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Noisy Dynamical Systems

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Abstract

The study of fluctuations in dynamical systems had rather humble beginnings. With the advent of the light microscope in the late 1600s, many investigators noticed that small particles suspended in liquid were continually moving, even though no macroscopic movements of the fluid could be detected [492]. Initially, it was thought that these movements indicated that the particles were alive.

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Notes

  1. 1.

    Robert Brown (1773–1858), Scottish botanist.

  2. 2.

    Robert Brown was not the first person to observe these movements. In his original paper, Brown cites ten investigators who had earlier observed these movements; subsequently, a scientific historian added the name J. Ingen-Housz to this list [626]. However, it is clear that Robert Brown was the first to investigate the origins of this phenomenon in a systematic manner, and consequently, he deserves to have his name attached to these particle motions [492].

  3. 3.

    Albert Einstein (1879–1955), German-born American physicist.

  4. 4.

    Paul Langevin (1872–1946), French physicist.

  5. 5.

    Kiyoshi It\(\bar{\mathrm{o}}\) (1915–2008), Japanese mathematician.

  6. 6.

    Ruslan Leont’evich Stratonovich (1930–1997), Russian physicist and engineer.

  7. 7.

    Adriaan Daniël Fokker (1887–1972), Dutch physicist and musician.

  8. 8.

    When the noise is multiplicative and/or the equation is nonlinear, the Ito and Stratonovich formulations are not typically equivalent.

  9. 9.

    Osborne Reynolds (1842–1912), Anglo-Irish physicist.

  10. 10.

    George Gabriel Stokes (1819–1903), Irish mathematician, physicist, politician, and theologian.

  11. 11.

    Haldan Keffer Hartline (1903–1983), American physiologist.

  12. 12.

    L.A. Scott Kelso (d. 1947), Irish-American neuroscientist.

  13. 13.

    The noise intensity can be measured in a variety of ways, including its variance, root-mean-square value, its standard deviation, and the coefficient of variation, which is the standard deviation divided by the mean.

  14. 14.

    Peter Swain (b. 1970), British-Canadian systems biologist.

  15. 15.

    A 1,000-nt protein corresponds to a protein sequence containing ≈ 333 amino acids, since each amino acid is encoded in the gene by three consecutive nucleotides.

  16. 16.

    Otto Herbert Schmitt (1913–1998), American inventor, engineer, and biophysicist.

  17. 17.

    Frank Edward Moss (1934–2011), American physicist, pioneered applications of stochastic resonance to biology.

  18. 18.

    Intermittency is the term used by physicists to describe the burstiness of biological signals discussed in Section 15.5.2. Technically, the type of intermittency we describe here is referred to as on–off intermittency in order to distinguish it from other forms of intermittency [503].

  19. 19.

    The law of large numbers states that the arithmetic mean of a very large sum of independent observations of a random variable x(s) is equal to the mean of x(s) [162, 486].

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Authors and Affiliations

  1. W.M. Keck Science Department, The Claremont Colleges, Claremont, CA, USA

    John Milton

  2. Graduate School of Mathematics, Nagoya University, Nagoya, Japan

    Toru Ohira

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Milton, J., Ohira, T. (2014). Noisy Dynamical Systems. In: Mathematics as a Laboratory Tool. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-9096-8_13

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