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Escape from Domains of Attraction for Systems Perturbed by Noise

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Nonlinear Phenomena in Physics and Biology

Part of the book series: NATO Advanced Study Institutes Series ((NSSB,volume 75))

Abstract

Deterministic theories have been remarkably successful in interpreting and explaining the world, although more precise formulations will always involve random effects. An outstanding example is classical mechanics, whose utility is hardly impaired by the existence of quantum effects. Intuitively, we may think of the deterministic trajectories as being smeared out by the random effects. Even if the smearing is large, the qualitative behavior may still be captured by the deterministic result.

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References

  • Abramowitz, M., and Stegun, I.A., 1964, “Handbook of Mathematical Functions,” Dover Pub., New York,

    Google Scholar 

  • de Groen, P.P.N., 1980, The nature of resonance in a singular perturbation problem of turning point type, SIAM J. Math. Anal., 11: 1.

    Article  MathSciNet  MATH  Google Scholar 

  • Feller, W., 1966, “An Introduction to Probability Theory, Vol. II,” John Wiley, New York. Gikhman, I.I., and Skorohod, A.V., 1973, “Stochastic Differential Equations,” Springer-Verlag, Berlin.

    Google Scholar 

  • Gould, S.J., and Eldredge, N., 1977, Punctuated equilibria: the tempo and mode of evolution reconsidered, Paleobiology, 3: 115.

    Google Scholar 

  • Holling, C.S., 1973, Resilience and stability of ecological systems, Ann. Rev. Ecol. Syst., 4: 1.

    Article  Google Scholar 

  • Kamin, S., 1979, On elliptic singular perturbation problems with turning points, SIAM J. Math. Anal., 10: 447.

    Article  MathSciNet  MATH  Google Scholar 

  • Karlin, S., and Taylor, H.M., 1975, “A First Course in Stochastic Processes,” 2nd ed., Academic Press, New York.

    MATH  Google Scholar 

  • Kopell, N., 1979, A geometric approach to boundary layer problems exhibiting resonance, SIAM J. Appl. Math., 37: 436.

    Article  MathSciNet  MATH  Google Scholar 

  • Kramers, H.A., 1940, Brownian motion in a field of force and the diffusion model of chemical reactions, Physica, 7: 284.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  • Ludwig, D., 1974, Persistence of dynamical systems under random perturbations, SIAM Rev., 17: 605.

    Article  MathSciNet  Google Scholar 

  • Mangel, M., 1979, Small fluctuations in systems with multiple steady states, SIAM J. Appl. Math., 36: 544.

    Article  MathSciNet  MATH  Google Scholar 

  • Mangel, M., and Ludwig, D., 1977, Probability of extinction in a stochastic competition, SIAM J. Appl. Math., 33: 256.

    Article  MathSciNet  MATH  Google Scholar 

  • Matkowsky, B., and Schuss, Z., 1977, The exit problem, SIAM J. Appl. Math., 33: 230.

    Article  MathSciNet  MATH  Google Scholar 

  • Schuss, Z., 1980, Singular perturbation methods in stochastic differential equations of mathematical physics, SIAM Rev., 22: 119.

    Article  MathSciNet  MATH  Google Scholar 

  • Schuss, Z., and Matkowsky, B., 1979, The exit problem: a new approach to diffusion across potential barriers, SIAM J. Appl. Math., 35: 604.

    Article  MathSciNet  Google Scholar 

  • Strook, D.W., and Varadhan, S.R.S., 1979, “Multidimensional Diffusion Processes,” Springer-Verlag, Berlin.

    Google Scholar 

  • Turelli, M., 1977, Random environments and stochastic calculus, Th. Pop. Biol., 12: 140.

    Article  MathSciNet  MATH  Google Scholar 

  • Venttsell, A.D., and Freidlin, M.I., 1970, On small random perturbations of dynamic systems, Uspehi Mat. Nauk, 25: 3. (Russian Math, Surveys, 25:1.)

    Google Scholar 

  • Williams, M., preprints, Asymptotic one-dimensional exit time distributions, and Another look at Ackerberg-Malley resonance, Dept. of Math., Virginia Polytechs. Inst., Blacksburg, Virginia 24061.

    Google Scholar 

  • Williams, R.G., 1978, The stochastic exit problem for dynamical systems, Ph.D. Thesis, Dept. of Appl. Math., Calif. Inst, of Tech.

    Google Scholar 

  • Wright, S., 1932, The roles of mutation, inbreeding, cross-breeding and selection in evolution, Proc. VI Int. Cong. Genet., 1: 356.

    Google Scholar 

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© 1981 Plenum Press, New York

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Ludwig, D. (1981). Escape from Domains of Attraction for Systems Perturbed by Noise. In: Enns, R.H., Jones, B.L., Miura, R.M., Rangnekar, S.S. (eds) Nonlinear Phenomena in Physics and Biology. NATO Advanced Study Institutes Series, vol 75. Springer, Boston, MA. https://doi.org/10.1007/978-1-4684-4106-2_14

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  • DOI: https://doi.org/10.1007/978-1-4684-4106-2_14

  • Publisher Name: Springer, Boston, MA

  • Print ISBN: 978-1-4684-4108-6

  • Online ISBN: 978-1-4684-4106-2

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