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Origins of Discontinuity

  • Mike R. Jeffrey
Chapter

Abstract

Discontinuities occur when light refracts, when neurons or electronic switches activate, and when collisions or decisions or mitosis or myriad other processes enact a change of regime. We observe them in empirical laws, in the structure of solid bodies, and also in the series expansions of certain mathematical functions.

References

  1. 3.
    M. Abramowitz and I. Stegun. Handbook of Mathematical Functions. Dover, 1964.Google Scholar
  2. 4.
    M. A. Aizerman and F. R. Gantmakher. On the stability of equilibrium positions in discontinuous systems. Prikl. Mat. i Mekh., 24:283–93, 1960.zbMATHGoogle Scholar
  3. 5.
    M. A. Aizerman and E. S. Pyatnitskii. Fundamentals of the theory of discontinuous systems I,II. Automation and Remote Control, 35:1066–79, 1242–92, 1974.Google Scholar
  4. 6.
    J. C. Alexander and T. I. Seidman. Sliding modes in intersecting switching surfaces, i: Blending. Houston J. Math, 24(3):545–569, 1998.MathSciNetzbMATHGoogle Scholar
  5. 7.
    J. C. Alexander and T. I. Seidman. Sliding modes in intersecting switching surfaces, ii: Hysteresis. Houston J. Math, 25(1):185–211, 1999.MathSciNetzbMATHGoogle Scholar
  6. 8.
    G. Amontons. De la resistance causée dans les machines, tant par les frottemens des parties qui les composent, que par roideur des cordes qu’on y employe, et la maniere de calculer l’un et l’autre. Histoire de l’Académie royale des sciences, 1699.Google Scholar
  7. 9.
    A. A. Andronov and S. E. Khaikin. Theory of Oscillations, Part I. Moscow: ONTI NKTN (in Russian), 1937.Google Scholar
  8. 10.
    A. A. Andronov, A. A. Vitt, and S. E. Khaikin. Theory of oscillations. Moscow: Fizmatgiz (in Russian), 1959.Google Scholar
  9. 11.
    Aristotle. The Physics. 1837 Bekker edition of Aristotle’s Works, 3rd century BC.Google Scholar
  10. 12.
    Robin Waterfield (Translator)] Aristotle [David Bostock (Editor). Physics. Oxford World’s Classics, 2008.Google Scholar
  11. 16.
    J. B. Barbour. The Discovery of Dynamics. Oxford University Press, 2001.Google Scholar
  12. 19.
    C. M. Bender and S. A. Orszag. Advanced mathematical methods for scientists and engineers I. Asymptotic methods and perturbation theory. Springer-Verlag, New York, 1999.Google Scholar
  13. 24.
    F P Bowden and D Tabor. The friction and Lubrication of Solids. Oxford University Press, 1950.Google Scholar
  14. 28.
    J. A. bu Qahouq, H. Mao, H. J. Al-Atrash, and I. Batarseh. Maximum efficiency point tracking (mept) method and digital dead time control implementation. IEEE Transactions on Power Electronics, 21(5):1273–1281, 2006.Google Scholar
  15. 34.
    P. Collett and J-P. Eckmann. Iterated Maps on the Interval as Dynamical Systems. Boston: Birkhauser, 1980.Google Scholar
  16. 41.
    Charles Darwin. Origin of Species. John Murray, London, 1859.Google Scholar
  17. 42.
    R. Descartes. Traité du monde et de la lumière. modern reprint: Abaris Books, 1979, 1677.Google Scholar
  18. 55.
    D. Dowson. History of Tribology. Professional Engineer Pulbishing, London, 1998.Google Scholar
  19. 58.
    J. Eggers. Singularities in droplet pinching with vanishing viscocity. SIAM J. Appl. Math., 60(6):1997–2008, 2000.MathSciNetzbMATHGoogle Scholar
  20. 67.
    H. Feinberg. Physicists: Epoch And Personalities, volume 4 of History of Modern Physical Sciences. World Scientific, Singapore, 2011.Google Scholar
  21. 70.
    A. F. Filippov. Differential equations with discontinuous right-hand side. American Mathematical Society Translations, Series 2, 42:19–231, 1964.Google Scholar
  22. 71.
    A. F. Filippov. Differential Equations with Discontinuous Righthand Sides. Kluwer Academic Publ. Dortrecht, 1988 (Russian 1985).Google Scholar
  23. 73.
    I. Flügge-Lotz. Discontinuous Automatic Control. Princeton University Press, 1953.Google Scholar
  24. 74.
    A. Kh. Gelig, G. A. Leonov, and V. A. Iakubovich. Stability of nonlinear systems with nonunique equilibrium position. Moscow, Izdatel’stvo Nauka (in Russian), page 400, 1978.Google Scholar
  25. 75.
    P. Glendinning. Robust new routes to chaos in differential equations. Physics Letters A, 168:40–46, 1992.MathSciNetGoogle Scholar
  26. 94.
    A. Guran, F. Pfeiffer, and K. Popp, editors. Dynamics with Friction: Modeling, Analysis and Experiment I & II, volume 7 of Series B. World Scientific, 1996.Google Scholar
  27. 95.
    O. Hájek. Discontinuous differential equations, I. J. Differential Equations, 32(2):149–170, 1979.MathSciNetzbMATHGoogle Scholar
  28. 96.
    O. Hájek. Discontinuous differential equations, II. J. Differential Equations, 32(2):171–185, 1979.MathSciNetzbMATHGoogle Scholar
  29. 97.
    W. R. Hamilton. Third supplement to an essay on the theory of systems of rays. Transactions of the Royal Irish Academy, presented in 1832, 17:1–144, 1837.Google Scholar
  30. 103.
    A. V. Hill. The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves. Proc. Physiol. Soc., 40:iv–vii, 1910.Google Scholar
  31. 105.
    N. Hinrichs, M. Oestreich, and K. Popp. On the modelling of friction oscillators. J. Sound Vib., 216(3):435–459, 1998.Google Scholar
  32. 110.
    I. M. Hutchings. Leonardo da vinci’s studies of friction. Wear, 360–361:51–66, 2016.Google Scholar
  33. 127.
    M. A. Kiseleva and N. V. Kuznetsov. Coincidence of the Gelig-Leonov-Yakubovich, Filippov, and Aizerman-Pyatnitskiy definitions. Vestnik St. Petersburg University: Mathematics, 48(2):66–71, 2015.MathSciNetzbMATHGoogle Scholar
  34. 128.
    Klymkowsky and Cooper. Biofundamentals, 2016.Google Scholar
  35. 132.
    J. Krim. Friction at macroscopic and microscopic length scales. Am. J. Phys., 70:890–897, 2002.Google Scholar
  36. 136.
    V. Kulebakin. On theory of vibration controller for electric machines. Theor. Exp. Electon (in Russian), 4, 1932.Google Scholar
  37. 141.
    J. Larmor. Sir George Gabriel Stokes: Memoir and Scientific Correspondence, volume 1. Cambridge University Press, 1907.Google Scholar
  38. 142.
    A. Le Bot and E. Bou Chakra. Measurement of friction noise versus contact area of rough surfaces weakly loaded. Tribology Letters, 37:273–281, 2010.Google Scholar
  39. 143.
    G. W. (Translated Leibniz, edited by P., and J.) Bennett. New Essays on Human Understanding c.1704, volume IV:16. New York: Cambridge University Press, 1981.Google Scholar
  40. 149.
    E. Lindelöf. Sur l’application de la méthode des approximations successives aux équations différentielles ordinaires du premier ordre. C. R. Hebd. Seances Acad. Sci, 114:454–457, 1894.zbMATHGoogle Scholar
  41. 150.
    C. Linnaeus. Philosophia Botanica, chapter III, 77. Typis Joannis Thomae de Trattner, 1st edition, 1751.Google Scholar
  42. 153.
    H Lloyd. Further experiments on the phaenomena presented by light in its passage along the axes of biaxial crystals. Lond Ed Phil Mag, 2:207–210, 1833.Google Scholar
  43. 154.
    H Lloyd. On the phaenomena presented by light in its passage along the axes of biaxial crystals. Lond Ed Phil Mag, 2:112–120, 1833.Google Scholar
  44. 156.
    R. Lozi. Un attracteur étrange (?) du type attracteur de Hénon. Journal de Physique, 39:C5–9, 1978.Google Scholar
  45. 157.
    A.J.S. Machado, S.T. Renosto, C.A.M. dos Santos, L.M.S. Alves, and Z. Fisk. Superconductors - Materials, Properties and Applications, chapter Defect Structure Versus Superconductivity in MeB2 Compounds (Me = Refractory Metals) and One-Dimensional Superconductors. InTech, 2012.Google Scholar
  46. 164.
    NASA. Voyager recent history, http://voyager.gsfc.nasa.gov/heliopause/recenthist.html, last viewed Aug 2016, 2012.
  47. 165.
    J. Nash. The imbedding problem for Riemannian manifolds. Annals of Mathematics, 63(1):20–63, 1956.MathSciNetzbMATHGoogle Scholar
  48. 167.
    I. Newton. PhilosophiæNaturalis Principia Mathematica. 1729.Google Scholar
  49. 168.
    G. Nikolsky. On automatic stability of a ship on a given course. Proc Central Commun Lab (in Russian), 1:34–75, 1934.Google Scholar
  50. 173.
    D. Paillard and F. Parrenin. The antarctic ice sheet and the triggering of deglaciations. Earth and Planetary Science Letters, 227:263–271, 2004.Google Scholar
  51. 174.
    B. N. J. Persson. Sliding Friction: Physical Principles and Applications. Springer, 1998.Google Scholar
  52. 178.
    T. Putelat, J. H. P. Dawes, and J. R. Willis. On the microphysical foundations of rate-and-state friction. Journal of the Mechanics and Physics of Solids, 59(5):1062–1075, 2011.MathSciNetzbMATHGoogle Scholar
  53. 179.
    M. Radiguet, D. S. Kammer, P. Gillet, and J.-F. Molinari. Survival of heterogeneous stress distributions created by precursory slip at frictional interfaces. PRL, 111(164302):1–4, 2013.Google Scholar
  54. 185.
    L. Sherwood. Human Physiology, From Cells to Systems. Cengage Learning, 2012.Google Scholar
  55. 186.
    J. Shi, J. Guldner, and V. I. Utkin. Sliding mode control in electro-mechanical systems. CRC Press, 1999.Google Scholar
  56. 190.
    D. J. W. Simpson and R. Kuske. Stochastically perturbed sliding motion in piecewise-smooth systems. Discrete Contin. Dyn. Syst. Ser. B, 19(9):2889–2913, 2014.MathSciNetzbMATHGoogle Scholar
  57. 193.
    M. Sorensen, S. DeWeerth, G Cymbalyuk, and R. L. Calabrese. Using a hybrid neural system to reveal regulation of neuronal network activity by an intrinsic current. Journal of Neuroscience, 24(23):5427–5438, 2004.Google Scholar
  58. 197.
    G G Stokes. On the discontinuity of arbitrary constants which appear in divergent developments. Trans Camb Phil Soc, 10:106–128, 1864.Google Scholar
  59. 200.
    C. Studer. Numerics of Unilateral Contacts and Friction, volume 47 of Lecture Notes in Applied and Computational Mechanics. Springer Verlag, 2009.Google Scholar
  60. 201.
    D. Tabor. Triobology - the last 25 years. a personal view. Tribology International, 28(1):7–10, 1995.Google Scholar
  61. 203.
    M. A. Teixeira. Structural stability of pairings of vector fields and functions. Boletim da S.B.M., 9(2):63–82, 1978.MathSciNetzbMATHGoogle Scholar
  62. 204.
    M. A. Teixeira. On topological stability of divergent diagrams of folds. Math. Z., 180:361–371, 1982.MathSciNetzbMATHGoogle Scholar
  63. 208.
    G.A. Tomlinson. A molecular theory of friction. Philos. Mag., 7(7):905–939, 1929.Google Scholar
  64. 209.
    Y. Tsypkin. Theory of Relay Control Systems. (in Russian), Moscow: Gostechizdat, 1955.Google Scholar
  65. 212.
    V. I. Utkin. Sliding modes in control and optimization. Springer-Verlag, 1992.Google Scholar
  66. 214.
    A. J. Vander, D. Luciano, and J. Sherman. Movement of Molecules Across Cell Membranes. Human Physiology: The Mechanism of Body Function. McGraw-Hill Higher Education, 2001.Google Scholar
  67. 217.
    W. R. Webber and F. B. McDonald. Recent voyager 1 data indicate that on 25 august 2012 at a distance of 121.7 au from the sun, sudden and unprecedented intensity changes were observed in anomalous and galactic cosmic rays. Geophysical Research Letters, 40(9):1665–1668, 2013.Google Scholar
  68. 221.
    A. J. Weymouth, D. Meuer, P. Mutombo, T. Wutscher, M. Ondracek, P. Jelinek, and F. J. Giessibl. Atomic structure affects the directional dependence of friction. PRL, 111(126103):1–4, 2013.Google Scholar
  69. 222.
    J. Wojewoda, S. Andrzej, M. Wiercigroch, and T. Kapitaniak. Hysteretic effects of dry friction: modelling and experimental studies. Phil. Trans. R. Soc. A, 366:747–765, 2008.MathSciNetzbMATHGoogle Scholar
  70. 223.
    J. Woodhouse, T. Putelat, and A. McKay. Are there reliable constitutive laws for dynamic friction? Phil. Trans. R. Soc. A, 373(20140401):1–21, 2015.Google Scholar
  71. 224.
    V. A. Yakubovich, G. A. Leonov, and A. K. Gelig. Stability of stationary sets in control systems with discontinuous nonlinearities. Singapore: World Scientific, 2004.Google Scholar

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Authors and Affiliations

  • Mike R. Jeffrey
    • 1
  1. 1.Department of Engineering MathematicsUniversity of BristolBristolUK

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