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Difference Equations Everywhere: Some Motivating Examples

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Difference Equations, Discrete Dynamical Systems and Applications (ICDEA 2017)

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Abstract

This work collects several situations where discrete dynamical systems or difference equations appear. Most of them are different from the examples used in textbooks and from the usual mathematical models appearing in Biology or Economy. The examples are presented in detail, including some appropriate references. Although most of them are known, the fact of collecting all together aims to be a source of motivation for studying DDS and difference equations and to facilitate teaching these subjects.

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References

  1. Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions, with Formulas, Graphs, and Mathematical Tables. Dover Publications (1965)

    Google Scholar 

  2. Arndt, J., Haenel, C.: Pi\(-\)unleashed. Translated from the German in 1998 by C, p. 2001. Springer, Berlin, Lischka and Lischka, D., Second ed (2001)

    Google Scholar 

  3. Backhouse, N.: Pancake Functions and Approximations to \(\pi \). Note 79.36, Math. Gazette 79, 371–374 (1995)

    Google Scholar 

  4. Balibrea, F., Linero, A.: Some new results and open problems on periodicity of difference equations. In: Iteration theory (ECIT 2004), 15–38, Grazer Math. Ber., 350, Karl-Franzens-Univ. Graz, Graz (2006)

    Google Scholar 

  5. Barnes, J.: Nice numbers. Birkhäuser/Springer, Cham (2016)

    Book  Google Scholar 

  6. Bastien, G., Rogalski, M.: Global behavior of the solutions of Lyness’ difference equation \(u_{n+2}u_n=u_{n+1}+a\). J. Differ. Equ. Appl. 10, 977–1003 (2004)

    Google Scholar 

  7. Boros, G., Moll, V.H.: Landen transformations and the integration of rational functions. Math. Comput. 71, 649–668 (2001)

    Article  MathSciNet  Google Scholar 

  8. Borwein, J.M.: The Life of Pi: from archimedes to ENIAC and beyond. Part III. In: Sidoli, N., Van Brummelen, G. (Eds.), Surveys and Studies in the Ancient Greek and Medieval Islamic Mathematical Sciences in Honor of J. L. Berggren. Springer, Heidelberg (2014)

    Google Scholar 

  9. Borwein, J.M., Borwein, P.B., Pi and the AGM: A study in analytic number theory and computational complexity. Canadian Mathematical Society Series of Monographs and Advanced Texts. A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York (1987)

    Google Scholar 

  10. Borwein, J.M., Borwein, P.B., Bailey, D.H.: Ramanujan, modular equations, and approximations to pi, or How to compute one billion digits of pi. Amer. Math. Monthly 96, 201–219 (1989)

    Article  MathSciNet  Google Scholar 

  11. Borwein, J.M., Macklem, M.S.: The (digital) life of Pi. Austral. Math. Soc. Gaz. 33, 243–248 (2006)

    MathSciNet  MATH  Google Scholar 

  12. Brent, R.P.: Fast multiple-precision evaluation of elementary functions. J. Assoc. Comput. Mach. 23, 242–251 (1976)

    Article  MathSciNet  Google Scholar 

  13. Cánovas, J.S., Linero, A., Soler. G.: A characterization of \(k\)-cycles. Nonlinear Anal. 72, 364–372 (2010)

    Google Scholar 

  14. Carlson, B.C.: Algorithms involving arithmetic and geometric means. Amer. Math. Monthly 78, 496–505 (1971)

    Article  MathSciNet  Google Scholar 

  15. Chabert, J.-L.: A history of algorithms. From the pebble to the microchip. Translated from the 1994 French original by Chris Weeks. Springer, Berlin (1999)

    Google Scholar 

  16. Chamberland, M.: A continuous extension of the \(3x + 1\) problem to the real line. Dynam. Contin. Discrete Impuls Syst. 2, 495–509 (1996)

    Google Scholar 

  17. Chamberland, M., Moll, V.H.: Dynamics of the degree six Landen transformation. Discrete Dyn. Syst. 15, 905–919 (2006)

    Article  MathSciNet  Google Scholar 

  18. Christopher, C.J., Lloyd, N.G.: Polynomial systems: a lower bound for the Hilbert numbers. Proc. Roy. Soc. London Ser. A 450, 219–224 (1995)

    Article  MathSciNet  Google Scholar 

  19. Cima, A., Gasull, A., Mañosa, V.: Global periodicity and complete integrability of discrete dynamical systems. J. Diff. Equ. Appl. 12, 697–716 (2006)

    Article  MathSciNet  Google Scholar 

  20. Cima, A., Gasull, A., Mañosas, F.: On Coxeter recurrences. J. Diff. Equ. Appl. 18, 1457–1465 (2012)

    Article  MathSciNet  Google Scholar 

  21. Chiarella, C., Bischi, G.I., Gardini, L. (eds.): Nonlinear Dynamics in Economics and Finance. Springer, Berlin (2009)

    Google Scholar 

  22. Cohen, A.M., Cutts, J.F., Fielder, R., Jones, D.E., Ribbans, J., Stuart, E.: Numerical Analysis. McGraw-Hill, London, New York (1973)

    Google Scholar 

  23. Cox, D.A.: The Arithmetic-Geometric mean of Gauss. L’Einseignement Mathématique 30, 275–330 (1984)

    MathSciNet  MATH  Google Scholar 

  24. Coxeter, H.S.M.: Frieze patterns. Acta Arith. 18, 297–310 (1971)

    Article  MathSciNet  Google Scholar 

  25. Csörnyei, M., Laczkovich, M.: Some periodic and non-periodic recursions. Monatshefte für Mathematik 132, 215–236 (2001)

    Article  MathSciNet  Google Scholar 

  26. Cull, P., Flahive, M., Robson, R.: Difference Equations. From rabbits to chaos. Undergraduate Texts in Mathematics. Springer, New York (2005)

    Google Scholar 

  27. Dalzell, D.P.: On \(22/7.\) J. London Math. Soc. 19, 133–134 (1944)

    Google Scholar 

  28. Dalzell, D.P.: On \(22/7\) and \(355/113.\) Eureka: the Archimedians J. 34, 10–13 (1971)

    Google Scholar 

  29. Delahaye, J.-P.: Le fascinant nombre \(\pi .\) Bibliothèque Scientifique. Belin-Pour la Science, Paris (1997)

    Google Scholar 

  30. Elaydi, S.N.: An Introduction to Difference Equations. Second edition. Undergraduate Texts in Mathematics. Springer, New York (1999)

    Google Scholar 

  31. Erickson, J.: Algorithms, Etc. 2015. http://jeffe.cs.illinois.edu/teaching/algorithms/

  32. Eymard, P., Lafon, J.-P.: The number \(\pi .\) Traduït de la versió francesa de 1999 per S. S. Wilson. American Mathematical Society, Providence, RI (2004)

    Google Scholar 

  33. Feller, W.: An Introduction to Probability Theory and its Applications, vol. I, 3rd edn. John Wiley & Sons Inc, New York, London-Sydney (1968)

    MATH  Google Scholar 

  34. Fulford, G., Forrester, P., Jones, A.: Modelling with Differential and Difference Equations. Australian Mathematical Society Lecture Series, 10. Cambridge University Press, Cambridge (1997)

    Google Scholar 

  35. Gasull, A., Mañosa, V., Xarles, X.: Rational periodic sequences for the Lyness recurrence. Discrete Contin. Dyn. Syst. A 32, 587–604 (2012)

    Article  MathSciNet  Google Scholar 

  36. Gauss, C.F.: Arithmetisch geometrisches Mittel, Werke, Bd. 3 361–403. Königlichen Gesell. Wiss., Göttingen 1876, pp. 361–403

    Google Scholar 

  37. Goldberg, S.: Introduction to Difference Equations, with Illustrative Examples from Economics, Psychology, and Sociology. John Wiley & Sons, Inc., New York; Chapman & Hall, Ltd., London (1958)

    Google Scholar 

  38. Guillera, J.: History of the formulas and algorithms for \(\pi \), Gems in experimental mathematics, Contemp. Math., 517, Amer. Math. Soc., Providence, RI, pp. 173–188 (2010)

    Google Scholar 

  39. Han, M., Li, J.: Lower bounds for the Hilbert number of polynomial systems. J. Differ. Eq. 252, 3278–3304 (2012)

    Article  MathSciNet  Google Scholar 

  40. Havil, J.: Gamma: Exploring Euler’s Constant. Princeton University Press, Princeton, NJ (2003)

    MATH  Google Scholar 

  41. Johnson, T.: A quartic system with twenty-six limit cycles. Exp. Math. 20, 323–328 (2011)

    Article  MathSciNet  Google Scholar 

  42. Johnson, T., Tucker, W.: An improved lower bound on the number of limit cycles bifurcating from a Hamiltonian planar vector field of degree 7. Internat. J. Bifur. Chaos Appl. Sci. Engrg. 20, 1451–1458 (2010)

    Google Scholar 

  43. Kifowit, S.J.: More Proofs of Divergence of the Harmonic Series, Preprint (2006). http://stevekifowit.com/pubs/

  44. Kifowit, S.J., Stamps, T.A.: The harmonic series diverges again and again. AMATYC Rev. 27, 31–43 (2006)

    Google Scholar 

  45. Kifowit, S.J., Stamps, T.A.: Serious About the Harmonic Series II, 31st Annual Conference of the Illinois Mathematics Association of Community Colleges; Monticello, IL (2006). http://stevekifowit.com/pubs/

  46. Kot, M.: Elements of Mathematical Ecology. Cambridge University Press, Cambridge (2001)

    Book  Google Scholar 

  47. Kulenović, M.R.S., Ladas, G.: Dynamics of Second Order Rational Difference Equations: With Open Problems and Conjectures. Chapman & Hall/CRC, Boca Raton, FL (2002)

    MATH  Google Scholar 

  48. Kurshan, R.P., Gopinath, B.: Recursively generated periodic sequences. Canad. J. Math. 26, 1356–1371 (1974)

    Article  MathSciNet  Google Scholar 

  49. Lagarias, J.C.: Euler’s constant: Euler’s work and modern developments. Bul. AMS 50, 527–628 (2013)

    Article  MathSciNet  Google Scholar 

  50. Lagarias, J.C.: The \(3x + 1\) problem and its generalizations. Am. Math. Mont. 92, 3–23 (1985)

    Google Scholar 

  51. Lagarias, J.C.: The Ultimate Challenge: the \(3x+1\) Problem. American Mathematical Society, Providence, R.I. (2010)

    Google Scholar 

  52. Landen, J.: An investigation of a general theorem for finding the length of any arc of any conic hyperbola, by means of two elliptic arcs, with some other new and useful theorems deduced therefrom. Philos. Trans. Royal Soc. London 65, 283–289 (1775)

    Google Scholar 

  53. Lefort, J.: La tour de Hanoï, in French. Dossier Pour la Science, pp. 91–93 (2008)

    Google Scholar 

  54. Lefort, J.: Le baguenaudier et ses variantes in French. Dossier Pour la Science, pp. 94–97 (2008)

    Google Scholar 

  55. Letherman, S., Schleicher, D., Wood, R.: The \((3n + 1)\) -problem and holomorphic dynamics. Exp. Math. 8, 241–252 (1999)

    Google Scholar 

  56. Li, J.: Hilbert’s 16th problem and bifurcations of planar vector fields. Inter. J. Bifur. Chaos 13, 47–106 (2003)

    Article  MathSciNet  Google Scholar 

  57. Li, J., Chan, H., Chung, K.: Some lower bounds for \(H(n)\) in Hilbert’s 16th problem. Qual. Theory Dyn. Syst. 3, 345–360 (2003)

    Google Scholar 

  58. Liang, H., Torregrosa, J.: Parallelization of the Lyapunov constants and cyclicity for centers of planar polynomial vector fields. J. Differ. Eq. 259, 6494–6509 (2015)

    Article  MathSciNet  Google Scholar 

  59. Llibre, J., Piñol, C.: A gravitational approach to the Titius-Bode law. Astron. J. 93, 1272–1279 (1987)

    Article  Google Scholar 

  60. Lucas, S.K.: Integral proofs that \(355/113>\pi .\) Austral. Math. Soc. Gaz. 32, 263–266 (2005)

    Google Scholar 

  61. Lucas, S.K.: Approximations to \(\pi \) derived from integrals with nonnegative integrands. Amer. Math. Monthly 116, 166–172 (2009)

    Google Scholar 

  62. Manna, D., Moll, V.H.: A simple example of a new class of Landen transformations. Amer. Math. Mon. 114, 232–241 (2007)

    Article  MathSciNet  Google Scholar 

  63. Manna, D., Moll, V.H.: Landen survey. Probability, geometry and integrable systems, 287–319, Math. Sci. Res. Inst. Publ., 55, Cambridge Univ. Press, Cambridge (2008)

    Google Scholar 

  64. Mordell, L.J.: The infinity of rational solutions of \(y^2=x^3+k.\) J. London Math. Soc. 41, 523–525 (1966)

    Google Scholar 

  65. Murray, J.D.: Mathematical biology. 2 Edn. Biomathematics, 19. Springer, Berlin (1993)

    Google Scholar 

  66. Nelsen, R.B.: Proofs without Words: Exercises in Visual Thinking, Mathematical Association of America (1997)

    Google Scholar 

  67. Nelsen, R.B.: Proofs without Words II: More Exercises in Visual Thinking, Mathematical Association of America (2000)

    Google Scholar 

  68. Newell, A.C., Shipman, P.D.: Plants and Fibonacci. J. Stat. Phys. 121, 937–968 (2005)

    Google Scholar 

  69. Nield, D.A.: (misspelled Neild in the paper). Rational approximations to pi. New Zealand Math. Mag. 18, 99–100 (1981/82)

    Google Scholar 

  70. Offner, C.D.: Computing the Digits in \(\pi \). Preprint (2015). https://www.cs.umb.edu/~offner/files/pi.pdf

  71. Prohens, R., Torregrosa, J.: New lower bounds for the Hilbert numbers using reversible centers, in preparation (2017)

    Google Scholar 

  72. Puu, T., Panchuk, A. (eds.): Nonlinear Economic Dynamics. Nova Science Publisher, New York (2011)

    Google Scholar 

  73. Ralston, A.: A First Course in Numerical Analysis. McGraw-Hill Book Co., New York, Toronto, London (1965)

    MATH  Google Scholar 

  74. Salamin, E.: Computation of \(\pi \) using arithmetic-geometric mean. Math. Comp. 30, 565–570 (1976)

    Google Scholar 

  75. Silverman, J.H., Tate, J.T.: Rational Points on Elliptic Curves, Undergraduate Texts in Mathematics. Springer, New York (1992)

    Book  Google Scholar 

  76. Stewart, I.: Cooking the classics. Math. Intell. 33, 61–71 (2011)

    Google Scholar 

  77. Stokey, N.L., Jr. Lucas, R.E.: Recursive Methods in Economic Dynamics. With the Collaboration of Edward C. Prescott. Harvard University Press, Cambridge, MA (1989)

    Google Scholar 

  78. Sun, X.B., Han, M.: On the number of limit cycles of a \(Z_4\)-equivariant quintic near-Hamiltonian system. Acta Math. Sin. (Engl. Ser.) 31, 1805–1824 (2015)

    Google Scholar 

  79. Zeeman, E.C.: Geometric unfolding of a difference equation, Hertford College, Oxford (1996). Unpublished paper. A video of the distinguished lecture, with the same title, at PIMS on March 21, 2000, is. http://www.pims.math.ca/resources/multimedia/video. The slides can be obtained at: http://zakuski.utsa.edu/~gokhman/ecz/geometric_unfolding.pdf

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Acknowledgements

The author is partially supported by Spanish Ministry of Economy and Competitiveness through grants MINECO MTM2013-40998-P and MTM2016-77278-P FEDER and by Generalitat de Catalunya through the SGR program.

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  1. Departament de Matemàtiques, Universitat Autònoma de Barcelona, Catalonia, Spain

    Armengol Gasull

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  1. Armengol Gasull
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Correspondence to Armengol Gasull .

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  1. Department of Mathematics, Trinity University, San Antonio, TX, USA

    Saber Elaydi

  2. Institut für Mathematik, Alpen-Adria Universität Klagenfurt, Klagenfurt, Austria

    Christian Pötzsche

  3. Department of Mathematics, West University of Timişoara, Timişoara, Romania

    Adina Luminiţa Sasu

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Gasull, A. (2019). Difference Equations Everywhere: Some Motivating Examples. In: Elaydi, S., Pötzsche, C., Sasu, A. (eds) Difference Equations, Discrete Dynamical Systems and Applications. ICDEA 2017. Springer Proceedings in Mathematics & Statistics, vol 287. Springer, Cham. https://doi.org/10.1007/978-3-030-20016-9_5

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