Skip to main content

The Birth of Chaos

  • Conference paper
  • First Online:

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 35))

Abstract

The word chaos has become firmly embedded in the literature on dynamical systems. Indeed, James Gleick’s book, Chaos Theory [17], established that term as a description of the entire subject in the public mind. Nonetheless, there is no authoritative technical meaning of “chaos” in dynamical systems. Li and Yorke first used the word in the title of their paper “Period three implies chaos” [31], but it does not appear in the text.

This is a preview of subscription content, log in via an institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   169.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD   219.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

References

  1. Arnol’d, V.I.: Singularity Theory. London Mathematical Society Lecture Note Series, vol. 53. Cambridge University Press, Cambridge (1981). Selected papers, Translated from the Russian, With an introduction by C. T. C. Wall

    Google Scholar 

  2. Arnold, V.I., Afrajmovich, V.S., Il’yashenko, Yu.S., Shil’nikov, L.P.: Bifurcation Theory and Catastrophe Theory. Springer, Berlin (1999). Translated from the 1986 Russian original by N. D. Kazarinoff, Reprint of the 1994 English edition from the series Encyclopaedia of Mathematical Sciences [ıt Dynamical systems. V, Encyclopaedia Math. Sci., 5, Springer, Berlin, 1994; MR1287421 (95c:58058)]

    Google Scholar 

  3. Barrow-Green, J.: Poincaré and the Three Body Problem. History of Mathematics, vol. 11. American Mathematical Society, Providence (1997)

    Google Scholar 

  4. Benedicks, M., Carleson, L.: The dynamics of the Hénon map. Ann. Math. (2) 133(1), 73–169 (1991)

    Google Scholar 

  5. Benoît, É.: Systèmes lents-rapides dans \({\mathbf{R}}^{3}\) et leurs canards. In: Third Schnepfenried Geometry Conference, vol. 2 (Schnepfenried, 1982). Astérisque, vol. 109, pp. 159–191. Soc. Math. France, Paris (1983)

    Google Scholar 

  6. Bold, K., Edwards, C., Guckenheimer, J., Guharay, S., Hoffman, K., Hubbard, J., Oliva, R., Weckesser, W.: The forced van der Pol equation. II. Canards in the reduced system. SIAM J. Appl. Dyn. Syst. 2(4), 570–608 (2003, electronic)

    Google Scholar 

  7. Cartwright, M.L.: Balthazar van der Pol. J. Lond. Math. Soc. 35, 367–376 (1960)

    Article  MathSciNet  MATH  Google Scholar 

  8. Cartwright, M.: Some points in the history of the theory of nonlinear oscillations. Bull. Inst. Math. Appl. 10(9–10), 329–333 (1974)

    MathSciNet  Google Scholar 

  9. Cartwright, M.L., Littlewood, J.E.: On non-linear differential equations of the second order. I. The equation \(\ddot{y} - k(1 - {y}^{2})y + y = b\lambda k\;\mathrm{cos}(\lambda t + a),k\) large. J. Lond. Math. Soc. 20, 180–189 (1945)

    Google Scholar 

  10. Cartwright, M.L., Littlewood, J.E.: On non-linear differential equations of the second order. II. The equation \(\ddot{y} + kf(y)\dot{y} + g(y,k) = p(t) = p_{1}(t) + kp_{2}(t)\); k > 0, f(y) ≧ 1. Ann. Math. (2) 48, 472–494 (1947)

    Google Scholar 

  11. Cartwright, M.L., Littlewood, J.E.: Errata: On non-linear differential equations of the second order. II. Ann. Math. (2) 49, 1010 (1948)

    Google Scholar 

  12. Cartwright, M.L., Littlewood, J.E.: Addendum to ‘On non-linear differential equations of the second order. II’. Ann. Math. (2) 50, 504–505 (1949)

    Google Scholar 

  13. Diener, M.: The canard unchained or how fast/slow dynamical systems bifurcate. Math. Intelligencer 6(3), 38–49 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  14. Dumortier, F., Roussarie, R.: Canard cycles and center manifolds. Mem. Am. Math. Soc. 121(577), x + 100 (1996). With an appendix by Cheng Zhi Li

    Google Scholar 

  15. Eckmann, J.-P., Ruelle, D.: Ergodic theory of chaos and strange attractors. Rev. Mod. Phys. 57(3, part 1), 617–656 (1985)

    Google Scholar 

  16. Fenichel, N.: Geometric singular perturbation theory for ordinary differential equations. J. Differ. Equ. 31(1), 53–98 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  17. Gleick, J.: Making a new science. In: Chaos. Penguin Books, New York (1987).

    MATH  Google Scholar 

  18. Grasman, J.: Asymptotic Methods for Relaxation Oscillations and Applications. Applied Mathematical Sciences, vol. 63. Springer, New York (1987)

    Google Scholar 

  19. Guckenheimer, J., Hoffman, K., Weckesser, W.: Global bifurcations of periodic orbits in the forced van der Pol equation. In: Global Analysis of Dynamical Systems, pp. 261–276. Inst. Phys., Bristol (2001)

    Google Scholar 

  20. Guckenheimer, J., Hoffman, K., Weckesser, W.: The forced van der Pol equation. I. The slow flow and its bifurcations. SIAM J. Appl. Dyn. Syst. 2(1), 1–35 (2003, electronic)

    Google Scholar 

  21. Guckenheimer, J., Wechselberger, M., Young, L.-S.: Chaotic attractors of relaxation oscillators. Nonlinearity 19(3), 701–720 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  22. Haiduc, R: Horseshoes in the forced van der Pol equation. Thesis (Ph.D.)–Cornell University. ProQuest LLC, Ann Arbor (2005)

    Google Scholar 

  23. Haiduc, R.: Horseshoes in the forced van der Pol system. Nonlinearity 22(1), 213–237 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  24. Hénon, M.: A two-dimensional mapping with a strange attractor. Commun. Math. Phys. 50(1), 69–77 (1976)

    Article  MATH  Google Scholar 

  25. Hirsch, M.W., Smale, S., Devaney, R.L.: Differential Equations, Dynamical Systems, and an Introduction to Chaos. Pure and Applied Mathematics (Amsterdam), vol. 60, 2nd edn. Elsevier/Academic, Amsterdam (2004)

    Google Scholar 

  26. Jakobson, M.V.: Absolutely continuous invariant measures for one-parameter families of one-dimensional maps. Commun. Math. Phys. 81(1), 39–88 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  27. Jones, C.K.R.T.: Geometric singular perturbation theory. In: Dynamical Systems (Montecatini Terme, 1994). Lecture Notes in Mathematics, vol. 1609, pp. 44–118. Springer, Berlin (1995)

    Google Scholar 

  28. Krupa, M., Szmolyan, P.: Extending geometric singular perturbation theory to nonhyperbolic points—fold and canard points in two dimensions. SIAM J. Math. Anal. 33(2), 286–314 (2001, electronic)

    Google Scholar 

  29. Levi, M.: Qualitative analysis of the periodically forced relaxation oscillations. Mem. Am. Math. Soc. 32(244), vi + 147 (1981)

    Google Scholar 

  30. Levinson, N.: A second order differential equation with singular solutions. Ann. Math. (2) 50, 127–153 (1949)

    Google Scholar 

  31. Li, T.Y., Yorke, J.A.: Period three implies chaos. Am. Math. Monthly 82(10), 985–992 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  32. Littlewood, J.E.: Errata: On non-linear differential equations of the second order. III. The equation \(\ddot{y} - k\left (1 - {y}^{2}\right )\dot{y} + y = b\mu k\cos (\mu t+\alpha )\) for large k, and its generalizations cos(μ t + α) for large k, and its generalizations. Acta Math. 98(1–4), 110 (1957)

    Google Scholar 

  33. Littlewood, J.E.: On non-linear differential equations of the second order. III. The equation \(\ddot{y} - k(1 - {y}^{2})\dot{y} + y = b\mu k\cos (\mu t+\alpha )\) for large k, and its generalizations. Acta Math. 97, 267–308 (1957)

    Google Scholar 

  34. Littlewood, J.E.: On non-linear differential equations of the second order. IV. The general equation \(\ddot{y} + kf(y)\dot{y} + g(y) = bkp(\phi ),\;\phi = t+\alpha\). Acta Math. 98, 1–110 (1957)

    Google Scholar 

  35. Lorenz, E.N.: Deterministic nonperiodic flow. J. Atmos. Sci. 20, 130–141 (1963)

    Article  Google Scholar 

  36. Mañé, R.: Ergodic Theory and Differentiable Dynamics. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 8. Springer, Berlin (1987). Translated from the Portuguese by Silvio Levy

    Google Scholar 

  37. McMurran, S.L., Tattersall, J.J.: The mathematical collaboration of M. L. Cartwright and J. E. Littlewood. Am. Math. Monthly 103(10), 833–845 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  38. McMurran, S.L., Tattersall, J.J.: Cartwright and Littlewood on van der Pol’s equation. In: Harmonic Analysis and Nonlinear Differential Equations (Riverside, CA, 1995). Contemp. Math., vol. 208, pp. 265–276. Amer. Math. Soc., Providence (1997)

    Google Scholar 

  39. Mishchenko, E.F., Kolesov, Yu.S., Kolesov, A.Yu., Rozov, N.Kh.: Asymptotic Methods in Singularly Perturbed Systems. Monographs in Contemporary Mathematics. Consultants Bureau, New York (1994). Translated from the Russian by Irene Aleksanova

    Book  MATH  Google Scholar 

  40. Mishchenko, E.F., Rozov, N.Kh.: Differential Equations with Small Parameters and Relaxation Oscillations. Mathematical Concepts and Methods in Science and Engineering, vol. 13. Plenum Press, New York (1980). Translated from the Russian by F. M. C. Goodspeed

    Google Scholar 

  41. Mora, L., Viana, M.: Abundance of strange attractors. Acta Math. 171(1), 1–71 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  42. Plykin, R.V.: Sources and sinks of A-diffeomorphisms of surfaces. Mat. Sb. (N.S.) 94(136), 243–264, 336 (1974)

    Google Scholar 

  43. Poincaré, H.: Sur les probelème des trois corps et les équations de la dynamique. Acta Math. 13, 1–270 (1890)

    MATH  Google Scholar 

  44. Ruelle, D.: Ergodic theory of differentiable dynamical systems. Inst. Hautes Études Sci. Publ. Math. (50), 27–58 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  45. Ruelle, D., Takens, F.: On the nature of turbulence. Commun. Math. Phys. 20, 167–192 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  46. Smale, S.: Differentiable dynamical systems. Bull. Am. Math. Soc. 73, 747–817 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  47. Smale, S.: Diffeomorphisms with many periodic points. In: Differential and Combinatorial Topology (A Symposium in Honor of Marston Morse), pp. 63–80. Princeton University Press, Princeton (1965)

    Google Scholar 

  48. Smale, S.: Essays on dynamical systems, economic processes, and related topics. In: The Mathematics of Time. Springer, New York (1980).

    MATH  Google Scholar 

  49. Smale, S.: Finding a horseshoe on the beaches of Rio. Math. Intelligencer 20(1), 39–44 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  50. Tattersall, J., McMurran, S.: An interview with Dame Mary L. Cartwright, D.B.E., F.R.S. College Math. J. 32(4), 242–254 (2001)

    Google Scholar 

  51. Van der Pol, B.: A theory of the amplitude of free and forced triode vibrations. Radio Rev. 1, 701–710 (1920)

    Google Scholar 

  52. Van der Pol, B.: On “relaxation oscillations”. I. Phil. Mag. 2, 978–992 (1926)

    Google Scholar 

  53. Van der Pol, B., Van der Mark, J.: Frequency demultiplication. Nature 120, 363–364 (1927)

    Google Scholar 

  54. Wang, Q., Young, L.-S.: Nonuniformly expanding 1D maps. Commun. Math. Phys. 264(1), 255–282 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  55. Wang, Q., Young, L.-S.: Toward a theory of rank one attractors. Ann. Math. (2) 167(2), 349–480 (2008)

    Google Scholar 

  56. Wilczak, D.: Uniformly hyperbolic attractor of the Smale-Williams type for a Poincaré map in the Kuznetsov system. SIAM J. Appl. Dyn. Syst. 9(4), 1263–1283 (2010). With online multimedia enhancements.

    Google Scholar 

  57. Yorke, J., Sauer, T.: Chaos. Scholarpedia. http://www.scholarpedia.org/article/Chaos

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to John Guckenheimer .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2013 Springer Basel

About this paper

Cite this paper

Guckenheimer, J. (2013). The Birth of Chaos. In: Johann, A., Kruse, HP., Rupp, F., Schmitz, S. (eds) Recent Trends in Dynamical Systems. Springer Proceedings in Mathematics & Statistics, vol 35. Springer, Basel. https://doi.org/10.1007/978-3-0348-0451-6_1

Download citation

Publish with us

Policies and ethics