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.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
V.M. Alekseev and M.V. Yakobson, Phys. Reports 75 (1981) 287.
G. Chaitin, Information, Randomness and Incompleteness (World Scientific, 1990).
B.V. Chirikov, F.M. Izrailev and D.L. Shepelyansky, Physica D 33 (1988) 77.
B.V. Chirikov, Time-Dependent Quantum Systems, Proc. Les Houches Summer School on Chaos and Quantum Physics (Elsevier, 1990).
A. Lichtenberg, M. Lieberman, Regular and Stochastic Motion (Springer, 1983).
G.M. Zaslavsky, Chaos in Dynamic Systems (Harwood, 1985).
B.V. Chirikov, Phys. Reports 52 (1979) 263.
B.V. Chirikov and V.V. Vecheslavov, Astron. Astroph. 221 (1989) 146.
G. Casati et al., Phys. rev. A 36 (1987) 3501.
A.A. Chernikov, R.Z. Sagdeev and G.M. Zaslavsky, Physica D 33 (1988) 65.
B.V. Chirikov, Foundations of Physics 16 (1986) 39.
M. Eisenman et al., Lecture Notes in Physics 38 (1975) 112; J.von Hemmen, ibid., 93 (1979) 232.
B.G. Konopelchenko, Nonlinear Integrable Equations, Lecture Notes in Physics 270 (1987).
B.V. Chirikov and V.V. Vecheslavov, KAM integrability, in: Analysis etc. (Academic Press, 1990) p.219.
V.I. Arnold and A. Avez, Ergodic Problems in Classical Mechanics (Benjamin, 1968).
B.V. Chirikov, Proc. Roy. Soc. Lond. A 413 (1987) 145.
A. Rechester et al. Phys. Rev. A 23 (1981) 2664.
B.V. Chirikov, D.L. Shepelyansky, Radiofizika 29 (1986) 1041.
F. Vivaldi, private communication.
I. Kornfeld, S. Fromin and Ya. Sinai, Ergodic Theory (Springer, 1982).
R. MacKay, Physica d 7 (1983) 283.
B.V. Chirikov and D.L. Shepelyansky, Physica D 13 (1984) 395.
R. Artuso, G. Casati and D.L. Shepelyansky, 1990 (to appear).
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.
S. Channon and J. Lebowitz, Ann. N.Y. Acad. Sci 357 (1980) 108.
G. Paladin and A. Vulpiani, Phys. Reports 156 (1987) 147.
C. Karney. Physica D 8 (1983) 360.
P. Grassberger and I. Procaccia, Physica D 13 (1984) 34.
J. Bene, P. Szépfalusy and A. Fülöp, A generic dynamical phase transition in chaotic Hamiltonian systems, Phys. Rev. Lett. (to appear).
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).
B.V. Chirikov, Intrinsic Stochasticity, Proc. Int. Conf. on Plasma Physics, Lausanne, 1984, Vol. II, p.761.
G. Schmidt and J. Bialek, Physica D 5 (1982) 397.
J. Greene, J.Math.Phys. 9 (1968) 760; 20 (1979) 1183.
M. Feigenbaum, J.Stat.Phys. 19 (1978) 25; 21 (1979) 669.
S. Ostlund et al.,Physica D 8 (1983) 303.
E.M. Lifshits et al., Zh.Eksp.Teor.Fiz. 59 (1970) 322; ibid (Pisma) 38 (1983) 79; J. Barrow, Phys. Reports 85 (1982) 1.
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).
I. Dana et al., Phys.Rev.Lett. 62 (1989) 233.
R. MacKay et al., Physica D 13 (1984) 55.
J. Hanson et al. J.Stat.Phys. 39 (1985) 327.
B.V. Chirikov, Lecture Notes in Physics 179 (1983) 29.
J. Meiss and E. Ott, Phys.Rev.Lett. 55 (1985) 2741; Physica D 20 (1986) 387.
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.
H. Mori et al, Prog.Theor.Phys. Suppl., 1989, No.99, p.1.
J. Greene et al. Physica D 21 (1986) 267.
C. Karney et al, ibid 4 (1982) 425.
Y. Ichikawa et al., ibid 29 (1987) 247.
P. de Gennes, Scaling Concepts in Polymer Physics. (Cornell Univ. Press., 1979).
D. Umberger and D. Farmer, Phys.Rev.Lett. 55 (1985) 661; G. Grebogi et al., Phys.Lett. A 110 (1985) 1.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights 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