Noisy heteroclinic networks. (English) Zbl 1231.34100
The effect of small noise perturbations of a smooth continuous time dynamical system in the neighbourhood of its heteroclinic network is studied here. The limiting process is precisely described and implications and possible extensions of the results are informally considered.
Reviewer: Andrew Dale (Durban)
MSC:
| 34F05 | Ordinary differential equations and systems with randomness |
| 34C37 | Homoclinic and heteroclinic solutions to ordinary differential equations |
| 60J60 | Diffusion processes |
| 60F17 | Functional limit theorems; invariance principles |
| 92B20 | Neural networks for/in biological studies, artificial life and related topics |
| 34D10 | Perturbations of ordinary differential equations |
References:
| [1] | Armbruster, D.; Stone, E.; Kirk, V., Noisy heteroclinic networks, Chaos, 13, 1, 71-86 (2003) · Zbl 1080.37510 · doi:10.1063/1.1539951 |
| [2] | Bakhtin, Y., Exit asymptotics for small diffusion about an unstable equilibrium, Stoch. Process. Appl., 118, 5, 839-851 (2008) · Zbl 1138.60052 · doi:10.1016/j.spa.2007.06.003 |
| [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 Publication · Zbl 0944.60003 |
| [4] | Blagoveščenskiĭ, J. N., Diffusion processes depending on a small parameter, Teor. Verojatnost. i Primenen., 7, 135-152 (1962) · Zbl 0112.09504 |
| [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 Szücs · Zbl 0922.60006 |
| [6] | Karatzas, I., Shreve, S.E.: Brownian motion and stochastic calculus. In: Graduate Texts in Mathematics, vol. 113. Springer, New York (1988) · Zbl 0638.60065 |
| [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 Mendoza · Zbl 0878.58020 |
| [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) · Zbl 0473.60067 · doi:10.1007/BF02761819 |
| [9] | Krupa, M., Robust heteroclinic cycles, J. Nonlinear Sci., 7, 2, 129-176 (1997) · Zbl 0879.58054 · doi:10.1007/BF02677976 |
| [10] | Rabinovich, M. I.; Huerta, R.; Afraimovich, V., Dynamics of sequential decision making, Phys. Rev. Lett., 97, 18, 188103 (2006) · doi:10.1103/PhysRevLett.97.188103 |
| [11] | Skorohod, A. V., Limit theorems for stochastic processes, Teor. Veroyatnost. i Primenen., 1, 289-319 (1956) · Zbl 0074.33802 |
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