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Transient Dynamics

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Mathematics as a Laboratory Tool

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Abstract

Scientists, particularly those who investigate physiological and/or biomechanical systems, use inputs to explore the nature of biological dynamical systems (e.g., the black box shown in Figure 6.1). Since in the laboratory, we most often investigate dynamical systems that are stable (at least in some sense), we expect the observed responses to an input to contain two components:

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Notes

  1. 1.

    The reader should note that we have written this equation to be dimensionally correct.

  2. 2.

    This is an example of a situation in which a system attains a steady state but is not at equilibrium.

  3. 3.

    An even gentler change can be obtained with a function that is not only continuous at the point where it begins to increase but differentiable as well. A parabolic forcing function would accomplish this.

  4. 4.

    Paul Adrien Maurice Dirac (1902–1984), English theoretical physicist.

  5. 5.

    George Green (1793–1841), British mathematical physicist.

  6. 6.

    E. Oran Brigham (b. 1919), American applied mathematician.

  7. 7.

    Several applets demonstrating this graphical approach for computing the convolution integral can be readily located on the Internet; see, for example, http://en.wikipedia.org/wiki/Convolution.

  8. 8.

    A geometric series is defined by \(S_{n} =\sum _{ i=0}^{n}m^{i}\), which converges for \(\vert m\vert < 1\) to \(1/(1 - m)\).

  9. 9.

    Wilfrid Rall (b. 1922), American neuroscientist.

  10. 10.

    Uwe an der Heiden (b. 1942), German mathematician and philosopher.

  11. 11.

    Leah Edelstein-Keshet (PhD 1982), Israeli–Canadian mathematical biologist.

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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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  1. John Milton
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Milton, J., Ohira, T. (2014). Transient Dynamics. In: Mathematics as a Laboratory Tool. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-9096-8_6

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