Skip to main content

Patterns In Chaos

  • Chapter
Chaos, Order, and Patterns

Part of the book series: NATO ASI Series ((NSSB,volume 280))

Abstract

Classification of chaotic patterns in classical Hamiltonian systems is given as a series of levels with increasing disorder. Overview of critical phenomena in Hamiltonian dynamics is presented, including the renormalization chaos, based upon the fairly simple resonant theory. First estimates for the critical structure and related statistical anomalies in arbitrary dimensions are discussed.

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

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight 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

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. V.M. Alekseev and M.V. Yakobson, Phys. Reports 75 (1981) 287.

    Article  MathSciNet  ADS  Google Scholar 

  2. G. Chaitin, Information, Randomness and Incompleteness (World Scientific, 1990).

    Google Scholar 

  3. B.V. Chirikov, F.M. Izrailev and D.L. Shepelyansky, Physica D 33 (1988) 77.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  4. B.V. Chirikov, Time-Dependent Quantum Systems, Proc. Les Houches Summer School on Chaos and Quantum Physics (Elsevier, 1990).

    Google Scholar 

  5. A. Lichtenberg, M. Lieberman, Regular and Stochastic Motion (Springer, 1983).

    Google Scholar 

  6. G.M. Zaslavsky, Chaos in Dynamic Systems (Harwood, 1985).

    Google Scholar 

  7. B.V. Chirikov, Phys. Reports 52 (1979) 263.

    Article  MathSciNet  ADS  Google Scholar 

  8. B.V. Chirikov and V.V. Vecheslavov, Astron. Astroph. 221 (1989) 146.

    ADS  Google Scholar 

  9. G. Casati et al., Phys. rev. A 36 (1987) 3501.

    Article  ADS  Google Scholar 

  10. A.A. Chernikov, R.Z. Sagdeev and G.M. Zaslavsky, Physica D 33 (1988) 65.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  11. B.V. Chirikov, Foundations of Physics 16 (1986) 39.

    Article  MathSciNet  ADS  Google Scholar 

  12. M. Eisenman et al., Lecture Notes in Physics 38 (1975) 112; J.von Hemmen, ibid., 93 (1979) 232.

    Article  ADS  Google Scholar 

  13. B.G. Konopelchenko, Nonlinear Integrable Equations, Lecture Notes in Physics 270 (1987).

    Google Scholar 

  14. B.V. Chirikov and V.V. Vecheslavov, KAM integrability, in: Analysis etc. (Academic Press, 1990) p.219.

    Google Scholar 

  15. V.I. Arnold and A. Avez, Ergodic Problems in Classical Mechanics (Benjamin, 1968).

    Google Scholar 

  16. B.V. Chirikov, Proc. Roy. Soc. Lond. A 413 (1987) 145.

    Article  MathSciNet  ADS  Google Scholar 

  17. A. Rechester et al. Phys. Rev. A 23 (1981) 2664.

    Article  MathSciNet  ADS  Google Scholar 

  18. B.V. Chirikov, D.L. Shepelyansky, Radiofizika 29 (1986) 1041.

    ADS  Google Scholar 

  19. F. Vivaldi, private communication.

    Google Scholar 

  20. I. Kornfeld, S. Fromin and Ya. Sinai, Ergodic Theory (Springer, 1982).

    Google Scholar 

  21. R. MacKay, Physica d 7 (1983) 283.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  22. B.V. Chirikov and D.L. Shepelyansky, Physica D 13 (1984) 395.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  23. R. Artuso, G. Casati and D.L. Shepelyansky, 1990 (to appear).

    Google Scholar 

  24. B.V. Chirikov, D.L. Shepelyansky, Proc. 9th Int. Conf. on Nonlinear Oscillations, Kiev, 1981, Vol.2, p.421. (Kiev, Naukova Dumka, 1983) English translation available as preprint PPL-TRANS-133, Plasma Physics Lab., Princeton Univ., 1983.

    MathSciNet  Google Scholar 

  25. S. Channon and J. Lebowitz, Ann. N.Y. Acad. Sci 357 (1980) 108.

    Article  ADS  Google Scholar 

  26. G. Paladin and A. Vulpiani, Phys. Reports 156 (1987) 147.

    Article  MathSciNet  ADS  Google Scholar 

  27. C. Karney. Physica D 8 (1983) 360.

    Article  MathSciNet  ADS  Google Scholar 

  28. P. Grassberger and I. Procaccia, Physica D 13 (1984) 34.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  29. J. Bene, P. Szépfalusy and A. Fülöp, A generic dynamical phase transition in chaotic Hamiltonian systems, Phys. Rev. Lett. (to appear).

    Google Scholar 

  30. B.V. Chirikov and D.L. Shepelyansky, Chaos Border and Statistical Anomalies, in: Renormalization Group, D.V. Shirkov, D.I. Kazakov and A.A. Vladimirov (eds.), p. 221. (World Scientific, Singapore, 1988).

    Google Scholar 

  31. B.V. Chirikov, Intrinsic Stochasticity, Proc. Int. Conf. on Plasma Physics, Lausanne, 1984, Vol. II, p.761.

    Google Scholar 

  32. G. Schmidt and J. Bialek, Physica D 5 (1982) 397.

    Article  MathSciNet  ADS  Google Scholar 

  33. J. Greene, J.Math.Phys. 9 (1968) 760; 20 (1979) 1183.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  34. M. Feigenbaum, J.Stat.Phys. 19 (1978) 25; 21 (1979) 669.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  35. S. Ostlund et al.,Physica D 8 (1983) 303.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  36. E.M. Lifshits et al., Zh.Eksp.Teor.Fiz. 59 (1970) 322; ibid (Pisma) 38 (1983) 79; J. Barrow, Phys. Reports 85 (1982) 1.

    ADS  Google Scholar 

  37. B.V. Chirikov, The Nature and Properties of the Dynamic Chaos, Proc. 2d Int.Seminar “Group Theory Methods in Physics” (Zveingorod, 1982), Vol.1, p.553. (Harwood, 1985).

    Google Scholar 

  38. I. Dana et al., Phys.Rev.Lett. 62 (1989) 233.

    Article  MathSciNet  ADS  Google Scholar 

  39. R. MacKay et al., Physica D 13 (1984) 55.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  40. J. Hanson et al. J.Stat.Phys. 39 (1985) 327.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  41. B.V. Chirikov, Lecture Notes in Physics 179 (1983) 29.

    Article  MathSciNet  ADS  Google Scholar 

  42. J. Meiss and E. Ott, Phys.Rev.Lett. 55 (1985) 2741; Physica D 20 (1986) 387.

    Article  ADS  Google Scholar 

  43. P. Lévy, Théorie de l’addition des variables eléatoires. (Gauthier-Villiers, Paris, 1937); T. Geisel et al. Phys.Rev.Lett. 54 (1985) 616; R. Pasmanter, Fluid Dynamic Research 3 (1988) 320; R. Voss, Physica D 38 (1989) 362; G.M. Zaslavsky et al., Zh.Exper.Teor.Fiz. 96 (1989) 1563.

    Article  Google Scholar 

  44. H. Mori et al, Prog.Theor.Phys. Suppl., 1989, No.99, p.1.

    Article  Google Scholar 

  45. J. Greene et al. Physica D 21 (1986) 267.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  46. C. Karney et al, ibid 4 (1982) 425.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  47. Y. Ichikawa et al., ibid 29 (1987) 247.

    Article  ADS  Google Scholar 

  48. P. de Gennes, Scaling Concepts in Polymer Physics. (Cornell Univ. Press., 1979).

    Google Scholar 

  49. D. Umberger and D. Farmer, Phys.Rev.Lett. 55 (1985) 661; G. Grebogi et al., Phys.Lett. A 110 (1985) 1.

    Article  MathSciNet  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 1991 Plenum Press, New York

About this chapter

Cite this chapter

Chirikov, B.V. (1991). Patterns In Chaos. In: Artuso, R., Cvitanović, P., Casati, G. (eds) Chaos, Order, and Patterns. NATO ASI Series, vol 280. Springer, Boston, MA. https://doi.org/10.1007/978-1-4757-0172-2_5

Download citation

  • DOI: https://doi.org/10.1007/978-1-4757-0172-2_5

  • Publisher Name: Springer, Boston, MA

  • Print ISBN: 978-1-4757-0174-6

  • Online ISBN: 978-1-4757-0172-2

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics