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.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsReferences
M. Abramowitz and I. Stegun. Handbook of Mathematical Functions. Dover, 1964.
M. A. Aizerman and F. R. Gantmakher. On the stability of equilibrium positions in discontinuous systems. Prikl. Mat. i Mekh., 24:283–93, 1960.
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.
J. C. Alexander and T. I. Seidman. Sliding modes in intersecting switching surfaces, i: Blending. Houston J. Math, 24(3):545–569, 1998.
J. C. Alexander and T. I. Seidman. Sliding modes in intersecting switching surfaces, ii: Hysteresis. Houston J. Math, 25(1):185–211, 1999.
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.
A. A. Andronov and S. E. Khaikin. Theory of Oscillations, Part I. Moscow: ONTI NKTN (in Russian), 1937.
A. A. Andronov, A. A. Vitt, and S. E. Khaikin. Theory of oscillations. Moscow: Fizmatgiz (in Russian), 1959.
Aristotle. The Physics. 1837 Bekker edition of Aristotle’s Works, 3rd century BC.
Robin Waterfield (Translator)] Aristotle [David Bostock (Editor). Physics. Oxford World’s Classics, 2008.
J. B. Barbour. The Discovery of Dynamics. Oxford University Press, 2001.
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.
F P Bowden and D Tabor. The friction and Lubrication of Solids. Oxford University Press, 1950.
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.
P. Collett and J-P. Eckmann. Iterated Maps on the Interval as Dynamical Systems. Boston: Birkhauser, 1980.
Charles Darwin. Origin of Species. John Murray, London, 1859.
R. Descartes. Traité du monde et de la lumière. modern reprint: Abaris Books, 1979, 1677.
D. Dowson. History of Tribology. Professional Engineer Pulbishing, London, 1998.
J. Eggers. Singularities in droplet pinching with vanishing viscocity. SIAM J. Appl. Math., 60(6):1997–2008, 2000.
H. Feinberg. Physicists: Epoch And Personalities, volume 4 of History of Modern Physical Sciences. World Scientific, Singapore, 2011.
A. F. Filippov. Differential equations with discontinuous right-hand side. American Mathematical Society Translations, Series 2, 42:19–231, 1964.
A. F. Filippov. Differential Equations with Discontinuous Righthand Sides. Kluwer Academic Publ. Dortrecht, 1988 (Russian 1985).
I. Flügge-Lotz. Discontinuous Automatic Control. Princeton University Press, 1953.
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.
P. Glendinning. Robust new routes to chaos in differential equations. Physics Letters A, 168:40–46, 1992.
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.
O. Hájek. Discontinuous differential equations, I. J. Differential Equations, 32(2):149–170, 1979.
O. Hájek. Discontinuous differential equations, II. J. Differential Equations, 32(2):171–185, 1979.
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.
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.
N. Hinrichs, M. Oestreich, and K. Popp. On the modelling of friction oscillators. J. Sound Vib., 216(3):435–459, 1998.
I. M. Hutchings. Leonardo da vinci’s studies of friction. Wear, 360–361:51–66, 2016.
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.
Klymkowsky and Cooper. Biofundamentals, 2016.
J. Krim. Friction at macroscopic and microscopic length scales. Am. J. Phys., 70:890–897, 2002.
V. Kulebakin. On theory of vibration controller for electric machines. Theor. Exp. Electon (in Russian), 4, 1932.
J. Larmor. Sir George Gabriel Stokes: Memoir and Scientific Correspondence, volume 1. Cambridge University Press, 1907.
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.
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.
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.
C. Linnaeus. Philosophia Botanica, chapter III, 77. Typis Joannis Thomae de Trattner, 1st edition, 1751.
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.
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.
R. Lozi. Un attracteur étrange (?) du type attracteur de Hénon. Journal de Physique, 39:C5–9, 1978.
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.
NASA. Voyager recent history, http://voyager.gsfc.nasa.gov/heliopause/recenthist.html, last viewed Aug 2016, 2012.
J. Nash. The imbedding problem for Riemannian manifolds. Annals of Mathematics, 63(1):20–63, 1956.
I. Newton. PhilosophiæNaturalis Principia Mathematica. 1729.
G. Nikolsky. On automatic stability of a ship on a given course. Proc Central Commun Lab (in Russian), 1:34–75, 1934.
D. Paillard and F. Parrenin. The antarctic ice sheet and the triggering of deglaciations. Earth and Planetary Science Letters, 227:263–271, 2004.
B. N. J. Persson. Sliding Friction: Physical Principles and Applications. Springer, 1998.
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.
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.
L. Sherwood. Human Physiology, From Cells to Systems. Cengage Learning, 2012.
J. Shi, J. Guldner, and V. I. Utkin. Sliding mode control in electro-mechanical systems. CRC Press, 1999.
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.
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.
G G Stokes. On the discontinuity of arbitrary constants which appear in divergent developments. Trans Camb Phil Soc, 10:106–128, 1864.
C. Studer. Numerics of Unilateral Contacts and Friction, volume 47 of Lecture Notes in Applied and Computational Mechanics. Springer Verlag, 2009.
D. Tabor. Triobology - the last 25 years. a personal view. Tribology International, 28(1):7–10, 1995.
M. A. Teixeira. Structural stability of pairings of vector fields and functions. Boletim da S.B.M., 9(2):63–82, 1978.
M. A. Teixeira. On topological stability of divergent diagrams of folds. Math. Z., 180:361–371, 1982.
G.A. Tomlinson. A molecular theory of friction. Philos. Mag., 7(7):905–939, 1929.
Y. Tsypkin. Theory of Relay Control Systems. (in Russian), Moscow: Gostechizdat, 1955.
V. I. Utkin. Sliding modes in control and optimization. Springer-Verlag, 1992.
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.
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.
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.
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.
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.
V. A. Yakubovich, G. A. Leonov, and A. K. Gelig. Stability of stationary sets in control systems with discontinuous nonlinearities. Singapore: World Scientific, 2004.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer Nature Switzerland AG
About this chapter
Cite this chapter
Jeffrey, M.R. (2018). Origins of Discontinuity. In: Hidden Dynamics. Springer, Cham. https://doi.org/10.1007/978-3-030-02107-8_1
Download citation
DOI: https://doi.org/10.1007/978-3-030-02107-8_1
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-02106-1
Online ISBN: 978-3-030-02107-8
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)