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Mathematics as a Laboratory Tool pp 57–90Cite as

Stability

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Abstract

A reasonable starting point for the study of input–output relationships is to assume that the black box shown in Figure 4.1 contains a dynamical system operating at its fixed point. In a laboratory setting, this is equivalent to a system that is either at equilibrium or in a steady state.

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Notes

  1. 1.

    Our use of the term “potential” is in keeping with the concept of the Lyapunov function introduced in Section 4.5 [228]. Thus we use the same symbol, U(x), to denote the potential function and the Lyapunov function.

  2. 2.

    After Louis Albert Necker (1786–1861), Swiss crystallographer, zoologist, and geographer.

  3. 3.

    Brook Taylor (1685–1731), English mathematician.

  4. 4.

    The partial derivative of f(x, y) with respect to x is calculated by assuming that y is constant. For example, if f(x, y) = 6xy, then \(\partial f/\partial x = 6y\). Similarly, \(\partial f/\partial y = 6x\).

  5. 5.

    For another definition of phase and phase shift, see Section 10.5.

  6. 6.

    Historically, mathematicians favored the term focal point. However, the recent trend is to use the term spiral point [52, 121, 600, 646, 654]. The advantage of the latter term is that it describes what one sees in the phase plane. Most undergraduate students do not intuitively associate this phase-plane portrait with the word focus.

  7. 7.

    Developed independently by Alfred James Lotka (1880–1949), American biophysicist and biostatistician, and Vito Volterra (1860–1940), Italian mathematician and physicist.

  8. 8.

    Although it seems intuitive that density should be number of animals per unit area, the SI unit is in fact area−1, since, as the reader will recall from Section 1.2.3, a number of objects has no units in the SI system.

  9. 9.

    Balthasar van der Pol (1889–1959), Dutch physicist.

  10. 10.

    Formulated in a weaker version by Poincaré and later strengthened and proved rigorously by Ivar Otto Bendixson (1861–1935), Swedish mathematician.

  11. 11.

    Aleksandr Mikhailovich Lyapunov (1857–1918), Russian mathematician and physicist.

  12. 12.

    After William Rowan Hamilton (1805–1865), Irish physicist, astronomer, and mathematician.

  13. 13.

    In biology, the term dissipative system is used to describe a thermodynamic open system that is operating out of, and often far from, thermodynamic equilibrium and freely exchanges both energy and mass with its surroundings. From this point of view, the damped harmonic oscillator describes a transient response of a closed dynamical system. A simple example of a dissipative system is that of the water fountain we considered in Section 3.3.1. A remarkable property of dissipative systems is their ability to form the large-scale spatially coherent structures that characterize biology, including cells, tissues, and organs [78, 462]. In Chapter 15, we more carefully examine the thermodynamic differences between systems at equilibrium and those not at equilibrium.

  14. 14.

    Asymptotic stability means that as t → ∞, the trajectory approaches the fixed point itself. In contrast, stability means that as t → ∞, the trajectory approaches a neighborhood of the fixed point.

  15. 15.

    Georg Duffing (1861–1944), German engineer and inventor.

  16. 16.

    Ronald Ross (1857–1932), Indian-born British physician.

  17. 17.

    Brian Carey Goodwin (1931–2009), Canadian biomathematician.

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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). Stability. In: Mathematics as a Laboratory Tool. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-9096-8_4

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