusion.

Mathematics Subject Classification (2000)

60J60 34E10 60F17 


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  1. 1.
    Armbruster D., Stone E., Kirk V.: Noisy heteroclinic networks. Chaos 13(1), 71–86 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bakhtin Y.: Exit asymptotics for small diffusion about an unstable equilibrium. Stoch. Process. Appl. 118(5), 839–851 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Billingsley, P.: Convergence of probability measures. In: Wiley Series in Probability and Statistics: Probability and Statistics, 2nd edn. Wiley, New York (1999). A Wiley-Interscience PublicationGoogle Scholar
  4. 4.
    Blagoveščenskiĭ J.N.: Diffusion processes depending on a small parameter. Teor. Verojatnost. i Primenen. 7, 135–152 (1962)MathSciNetGoogle Scholar
  5. 5.
    Freidlin, M.I., Wentzell, A.D.: Random perturbations of dynamical systems. In: Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 260, 2nd edn. Springer, New York (1998). Translated from the 1979 Russian original by Joseph SzücsGoogle Scholar
  6. 6.
    Karatzas, I., Shreve, S.E.: Brownian motion and stochastic calculus. In: Graduate Texts in Mathematics, vol. 113. Springer, New York (1988)Google Scholar
  7. 7.
    Katok, A., Hasselblatt, B.: Introduction to the modern theory of dynamical systems. In: Encyclopedia of Mathematics and its Applications, vol. 54. Cambridge University Press, Cambridge (1995). With a supplementary chapter by Katok and Leonardo MendozaGoogle Scholar
  8. 8.
    Kifer Y.: The exit problem for small random perturbations of dynamical systems with a hyperbolic fixed point. Israel J. Math. 40(1), 74–96 (1981)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Krupa M.: Robust heteroclinic cycles. J. Nonlinear Sci. 7(2), 129–176 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Rabinovich M.I., Huerta R., Afraimovich V.: Dynamics of sequential decision making. Phys. Rev. Lett. 97(18), 188103 (2006)CrossRefGoogle Scholar
  11. 11.
    Skorohod A.V.: Limit theorems for stochastic processes. Teor. Veroyatnost. i Primenen. 1, 289–319 (1956)MathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  1. 1.School of MathematicsGeorgia TechAtlantaUSA

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