Strange Attractors

  • R. V. Plykin
  • E. A. Sataev
  • S. V. Shlyachkov
Part of the Encyclopaedia of Mathematical Sciences book series (EMS, volume 66)

Abstract

The term “strange attractor” was introduced in (Ruelle and Tokens 1971) to denote a limit set of a dynamical system that is not a manifold and is consequently not a fixed point, limit cycle, invariant torus and so on. It immediately gained wide prevalence among physicists. Nowadays the notion of “strange attractor” has more the nature of a paradigm than a rigorously defined mathematical object. A dynamical system is considered to have a strange attractor if the phase space of the system has a limit set consisting of trajectories with chaotic behaviour. With regard to chaos, there are a number of “physical” criteria for chaotic behaviour. Among these we can single out the presence of homoclinics, the continuity of the spectrum, the presence of a positive Lyapunov exponent, the fractionality of some dimension, the presence of an infinite series of bifurcations, and period doubling. There are an enormous nimber of works in which for a given physical system one of the above criteria or some other criterion is used to deduce the presence of a strange attractor in the system. An extensive bibliography of works of this kind can be found in the book (Neimark and Landa 1987).

Keywords

Entropy Convection Manifold Dition Dinates 

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References

  1. Afraimovich, V.S., Bykov, V.V., Shil’nikov, L.P. (1977): The origin and structure of the Lorenz attractor. Dokl. Akad. Nauk SSSR 234, 336–339. [English transi.: Sov. Phys., Dokl. 22, 253–255 (1977)] Zbl. 451. 76052Google Scholar
  2. Afraimovich, V.S., Bykov, V.V., Shil’nikov, L.P. (1982): On structurally unstable attracting limit sets of Lorenz attractor type. Tr. Mosk. Mat. 0.-va44, 150–212. [English transi.: Trans. Mosc. Math. Soc. 1983, No. 2, 153–216 (1983)] Zbl. 506. 58023Google Scholar
  3. Afraimovich, V.S., Pesin, Ya.B. (1987): Dimension of Lorenz type attractors. Sov. Sci. Rev., Sect. C, Math. Phys. Rev. 6, 169–241. Zbl. 628. 58031Google Scholar
  4. Alekseev, V.M. (1976): Symbolic Dynamics (Eleventh Mathematical School). Akad. Nauk Ukr. SSR, Kiev (Russian )Google Scholar
  5. Anosov, D.V. (1967): Geodesic flows on closed Riemannian manifolds of negative curvature. Tr. Mat. Inst. Steklova 90 (209 pp.) [English Transi.: Proc. Steklov Inst. Math. 90 (1969)] Zbl 163, 436Google Scholar
  6. Anosov, D.V., Aranson S. Kh., Bronshtein, I.U., Grines, V.Z. (1985): Dynamical Systems I: Smooth Dynamical Systems. Itogi Nauld Tekh., Ser. Sovrem. Probl. Mat., Fundam. Napravleniya 1, 151–242. [English transi. in: Encycl. Math. Sci. 1, 149–230. Springer-Verlag, Berlin Heidelberg New York 1988] Zbl. 605. 58001Google Scholar
  7. Aoki, N. (1982): Homeomorphisms without the pseudo-orbit tracing property. Nagoya Math. J. 88, 155–160. Zbl. 466. 54034Google Scholar
  8. Aoki, N. (1983): Homeomorphism with the pseudo-orbit tracing property. Tokyo J. Math. 6, No. 2, 329–334. Zbl. 537. 58034Google Scholar
  9. Arnol’d, V.I., Afraimovich, V.S., Il’yashenko, Yu.S., Shil’nikov, L.P. (1986): Dynamical Systems V: Bifurcation Theory. Itogi Nauki Tekh., Ser. Sovrem. Probl. Mat., Fundam. Napravleniya 5, 5–218. [English transi. in: Encycl. Math. Sci. 5. Springer-Verlag, Berlin Heidelberg New York 1993 ]Google Scholar
  10. Barge, M. (1986): Horseshoe maps and inverse limits. Pac. J. Math. 121, No. 1, 29–39. Zbl. 601. 58049Google Scholar
  11. Belykh, V.P. (1982): Models of discrete systems of phase synchronization. Chap. 10 in Shakhgil’dyan, V.V., Belyustinaya, L.N. (eds.): Radio and Communication, Moscow, 161–176 (Russian)Google Scholar
  12. Bowen R. (1979): Methods of Symbolic Dynamics Mir, Moscow (Russian) Brindley, J., Moroz, I.M. (1980): Lorenz attractor behavior in a continuously stratified baroclinic fluid. Phys. Lett. A77, No. 6, 441–444Google Scholar
  13. Bunimovich, L. (1983): Statistical properties of Lorenz attractors, in Barenblatt, G.I. (ed.): Nonlinear Dynamics and Turbulence. Pitman, Boston, 71–92. Zbl. 578. 58025Google Scholar
  14. Bunimovich, L.A., Sinai, Ya.G. (1980): Stochasticity of an attractor in the Lorenz model. In Gaponov-Grekhov, A.V. (ed.): Non-linear Waves, Nauka, Moscow, 212226 (Russian)Google Scholar
  15. Collet, P., Levy, Y. (1984): Ergodic properties of the Lozi mappings. Commun. Math. Phys. 93, No. 4, 461–482. Zbl. 553. 58019Google Scholar
  16. Dateyama, M. (1989): Homeomorphisms with Markov partitions. Osaka J. Math. 26, No. 2, 411–428. Zbl. 711. 58026Google Scholar
  17. Fedotov, A.G. (1980): On Williams solenoids and their realization in two-dimensional dynamical systems. Dokl. Akad. Nauk SSSR 252, 801–804. [English transi- Soy. Math., Dokl. 21, 835–839 (1980)] Zbl. 489. 58019Google Scholar
  18. Franke, J.E., Selgrade, J.F. (1976): Abstract co-limit sets, chain recurrent sets, and basic sets for flows. Proc. Am. Math. Soc. 60, 309–316. Zbl. 316. 58014Google Scholar
  19. Franke, J.E., Selgrade, J.F. (1977): Hyperbolicity and chain recurrence. J. Differ. Equations 26, No. 1, 27–36. Zbl. 329. 58012Google Scholar
  20. Franke, J.E., Selgrade, J.F. (1978): Hyperbolicity and cycles. Trans. Am. Math. Soc. 245, 251–262. Zbl. 396. 58023Google Scholar
  21. Gibbon, J.D., McGuines, M.J. (1980): A derivation of the Lorenz equation for some unstable dispersive physical systems. Phys. Lett. A77, No. 5, 295–299MathSciNetCrossRefGoogle Scholar
  22. Guckenheimer, J., Williams, R.F. (1979): Structural stability of Lorenz attractors.Google Scholar
  23. Publ. Math., Inst. Hautes Etud. Sci. 50, 59–72. Zbl. 436. 58018Google Scholar
  24. Haken, H. (1975): Analogy between higher instabilities in fluids and lasers. Phys. Lett. 53A, No. 1, 77–79MathSciNetCrossRefGoogle Scholar
  25. Hiraide, K. (1985): On homeomorphisms with Markov partitions. Tokyo J. Math. 8, No. 1, 219–229. Zbl. 639. 54030Google Scholar
  26. Hiraide, K. (1987a): Manifolds which do not admit expansive homeomorphisms with the pseudo-orbit tracing property. The theory of dynamical systems and its applications to nonlinear problems. Singapore World Sci. Publ., 32–34Google Scholar
  27. Hiraide, K. (1987b): Expansive homeomorphisms of compact surfaces are pseudoAnosov. Proc. Japan Acad., Ser. A. 63, No. 9, 337–338. Zbl. 663.58024; Osaka J. Math. 27, No. 1 (1990), 117–162. Zbl. 713. 58042Google Scholar
  28. Hiraide, K. (1988): Expansive homeomorphisms with the pseudo-orbit tracing property on compact surfaces. J. Math. Soc. Japan 40, No. 1, 123–137. Zbl. 666. 54023Google Scholar
  29. Hiraide, K. (1989): Expansive homeomorphisms with the pseudo-orbit tracing property of n-tori. J. Math. Soc. Japan 41, No. 3, 357–389. See also World Sci. Adv. Ser. Dyn. Syst. 5, 35–41 (1987); Zbl. 678. 54029Google Scholar
  30. Hirsch, M., Pugh., C.C., Shub, M. (1977): Invariant Manifolds. Lect. Notes Math. 583. Zbl. 355. 58009Google Scholar
  31. Hurley, M. (1984): Consequences of topological stability. J. Differ. Equations 54, No. 1, 60–72. Zbl. 493. 58013Google Scholar
  32. Hurley, M. (1986): Fixed points of topologically stable flows. Trans. Am. Math. Soc. 294, No. 4, 625–633. Zbl. 594. 58030Google Scholar
  33. Kamaev, D.A. (1977): On the topological indecomposability of certain invariant sets of A-diffeomorphisms. Usp. Mat. Nauk 32, No. 1, 158Google Scholar
  34. Kato. K. (1984): Pseudo-orbits and stabilities of flows, I, II. Mem. Fac. Sci., Kochi Univ., Ser. A 5, 45–62. Zbl. 536.58017; 6 (1985), 33–43. Zbl. 571. 58013Google Scholar
  35. Kato, K., Morimoto, A. (1979): Topological (1-stability of Axiom A flows with no 0-explosions. J. Differ. Equations 34, No. 3, 464–481. Zbl. 406. 58025Google Scholar
  36. Katok, A.B., Strelcyn, J.M. (1986): Invariant Manifolds, Entropy and Billiards; Smooth Maps with Singularities. Lect. Notes Math. 1222. Zbl. 658. 58001Google Scholar
  37. Klinshpont, N.Eh. (1989): Instability of the cellular structure of Lorenz attractors. In Plykin, R.V. (ed.): Klinshpont, N.Eh, 11–30 ( Russian )Google Scholar
  38. Knobloch, F. (1981): Chaos in a segmented disc dynamo Phys. Lett. A82, No. 9, 439–440MathSciNetCrossRefGoogle Scholar
  39. Komuro, M. (1984): One parameter flows with the pseudo-orbit tracing property. Monatsh. Math. 98, No. 3, 219–253. Zbl. 545. 58037Google Scholar
  40. Komuro, M. (1985): Lorenz attractors do not have the pseudo-orbit tracing property. J. Math. Soc. Japan 37, No. 3, 489–514. Zbl. 552. 58020Google Scholar
  41. Lewowicz, J. (1989): Expansive homeomorphisms of surfaces. Bol. Soc. Bras. Mat., Nova Ser. 20, No. 1, 113–133. Zbl. 753. 58022Google Scholar
  42. Lorenz, E.N. (1963): Deterministic nonperiodic flow. J. Atmosph. Sci. 20, No. 2, 130–141MathSciNetCrossRefGoogle Scholar
  43. Lorenz, E.N. (1979): On the prevalence of aperiodicity in simple systems. In: Global Analysis. Proc. Semin. Calgary 1978, Lect. Notes Math. 755, 53–75. Zbl. 438. 34038Google Scholar
  44. Lozi, R. (1978): Un attracteur étrange du type de Hénon. J. Phys. C. 39, No. 5, 9–10Google Scholar
  45. McLaughlin, J.B., Martin, P.C. (1974): Transition to turbulence of statically stressed fluids. Phys. Rev. Lett. 33, No. 20, 1189–1192CrossRefGoogle Scholar
  46. McLaughlin, J.B., Martin, P.C. (1975): Transition to turbulence in statically stressed fluid systems. Phys. Rev. A 12, No. 1, 186–203CrossRefGoogle Scholar
  47. Malkin, M.I. (1982): On the continuity of the entropy of discontinuous maps of the interval. In Leontovich-Andronova, E.A. (ed.): Metody Kachestvennoi Teorii Differents. Uravnenij: Mezhvuz. Temat. Sb. Nauchen. Tr., Gor’kij Gos. Univ., 1982, 35–47. [English transl.: Sel. Math. Sov. 8, No. 2, 131–139 (1989)] Zbl. 572. 54035Google Scholar
  48. Milnor, J. (1985): On the concept of attractor. Commun. Math. Phys. 99, No. 2, 177–195. Zbl. 595. 58028Google Scholar
  49. Morimoto, A. (1981): Some stabilities of group automorphisms. In: Manifolds and Lie groups, Prog. Math. 14, 283–299. Zbl. 483. 58007Google Scholar
  50. Neimark, Yu.I., Landa, P.S. (1987): Stochastic and Chaotic Oscillations. Nauka, Moscow. [English transl.: Kluwer, Dordrecht 1992] Zbl. 644. 58013Google Scholar
  51. Nitecki, Z. (1971): On semistability for diffeomorphisms. Invent. Math. 14, No. 2, 83–122. Zbl. 218. 58007Google Scholar
  52. Nusse, H.E., Yorke, J.A. (1988): Is every approximate trajectory of some process near an exact trajectory of a nearby process ? Commun. Math. Phys. 114, No. 3, 363–379. Zbl. 697. 58036Google Scholar
  53. Oka, M. (1990): Expansiveness of real flows. Tsukuba J. Math. 14, No. 1, 1–8. Zbl. 733. 54030Google Scholar
  54. O’Brien, T., Reddy, W. (1970): Each compact orientable surface of positive genus admits an expansive homeomorphism. Pac. J. Math. 35, No. 3, 737–742. Zbl. 206, 524Google Scholar
  55. Pedlosky, J., Frenzen, C. (1980): Chaotic and periodic behavior of finite amplitude baroclinic waves. J. Atmosph. Sci. 37, No. 6, 1177–1196MathSciNetCrossRefGoogle Scholar
  56. Pesin, Ya.B. (1977): Characteristic Lyapunov exponents and smooth ergodic theory. Usp. Mat. Nauk 32, No. 4, 55–111. [English transl.: Russ. Math. Surv. 32, No. 4, 55–114 (1977)] Zbl. 359. 58010Google Scholar
  57. Pesin, Ya.B. (1988): Dimension type characteristics for invariant sets of dynamical systems. Usp. Mat. Nauk 43, No. 4, 95–128. [English transl.: Russ. Math. Surv. 43, No. 4, 111–151 (1988)] Zbl. 684. 58024Google Scholar
  58. Plykin, R.V. (1974): Sources and sinks of A-diffeomorphisms of surfaces. Mat. Sb., Nov. Ser. 94, 243–264. [English transl.: Math. USSR, Sb. 23, 233–253 (1975)] Zbl. 324. 58013Google Scholar
  59. Plykin, R.V. (1984): On the geometry of hyperbolic attractors of smooth cascades. Usp. Mat. Nauk 39, No. 6, 75–113. [English transl.: Russ. Math. Surv. 39, No. 6, 85–131 (1984)] Zbl. 584. 58038Google Scholar
  60. Plykin, R.V. (1990): Some problems of attractors of differentiable dynamical systems. Int. Conf. Topology, Abstracts, Math. Inst., VarnaGoogle Scholar
  61. Rand, D. (1978): The topological classification of Lorenz attractors. Math. Proc. Camb Philos. Soc. 83, No. 3, 451–460. Zbl. 375. 58015Google Scholar
  62. Robbins, K.A. (1977): A new approach to subcritical instability and turbulent tran-Google Scholar
  63. sition in a simple dynamo. Math. Proc. Camb. Philos. Soc. 82, No. 2, 309–325 Robinson, C. (1989): Homoclinic bifurcation to a transitive attractor of Lorenz type.Google Scholar
  64. Nonlinearity 2,No. 4, 495–518. Zbl. 704.58031Google Scholar
  65. Ruelle, D. (1976): The Lorenz attractor and the problem of turbulence. In Temam, R. (ed.): Turbulence and Navier Stokes Equations. Lect. Notes Math. 565, 146158. Zbl. 355. 76036Google Scholar
  66. Ruelle, D., Tokens, F. (1971): On the nature of turbulence. Commun. Math. Phys. 20, No. 3, 167–192. Zbl. 223. 76041Google Scholar
  67. Rychlik, M. (1990): Lorenz attractors through Shil’nikov type bifurcation. I. Ergodic Theory Dyn. Syst. 10, No. 4, 793–821. Zbl. 715. 58027Google Scholar
  68. Sakai, K. (1987): Anosov maps on closed topological manifolds. J. Math. Soc. Japan 39, No. 3, 505–519. Zbl. 647. 58039Google Scholar
  69. Saltzman, B. (1962): Finite amplitude free convection as an initial value problem. I. J. Atmosph. Sci. 19, No. 2, 329–341CrossRefGoogle Scholar
  70. Sataev, E.A. (1992): Invariant measures for hyperbolic maps with singularities. Usp. Mat. Nauk 47, No. 1, 147–202. [English transl.: Russ. Math. Surv. 47, No. 1, 191251 (1992)]Google Scholar
  71. Shil’nikov, A.L. (1986): Bifurcations and chaos in the Marioka-Shimitsu model. In: Metody Kachestvennoi Teorii Differents. Uravnenij: Mezhvuz. Temat. Sb. Nauchen. Tr., Gor’kij Gos. Univ., 180–193 [English transl.: Sel. Math. Sov. 10, No. 2, 105–117 (1991)] Zbl. 745.58036; II. Metody Kachestvennoi Teorii i Teorii Bifurkatsii: Mezhvuz. Temat. Sb. Nauchen. Tr., Gor’kij Gos. Univ. 1989, 130–138Google Scholar
  72. Shil’nikov, L.P. (1963): On certain cases of the generation of periodic motions from singular trajectories. Mat. Sb., Nov. Ser. 61, 433–466. Zbl. 121, 75Google Scholar
  73. Shil’nikov, L.P. (1981): Theory of bifurcations and quasihyperbolic attractors. Usp. Mat. Nauk 36, No. 4, 240–241Google Scholar
  74. Shimomura, T. (1987): Topological entropy and the pseudo-orbit tracing property. In: The Theory of Dynamical Systems and its Applications to Nonlinear Problems. Singapore World Sci. Publ., 35–41Google Scholar
  75. Shlyachkov, S.V. (1985): A theorem on E-trajectories for Lorenz mappings. Funkts. Anal. Prilozh. 19, No. 3, 84–85. [English transl.: Funct. Anal. Appl. 19, 236–238 (1985)] Zbl. 605. 58029Google Scholar
  76. Shlyachkov, S.V. (1988a): Structurally stable families of discontinuous maps of the interval into itself. OIATEh, Obninsk; Dep. VINITI, No. 4113-B88 (Russian)Google Scholar
  77. Shlyachkov, S.V. (1988b): On the entropy of piecewise-monotone expanding maps of the interval. In Leontovich-Andronova, E.A. (ed.): Metody Kachestvennoi Teorii Differents. Uravnenij: Mezhvuz. Temat. Sb. Nauchen. Tr., Gor’kij Gos. Univ., 91–97Google Scholar
  78. Sinai, Ya.G. (1972): Gibbs measures in ergodic theory. Usp. Mat. Nauk 27, No. 4,21–64. [English transl.: Russ. Math. Surv. 27, No. 4, 21–69 (1972)] Zbl. 246.28008 Sinai, Ya.G., Shil’nikov, L.P. (eds.) ( 1981 ): Strange Attractors. Mir, Moscow (Rus-sian)Google Scholar
  79. Sinai. Ya.G., Vul, E.B. (1981): Hyperbolicity conditions for the Lorenz model. Physica D2, No. 1, 3–7MathSciNetMATHGoogle Scholar
  80. Smale, S. (1967): Differentiable dynamical systems. Bull. Am. Math. Soc. 73, No. 6, 747–817. Zbl. 202, 552Google Scholar
  81. Sparrow, C. (1982): The Lorenz equations: Bifurcations, chaos and strange at-tractors. Appl. Math. Sci. 41. Springer-Verlag, Berlin Heidelberg New York. Zbl. 504. 58001Google Scholar
  82. Thomas, R.F. (1982): Stability properties of one-parameter flows. Proc. Lond. Math. Soc., III Ser. 45, No. 3, 479–505. Zbl. 449. 28019Google Scholar
  83. Thomas, R.F. (1987): Entropy of expansive flows. Ergodic Theory Dyn. Syst. 7, No. 4, 611–625. Zbl. 612. 28015Google Scholar
  84. Ustinov, Yu.I. (1987): Algebraic invariants of topological conjugacy of solenoids. Mat. Zametki. 42, No. 1, 132–144. [English transl.: Math. Notes 42, 583–590 (1987)] Zbl. 713. 58043Google Scholar
  85. Walters, P. (1978): On the pseudo-orbit tracing property and its relationship to stability. The structure of attractors in dynamical systems, Proc. North Dakota 1977, Lect. Notes Math. 668, 231–244. Zbl. 403. 58019Google Scholar
  86. Welander, P. (1967): On the oscillatory instability of a differentially heated fluid loop. J. Fluid. Mech. 29, No. 1, 17–30. Zbl. 163, 207Google Scholar
  87. Williams, R.F. (1967): One-dimensional non-wandering sets. Topology 6, No. 4, 473–487. Zbl. 159, 537Google Scholar
  88. Williams, R.F. (1970a): Classification of one dimensional attractors. Global Analysis. Proc. Symp. Pure Math./4, 341–361. Zbl. 213, 504Google Scholar
  89. Williams, R.F. (1970b): The DA-maps of Smale and structural stability. Global Analysis. Proc. Symp. Pure Math. 14, 329–334. Zbl. 213, 503Google Scholar
  90. Williams, R.F. (1974): Expanding attractors. Publ. Math., Inst. Hautes Etud. Sci. 43, 169–204. Zbl. 279. 58013Google Scholar
  91. Williams, R.F. (1977): The structure of Lorenz attractors. Turbulence Seminar, Berkeley 1976/77. Lect. Notes. Math. 615, 94–116. Zbl. 363. 58005Google Scholar
  92. Williams, R.F. (1979): The structure of Lorenz attractors. Publ. Math., Inst. Hautes. Etud. Sci. 50, 73–100. Zbl. 484. 58021Google Scholar
  93. Yorke, J., Yorke, E. (1979): Metastable chaos: the transition to sustained chaotic behaviour in the Lorenz model. J. Stat. Phys. 21, No. 3, 263–278MathSciNetCrossRefGoogle Scholar
  94. Young, L.-S. (1985): Bowen-Ruelle measures for certain piecewise hyperbolic maps. Trans. Am. Math. Soc. 287, No. 1, 41–48. Zbl. 552. 58022Google Scholar
  95. Yuri, M. (1983): A construction of an invariant stable foliation by the shadowing lemma. Tokyo J. Math. 6, No. 2, 291–296. Zbl. 541. 58041Google Scholar
  96. Zhirov, A.Yu. (1989): Matrices of intersections of one-dimensional hyperbolic at-tractors on a two-dimensional sphere. Mat. Zametki 45, No. 6, 44–56. English transl Math. Notes 45, No. 6, 461–469 (1989)1 Zbl. 687. 58019Google Scholar

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Authors and Affiliations

  • R. V. Plykin
  • E. A. Sataev
  • S. V. Shlyachkov

There are no affiliations available

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