Abstract
We consider the effect of random perturbations on canards. We find the appropriate size of the random perturbations to produce a random selection of a regular duck versus a headless duck. The appropriate limit theorem, in the appropriate topology, is proved.
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Berglund, N., Gentz, B.: Geometric singular perturbation theory for stochastic differential equations. J. Differ. Equ. 191(1), 1–54 (2003)
Berglund, N., Gentz, B.: Noise-Induced Phenomena in Slow-Fast Dynamical Systems. Probability and its Applications. Springer, London (2006). A sample-paths approach
Corless, R.M., Gonnet, G.H., Hare, D.E.G., Jeffrey, D.J., Knuth, D.E.: On the Lambert W function. Adv. Comput. Math. 5(4), 329–359 (1996)
Dumortier, F., Roussarie, R.: Canard cycles and center manifolds. Mem. Am. Math. Soc. 121(577), x+100 (1996). With an appendix by C.Z. Li
Durham, J., Moehlis, J.: Feedback control of canards. Chaos (in press)
Eckhaus, W.: Relaxation oscillations including a standard chase on French ducks. In: Asymptotic Analysis, II. Lecture Notes in Math., vol. 985, pp. 449–494. Springer, Berlin (1983)
Ethier, S.N., Kurtz, T.G.: Markov Processes: Characterization and Convergence. Wiley, New York (1986)
Freidlin, M.I.: On stable oscillations and equilibriums induced by small noise. J. Stat. Phys. 103(1–2), 283–300 (2001)
Lee DeVille, R.E., Vanden-Eijnden, E., Muratov, C.B.: Two distinct mechanisms of coherence in randomly perturbed dynamical systems. Phys. Rev. E (3) 72(3), 031105–031110 (2005)
Makarov, V.A., Nekorkin, V.I., Velarde, M.G.: Spiking behavior in a noise-driven system combining oscillatory and excitatory properties. Phys. Rev. Lett. 86(15), 3431–3434 (2001)
Pomarede, J.-M.: A unified approach via graphs to Skorohod’s topologies on the function space. D. PhD thesis, Yale (1976)
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This material is based upon work supported by the National Science Foundation under Grant Nos. 0305925 and 0604249. The author would also like to thank Professor Jeff Moehlis of the Department of Mechanical and Environmental Engineering at UC Santa Barbara for a number of useful discussions about canards.
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Sowers, R.B. Random Perturbations of Canards. J Theor Probab 21, 824–889 (2008). https://doi.org/10.1007/s10959-008-0150-1
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DOI: https://doi.org/10.1007/s10959-008-0150-1