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Instability Hierarchies of Self-Organization Systems and Devices

  • Chapter
Synergetics

Abstract

This text on the interdisciplinary field of synergetics will be of interest to students and scientists in physics, chemistry, mathematics, biology, electrical, civil and mechanical engineering, and other fields. It continues the outline of basic concepts and methods presented in my book Synergetics. An Introduction, which has by now appeared in English, Russian, Japanese, Chinese, and German. I have written the present book in such a way that most of it can be read independently of my previous book, though occasionally some knowledge of that book might be useful.

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References, Further Reading, and Comments

1. Introduction

  • Haken: Synergetics, An Introduction, 3rd ed. ( Springer, Berlin, Heidelberg, New York 1983 ) This reference is referred to in the present book as [I]

    MATH  Google Scholar 

  • H. Haken, R. Graham: Synergetik — Die Lehre vom Zusammenwirken. Umschau 6, 191 (1971) H. Haken (ed.): Synergetics (Proceedings of a Symposium on Synergetics, Elmau 1972) ( Teubner, Stuttgart 1973 )

    Google Scholar 

  • H. Haken (ed.): Cooperative Effects, Progress in Synergetics (North-Holland, Amsterdam 1974) H. Haken: Cooperative effects in systems far from thermal equilibrium and in nonphysical system. Rev. Mod. Phys. 47, 67 (1975)

    Google Scholar 

  • A further source of references is the Springer Series in Synergetics, whose individual volumes are listed in the front matter of this book.

    Google Scholar 

  • For a popularisation see

    Google Scholar 

  • H. Haken: Erfolgsgeheimnisse der Natur ( Deutsche Verlagsanstalt, Stuttgart 1981 )

    Google Scholar 

  • K. G. Wilson: Phys. Rev. B4, 3174; 3184 (1971)

    ADS  MATH  Google Scholar 

  • K. G. Wilson, M. E. Fisher: Phys. Rev. Lett. 28, 248 (1972)

    ADS  Google Scholar 

  • J. Wegener: Phys. Rev. B5, 4529 (1972); B6, 1891 (1972)

    ADS  Google Scholar 

  • T. W. Burkhardt, J. M. J. van Leeuwen (eds.): Real-Space Renormalization, Topics Curr. Phys., Vol. 30 ( Springer, Berlin, Heidelberg, New York 1982 )

    Google Scholar 

  • Books and reviews on the subject are, for example

    Google Scholar 

  • K. G. Wilson, J. Kogut: Phys. Rep. 12C, 75 (1974)

    ADS  Google Scholar 

  • C. Domb, M. S. Green (eds.): Phase Transitions and Critical Phenomena. Internat. Series of Monographs in Physics, Vols. 1 — 6 (Academic, London 1972 — 76)

    Google Scholar 

  • S. K. Ma: Modern Theory of Critical Phenomena (Benjamin, Reading, MA 1976 )

    Google Scholar 

  • I. Taylor: Philos. Trans. R. Soc. London A223, 289 (1923)

    ADS  MATH  Google Scholar 

  • For recent and more detailed studies see, for example

    Google Scholar 

  • R. P. Fenstermacher, H. L. Swinney, J. P. Gollub: J. Fluid Mech. 94, 103 (1979)

    ADS  Google Scholar 

  • R. C. DiPrima: In Transition and Turbulence, ed. by R. E. Meyer ( Academic, New York 1981 )

    Google Scholar 

  • H. Benard: Rev. Gen. Sci. Pures Appl. 11, 1261, 1309 (1900)

    Google Scholar 

  • Lord Rayleigh: Philos. Mag. 32, 529 (1916)

    Google Scholar 

  • For more recent theoretical studies on linear stability see, e.g.

    Google Scholar 

  • S. Chandrasekhar: Hydrodynamic and Hydromagnetic Stability (Clarendon, Oxford 1961 ) For nonlinear treatments see

    Google Scholar 

  • A. Schluter, D. Lortz, F. Busse: J. Fluid Mech. 23, 129 (1965)

    MathSciNet  ADS  Google Scholar 

  • F. H. Busse: J. Fluid Mech. 30, 625 (1967)

    ADS  MATH  Google Scholar 

  • A. C. Newell, J. A. Whitehead: J. Fluid Mech. 38, 279 (1969)

    ADS  MATH  Google Scholar 

  • R. C. DiPrima, H. Eckhaus, L. A. Segel: J. Fluid Mech. 49, 705 (1971)

    ADS  MATH  Google Scholar 

  • F. H. Busse: J. Fluid Mech. 52, 1 (1972)

    ADS  Google Scholar 

  • H. Busse: Rep. Prog. Phys. 41, 1929 (1978)

    ADS  Google Scholar 

  • H. Haken: Phys. Lett. 46A, 193 (1973); and, in particular, Rev. Mod. Phys. 47, 67 (1975); and H. Haken: Synergetics, Springer Ser. Synergetics, Vol. 1, 3rd. ed. (Springer, Berlin, Heidelberg, New York 1983 )

    Google Scholar 

  • R. Graham: Phys. Rev. Lett. 31, 1479 (1973); Phys. Rev. 10, 1762 (1974) For recent experiments see also

    Google Scholar 

  • Ahlers, R. Behringer: Phys. Rev. Lett. 40, 712 (1978)

    ADS  Google Scholar 

  • G. Ahlers, R. Walden: Phys. Rev. Lett. 44, 445 (1981)

    ADS  Google Scholar 

  • P. Berge: In Dynamical Critical Phenomena and Related Topics, ed. by C. P. Enz, Lecture Notes Phys., Vol. 104 (Springer, Berlin, Heidelberg, New York 1979 ) p. 288

    Google Scholar 

  • F. H. Busse, R. M. Clever: J. Fluid Mech. 102, 75 (1981)

    ADS  MATH  Google Scholar 

  • M. Giglio, S. Musazzi, U. Perini: Phys. Rev. Lett. 47, 243 (1981)

    ADS  Google Scholar 

  • E. L. Koschmieder, S. G. Pallas: Int. J. Heat Mass Transfer 17, 991 (1974)

    Google Scholar 

  • J. P. Gollub, S. W. Benson: J. Fluid Mech. 100, 449 (1980)

    ADS  Google Scholar 

  • J. Maurer, A. Libchaber: J. Phys. Paris Lett. 39, 369 (1978); 40, 419 (1979); 41, 515 (1980)

    Google Scholar 

  • G. Pfister, I. Rehberg: Phys. Lett. 83A, 19 (1981)

    Google Scholar 

  • H. Haken (ed.): Chaos and Order in Nature Springer Ser. Synergetics, Vol. 11 (Springer, Berlin, Heidelberg, New York 1981), see in particular contributions by A. Libchaber and S. Fauve, E. 0. Schulz-DuBois et al., P. Berge, F. H. Busse

    Google Scholar 

  • H. Haken (ed.): Evolution of Order and Chaos, Springer Ser. Synergetics, Vol. 17 ( Springer, Berlin, Heidelberg, New York 1982 )

    Google Scholar 

  • H. L. Swinney, J. P. Gollub (eds.): Hydrodynamic Instabilities and the Transition to Turbulence, Topics Appl. Phys., Vol. 45 ( Springer, Berlin, Heidelberg, New York 1981 )

    Google Scholar 

  • L. D. Landau, E. M. Lifshitz: Course of Theoretical Physics, Vol. 6 ( Pergamon, London, New York 1959 )

    Google Scholar 

  • Chia-Shun-Yih: Fluid Mechanics ( University Press, Cambridge 1970 )

    Google Scholar 

  • C. Lin: Hydrodynamic Stability ( University Press, Cambridge 1967 )

    Google Scholar 

  • D. Joseph: Stability of Fluid Motions, Springer Tracts Nat. Phil., Vols. 27, 28 ( Springer, Berlin, Heidelberg, New York 1976 )

    Google Scholar 

  • R. Scorer: Clouds of the World (Lothian, Melbourne 1972) Early papers on laser theory including quantum fluctuations are H. Haken: Z. Phys. 181, 96 (1964); 190, 327 (1966)

    Google Scholar 

  • H. Risken: Z. Phys. 186, 85 (1965)

    ADS  Google Scholar 

  • R. D. Hempstead, M. Lax: Phys. Rev. 161, 350 (1967)

    ADS  Google Scholar 

  • W. Weidlich, H. Risken, H. Haken: Z. Phys. 201, 396 (1967)

    ADS  Google Scholar 

  • M. Scully, W. E. Lamb: Phys. Rev. 159, 208 (1967); 166, 246 (1968) H. Haken: Rev. Mod. Phys. 47, 67 (1975) Laser-phase transition analogy

    Google Scholar 

  • R. Graham, H. Haken: Z. Phys. 213, 420 (1968) R. Graham, H. Haken: Z. Phys. 237, 31 (1970)

    MathSciNet  Google Scholar 

  • V. De Giorgio, M. 0. Scully: Phys. Rev. A2, 117a (1970)

    Google Scholar 

  • R. Graham, H. Haken: Z. Phys. 213, 420 (1968)

    ADS  Google Scholar 

  • H. Risken, K. Nummedal: Phys. Lett. 26A, 275 (1968); J. Appl. Phys. 39, 4662 (1968) H. Haken, H. Ohno: Opt. Commun. 16, 205 (1976); Phys. Lett. 59A, 261 (1976)

    Google Scholar 

  • H. Knapp, H. Risken, H. D. Vollmer: Appl. Phys. 15, 265 (1978)

    ADS  Google Scholar 

  • Buttiker, H. Thomas: In Solutions and Condensed Matter Physics ed. by A. R. Bishop

    Google Scholar 

  • T. Schneider, Springer Ser. Solid-State Phys., Vol. 8 (Springer, Berlin, Heidelberg, New York 1981 ) p. 321

    Google Scholar 

  • J. Zorell: Opt. Commun. 38, 127 (1981)

    ADS  Google Scholar 

  • S. L. McCall: Phys. Rev. A9, 1515 (1974)

    ADS  Google Scholar 

  • R. Bonifacio, L. A. Lugiato: Opt. Commun. 19, 172 (1976)

    ADS  Google Scholar 

  • R. Sulomaa, S. Stenholm: Phys. Rev. A8, 2695 (1973)

    ADS  Google Scholar 

  • A. Kossakowski, T. Marzalek: Z. Phys. B23, 205 (1976)

    Google Scholar 

  • L. A. Lugiato, V. Benza, L. M. Narducci, J. D. Farina: Opt. Commun. 39, 405 (1981)

    ADS  Google Scholar 

  • A. Lugiato, V. Benza, L. M. Narducci: In Evolution of Order and Chaos, ed. by H. Haken Springer Ser. Synergetics, Vol. 17 (Springer, Berlin, Heidelberg, New York 1982 ) p. 120

    Google Scholar 

  • G. Velarde: ibid., p. 132

    Google Scholar 

  • L. A. Lugiato: In Progress in Optics (North-Holland, Amsterdam 1983)

    Google Scholar 

  • R. Bonifacio (ed.): Dissipative Systems in Quantum Optics, Topics Curr. Phys., Vol. 27 ( Springer, Berlin, Heidelberg, New York 1982 )

    Google Scholar 

  • H. Haken: Laser Theory, in Encyclopedia of Physics, Vol. XXV/2c, Light and Matter lc, (Springer, Berlin, Heidelberg, New York 1970) and reprint edition Laser Theory (Springer, Berlin, Heidelberg, New York 1983 )

    Google Scholar 

  • M. Sargent, M. 0. Scully, W. E. Lamb: Laser Physics ( Addison-Wesley, Reading, MA 1974 )

    Google Scholar 

  • F. Cap: Handbook on Plasma Instabilities, Vols. 1, 2 (Academic, New York 1976 and 1978 )

    Google Scholar 

  • A. B. Mikhailowskii: Theory of Plasma Instabilities, Vols. 1, 2 ( Consultants Bureau, New York, London 1974 )

    Google Scholar 

  • H. Wilhelmson, J. Weiland: Coherent Non-Linear Interaction of Waves in Plasmas ( Pergamon, Oxford 1977 )

    Google Scholar 

  • S. G. Thornhill, D. ter Haar: Phys. Rep. C43, 43 (1978)

    ADS  Google Scholar 

  • J. B. Gunn: Solid State Commun. 1, 88 (1963)

    ADS  Google Scholar 

  • B. Gunn: IBM Res. Develop. 8, 141 (1964)

    Google Scholar 

  • Nakamura: J. Phys. Soc. Jpn. 38, 46 (1975)

    ADS  Google Scholar 

  • C. Zener: Proc. R. Soc. London 145, 523 (1934)

    ADS  Google Scholar 

  • Esaki: Phys. Rev. 109, 603 (1958)

    ADS  Google Scholar 

  • R. Landauer: J. Appl. Phys. 33, 2209 (1962)

    ADS  Google Scholar 

  • R. Landauer, J. W. F. Woo: In Synergetics, ed. by H. Haken ( Teubner, Stuttgart 1973 ) p. 97

    Google Scholar 

  • C. E. Bottani, G. Caglioti, P. M. Ossi: J. Phys. F. 11, 541 (1981)

    ADS  Google Scholar 

  • C. Caglioti, A. F. Milone (eds.): Mechanical and Thermal Behaviour of Metallic Materials. Proc. Int. School of Physics Enrico Fermi (North-Holland, Amsterdam 1982 )

    Google Scholar 

  • J. S. Langer: In Fluctuations, Instabilities and Phase Transitions, ed. by T. Riste ( Plenum, New York 1975 ) p. 82

    Google Scholar 

  • J. S. Langer: Rev. Mod. Phys. 52, 1 (1980)

    ADS  Google Scholar 

  • M. T. Thompson, G. W. Hunt: A General Theory of Elastic Stability ( Wiley, London 1973 )

    MATH  Google Scholar 

  • Huseyn: Nonlinear Theory of Elastic Stability ( Nordhoff, Leyden 1975 )

    Google Scholar 

  • D. 0. Brush, B. D. Almroth: Buckling of Bars, Plates and Shells ( McGraw-Hill, New York 1975 )

    Google Scholar 

  • A. A. Andronov, A. A. Vitt, S. E. Kaikin: Theory of Oscillators (Pergamon, Oxford, London 1966) N. Minorsky: Nonlinear Oscillations (van Nostrand, Princeton 1962 )

    Google Scholar 

  • C. Hayashi: Nonlinear Oscillations in Physical Systems (McGraw-Hill, New York 1964) P. S. Lindsay: Phys. Rev. Lett. 47, 1349 (1981)

    ADS  Google Scholar 

  • C. H. Bray: J. Am. Chem. Soc. 43, 1262 (1921)

    Google Scholar 

  • B. P. Belousov: Sb. Ref. Radats. Med. Moscow (1959)

    Google Scholar 

  • V. A. Vavalin, A. M. Zhabotinsky, L. S. Yaguzhinsky: Oscillatory Processes in Biological and Chemical Systems ( Science Publ., Moscow 1967 ) p. 181

    Google Scholar 

  • A. N. Zaikin, A. M. Zhabotinsky: Nature 225, 535 (1970)

    ADS  Google Scholar 

  • A. M. Zhabotinsky, A. N. Zaikin: J.Theor. Biol. 40, 45 (1973)

    Google Scholar 

  • A. M. Turing: Philos. Trans. R. Soc. London B237, 37 (1952)

    ADS  Google Scholar 

  • Nicolis, I. Prigogine: Self-Organization in Non-Equilibrium Systems ( Wiley, New York 1977 )

    Google Scholar 

  • Haken: Z. Phys. B20, 413 (1975)

    Google Scholar 

  • G. F. Oster, A. S. Perelson: Arch. Rat. Mech. Anal. 55, 230 (1974)

    MathSciNet  Google Scholar 

  • S. Perelson: G. F. Oster: Arch. Rat. Mech. Anal. 57, 31 (1974/75)

    Google Scholar 

  • G. Nicolis: Adv. Chem. Phys. 19, 209 (1971)

    Google Scholar 

  • Change, E. K. Pye, A. M. Ghosh, B. Hess (eds.): Biological and Biochemical Oscillators ( Academic, New York 1973 )

    Google Scholar 

  • G. Nicolis, J. Portnow: Chem. Rev. 73, 365 (1973)

    Google Scholar 

  • R. M. Noyes, R. J. Field: Annu. Rev. Phys. Chem. 25, 95 (1975)

    ADS  Google Scholar 

  • J. J. Tyson: The Belousov-Zhabotinsky Reaction. Lecture Notes Biomath., Vol. 10 ( Springer, Berlin, Heidelberg, New York 1976 )

    Google Scholar 

  • P. C. Fife: Mathematical Aspects of Reacting and Diffusing Systems. Lecture Notes Biomath., Vol. 28 ( Springer, Berlin, Heidelberg, New York 1979 )

    Google Scholar 

  • A. Pacault, C. Vidal (eds.): Synergetics. Far from Equilibrium. Springer Ser. Synergetics, Vol. 3 ( Springer, Berlin, Heidelberg, New York 1979 )

    Google Scholar 

  • Vidal, A. Pacault (eds.): Nonlinear Phenomena in Chemical Dynamics. Springer Ser. Synergetics, Vol. 12 ( Springer, Berlin, Heidelberg, New York 1981 )

    Google Scholar 

  • T. H. Bullock, R. Orkand, A. Grinnell: Introduction to Nervous Systems ( Freeman, San Francisco 1977 )

    Google Scholar 

  • A. C. Scott: Neurophysics ( Wiley, New York 1977 )

    Google Scholar 

  • E. Basar: Biophysical and Physiological System Analysis ( Addison Wesely, Reading MA 1976 )

    Google Scholar 

  • M. Conrad, W. Gilitinger, M. Dal Chin (eds.): Physics and Mathematics of the Nervous System. Lecture Notes Biomath., Vol. 4 ( Springer, Berlin, Heidelberg, New York 1974 )

    Google Scholar 

  • A. V. Holden: Models of Stochastic Activity of Neurons. Lecture Notes Biomath., Vol. 12 ( Springer, Berlin, Heidelberg, New York 1976 )

    Google Scholar 

  • H. Shimizu: Adv. Biophys. 13, 195 (1979)

    Google Scholar 

  • A. M. Turing: Philos. Trans. R. Soc. London B237, 37 (1952)

    ADS  Google Scholar 

  • L. Wolpert: J. Theor. Biol. 25, 1 (1969)

    Google Scholar 

  • A. Gierer, H. Meinhardt: Kybernetik 12, 30 (1972); J. Cell. Sci. 15, 321 (1974) H. Haken, H. Olbrich: J. Math. Biol. 6, 317 (1978)

    Google Scholar 

  • J. P. Murray: J. Theor. Biol. 88, 161 (1981)

    Google Scholar 

  • C. Berding, H. Haken: J. Math. Biol. 14, 133 (1982)

    MathSciNet  MATH  Google Scholar 

  • A. Lotka: Proc. Nat. Acad. Sci. (USA) 6, 410 (1920)

    ADS  Google Scholar 

  • V. Volterra: Lecons sur la Theorie Mathematiques de la Lutte pour la Vie, Paris (1931)

    Google Scholar 

  • N. S. Goel, S. C. Maitra, E. W. Montroll: Rev. Mod. Phys. 43, 231 (1971)

    MathSciNet  ADS  Google Scholar 

  • T. N. E. Greville (ed.): Population Dynamics ( Academic, London 1972 )

    MATH  Google Scholar 

  • D. Ludwig: In Stochastic Population Theories, ed. by S. Levin, Lecture Notes Biomath., Vol. 3 ( Springer, Berlin, Heidelberg, New York 1974 )

    Google Scholar 

  • R. B. May: Nature 261, 459 (1976)

    ADS  Google Scholar 

  • M. Eigen: Naturwissenschaften 58, 465 (1971)

    ADS  Google Scholar 

  • M. Eigen, P. Schuster: Naturwissenschaften 64, 541 (1977); 65, 7 (1978); 65, 341 (1978)

    Google Scholar 

  • W. Ebeling, R. Feistel: Physik der Selbstorganisation and Evolution ( Akademie-Verlag, Berlin 1982 )

    Google Scholar 

  • F. M. Burnet: Immunology, Aging, and Cancer ( Freeman, San Francisco 1976 )

    Google Scholar 

  • C. DeLisi: Antigen Antibody Interactions, Lecture Notes Biomath., Vol. 8 ( Springer, Berlin, Heidelberg, New York 1976 )

    Google Scholar 

  • N. Dubin: A Stochastic Model for Immunological Feedback in Carcinogenesis. Lecture Notes Biomath., Vol. 9 ( Springer, Berlin, Heidelberg, New York 1976 )

    Google Scholar 

  • P. H. Richter: Pattern formation in the immune system. Lect. Math. Life Sci. 11, 89 (1979)

    Google Scholar 

  • R. W. Hockney, C. R. Jesshope: Parallel Computers ( Hilger, Bristol 1981 )

    MATH  Google Scholar 

  • K. S. Fu: Digital Pattern Recognition, 2nd ed. ( Springer, Berlin, Heidelberg, New York 1980 )

    Google Scholar 

  • K. S. Fu: Syntactic Pattern Recognition Applications (Springer, Berlin, Heidelberg, New York 1976 )

    Google Scholar 

  • K. S. Fu: In Pattern Formation by Dynamic Systems and Pattern Recognition, ed. by H. Haken, Springer Ser. Synergetics, Vol. 5 (Springer, Berlin, Heidelberg, New York 1979 ) p. 176

    Google Scholar 

  • T. Kohonen: Associative Memory — A System Theoretical Approach (Springer, Berlin, Heidelberg, New York 1978 )

    Google Scholar 

  • T. Kohonen: Self-Organization and Associative Memory, Springer Ser. Inf. Sci., Vol. 8 ( Springer, Berlin, Heidelberg, New York 1983 )

    Google Scholar 

  • H. Haken (ed.): Pattern Formation by Dynamic Systems and Pattern Recognition, Springer Ser. Synergetics, Vol. 5 ( Springer, Berlin, Heidelberg, New York 1979 )

    Google Scholar 

  • H. Haken: unpublished material

    Google Scholar 

  • Mensch, K. Kaasch, A. Kleinknecht, R. Schnopp: IIM/dp 80–5 Innovation Trends, and Switching between Full- and Under-Employment Equilibria. 1950 — 1978 Discussion Paper Series, International Institute of Management, Wissenschaftszentrum Berlin

    Google Scholar 

  • Haken: Synergetics, Springer Ser. Synergetics, Vol. 1, 3rd. ed. ( Springer, Berlin, Heidelberg, New York 1983 )

    Google Scholar 

  • W. Weidlich, G. Haag: Quantitative Sociology, Springer Ser. Synergetics, Vol. 14 ( Springer, Berlin, Heidelberg, New York 1983 )

    Google Scholar 

  • H. Haken: Erfolgsgeheimnisse der Natur ( Deutsche Verlagsanstalt, Stuttgart 1981 )

    Google Scholar 

  • Ch. J. Krebs: Ecology. The Experimental Analysis of Distribution and Abundance ( Harper and Row, New York 1972 )

    Google Scholar 

  • E. Rickleps: Ecology ( Nelson, London 1973 )

    Google Scholar 

  • E. Ash: Social Psychology ( Prentice Hall, New York 1952 ) p. 452

    Google Scholar 

  • W. Weidlich: Collect. Phenom. 1, 51 (1972)

    Google Scholar 

  • E. Noelle-Neumann: Die Schweigespirale (Piper, Munchen 1980) [English transl. (to appear 1983): The Spiral of Silence: Public Opinion — The Skin of Time (Chicago, University Press)]

    Google Scholar 

  • A. Wunderlin, H. Haken: Lecture Notes, Projekt Mehrebenenanalyse im Rahmen des Forschungsschwerpunkts Mathematisierung (Universitat Bielefeld 1980 )

    Google Scholar 

  • H. Haken: Erfolgsgeheimnisse der Natur ( Deutsche Verlagsanstalt, Stuttgart 1981 )

    Google Scholar 

  • W. Weidlich, G. Haag: Quantitative Sociology, Springer Ser. Synergetics, Vol. 14 ( Springer, Berlin, Heidelberg, New York 1983 )

    Google Scholar 

  • H. Haken: Synergetics, Springer Ser. Synergetics, Vol. 1, 3rd. ed. ( Springer, Berlin, Heidelberg, New York 1983 )

    Google Scholar 

  • Because the individual topics of Sects. 1.11 — 17 will be dealt with in detail in later chapters, we refer the reader to the corresponding references belonging to those chapters. Here we quote only those references which will not be quoted later.

    Google Scholar 

  • R. Courant, D. Hilbert: Methods of Mathematical Physics, Vols. 1, 2 ( Wiley, New York 1962 )

    Google Scholar 

  • P. M. Morse, H. Feshbach: Methods of Theoretical Physics, Vols. 1, 2 ( McGraw-Hill, New York 1953 )

    Google Scholar 

  • W. F. Elen: Differential Equations, Vols. 1, 2 ( MacMillan, London 1967 )

    Google Scholar 

  • E. A. Coddington, N. Levinson: Theory of Ordinary Differential Equations ( McGraw-Hill, New York 1955 )

    MATH  Google Scholar 

  • V. V. Nemytskii, V. V. Stepanov: Qualitative Theory of Differential Equations ( University Press, Princeton 1960 )

    MATH  Google Scholar 

  • W. Hirsch, S. Smale: Differential Equations, Dynamical Systems, and Linear Algebra ( Academic, New York 1974 )

    MATH  Google Scholar 

  • Z. Nitecki: Differentiable Dynamics ( MIT Press, Cambridge, MA 1971 )

    MATH  Google Scholar 

  • Abraham, J. E. Marsden: Foundations of Mechanics (Benjamin/Cummings, Reading, MA 1978 )

    Google Scholar 

  • Smale: The Mathematics of Time (Springer, Berlin, Heidelberg, New York 1980 )

    Google Scholar 

  • H. Haken: Synergetics, Springer Ser. Synergetics, Vol. 1, 3rd. ed. ( Springer, Berlin, Heidelberg, New York 1983 )

    Google Scholar 

  • J. L. Doob: Stochastic Processes ( Wiley, New York 1953 )

    MATH  Google Scholar 

  • M. Loeve: Probability Theory (van Nostrand, Princeton 1963 )

    Google Scholar 

  • R. von Mises: Mathematical Theory of Probability and Statistics ( Academic, New York 1964 )

    MATH  Google Scholar 

  • Yu. V. Prokhorov, Yu. A. Rozanov: Probability Theory, Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen, Vol. 157 ( Springer, Berlin, Heidelberg, New York 1968 )

    Google Scholar 

  • R. C. Dubes: The Theory of Applied Probability ( Prentice Hall, Englewood Cliffs, NJ 1968 )

    MATH  Google Scholar 

  • W. Feller: An Introduction to Probability Theory and Its Applications, Vol. 1 ( Wiley, New York 1971 )

    MATH  Google Scholar 

  • Kai Lai Chung: Elementary Probability Theory with Stochastic Processes (Springer, Berlin, Heidelberg, New York 1974 )

    Google Scholar 

  • Hida: Brownian Motion, Applications of Mathematics, Vol. 11 ( Springer, Berlin, Heidelberg, New York 1980 )

    Google Scholar 

  • L. D. Landau, E. M. Lifshitz: In Course of Theoretical Physics, Vol. 5 (Pergamon, London 1952)

    Google Scholar 

  • R. Kubo: Thermodynamics ( North-Holland, Amsterdam 1968 )

    Google Scholar 

  • D. N. Zubarev: Non-Equilibrium Statistical Thermodynamics ( Consultants Bureau, New York 1974 )

    Google Scholar 

  • H. Haken: Laser Theory, in Encylopedia of Physics, Vol. XXV/2c, Light and Matter lc (Springer, Berlin, Heidelberg, New York 1970) and reprint edition Laser Theory (Springer, Berlin, Heidelberg, New York 1983 )

    Google Scholar 

  • H. Haken: Synergetics, Springer Ser. Synergetics, Vol. 1, 3rd. ed. ( Springer, Berlin, Heidelberg, New York 1978 )

    Google Scholar 

  • H. Haken (ed.): Chaos and Order in Nature, Springer Ser. Synergetics, Vol. 11 ( Springer, Berlin, Heidelberg, New York 1981 )

    Google Scholar 

  • H. Haken: Order and Chaos, Springer Ser. Synergetics, Vol. 17 ( Springer, Berlin, Heidelberg, New York 1982 ) Approach

    Google Scholar 

  • H. Haken: unpublished material 1.12 How to Visualize Solutions

    Google Scholar 

  • Y. Choquet-Bruhat, C. DeWitt-Morette, M. Dillard-Bleick: Analysis, Manifolds and Physics ( Nort h-Holland, Amsterdam 1982 )

    MATH  Google Scholar 

  • R. D. Richtmyer: Principles of Advanced Mathematical Physics II (Springer, Berlin, Heidelberg, New York 1981 )

    Google Scholar 

  • H. Haken: Synergetics, Springer Ser. Synergetics, Vol. 1, 3rd. ed. ( Springer, Berlin, Heidelberg, New York 1983 )

    Google Scholar 

  • D’Arcy W. Thompson: On Growth and Form ( Cambridge University Press, London 1961 )

    Google Scholar 

  • See H. Haken: Synergetics Springer Ser. Synergetics, Vol. 1, 3rd. ed. (Springer, Berlin, Heidelberg, New York 1983) and references cited in later chapters. Here References are presented for

    Google Scholar 

  • V. I. Oseledec: A multiplicative ergodic theorem. Lyapunov characteristic number for dynamical systems. Tr. Mosk. Mat. Osc. 19, 179 (1968) [English transl.: Trans. Moscow Math. Soc. 19, 197 (1968)1

    Google Scholar 

  • Ya. B. Pesin: Characteristic Lyapunov Exponents and Smooth Ergodic Theory. Russ. Math. Surv. 32 (4), 55 (1977)

    Google Scholar 

  • D. Ruelle: “Sensitive Dependence on Initial Conditions and Turbulent Behavior of Dynamical Systems”, in Bifurcation Theory and Its Applications in Scientific Disciplines, ed. by O. Gurel, O. E. ROssler, New York Acad. of Sci. 316, (1979)

    Google Scholar 

  • D. Farmer: Physica 4D, 366 (1982)

    MathSciNet  MATH  Google Scholar 

  • Tomita: Phys. Rep. 86, 113 (1982)

    MathSciNet  ADS  Google Scholar 

  • See the references in Sect. 1.11.5 as well as those of later chapters

    Google Scholar 

  • H. Haken: Synergetics, Springer Ser. Synergetics, Vol. 1, 3rd. ed. ( Springer, Berlin, Heidelberg, New York 1983 )

    Google Scholar 

  • Discrete Maps are treated in Chap. 11. The Poincare map is discussed, e.g., in

    Google Scholar 

  • R. Abraham, J. E. Marsden: Foundations of Mechanics ( Benjamin/Cummings, Reading, MA 1978 )

    MATH  Google Scholar 

2. Linear Ordinary Differential Equations

  • E. C. G. Sudarshan, M. Mukunda: Classical Dynamics: A Modern Perspective ( Wiley, New York 1974 )

    MATH  Google Scholar 

  • R. D. Richtmyer: Principles of Advanced Mathematical Physics II (Springer, Berlin, Heidelberg, New York 1981 )

    Google Scholar 

  • H. Haken: Synergetics, Springer Ser. Synergetics, Vol. 1, 3rd. ed. ( Springer, Berlin, Heidelberg, New York 1983 )

    Google Scholar 

  • Haken: Prog. Theor. Phys. Suppl. 69, 30 (1980)

    MathSciNet  ADS  Google Scholar 

  • H. Haken: Unpublished material

    Google Scholar 

  • O. Duffing: Erzwungene Schwingungen bei veranderlicher Eigenfrequenz and ihre technische Bedeutung ( Vieweg, Braunschweig 1918 )

    Google Scholar 

  • C. Hayashi: Nonlinear Oscillations in Physical Systems (McGraw-Hill, New York 1964) compare Sect. 2.1.1 also

    Google Scholar 

  • R. Bellman, K. L. Cooke: Introduction to Matrix Analysis ( McGraw-Hill, New York 1960 )

    MATH  Google Scholar 

  • N. Dunford, J. T. Schwartz: Linear Operators Pure and Applied Mathematics, Vol. VII, Parts I — III (Wiley, Interscience, New York 1957) see Sect. 2.4.3. For the Lyapunov exponents see Sect. 1.14.6. Theorem on vanishing Lyapunov exponents:

    Google Scholar 

  • H. Haken: Phys. Lett. 94A, 71 (1983)

    MathSciNet  Google Scholar 

  • E. A. Coddington, N. Levinson: Theory of Ordinary Differential Equations ( McGraw-Hill, New York 1955 )

    MATH  Google Scholar 

  • Floquet: Sur les equations dtfferentielles lineaires a coefficients periodiques. Ann. Ecole Norm. Ser. 2 12, 47 (1883) Haken: Unpublished material

    Google Scholar 

3. Linear Ordinary Differential Equations with Quasiperiodic Coefficients

  • The results of this chapter, with the exception of Sect. 3.9, were obtained by the present author. The operator T in Eq. (3.1.6) was introduced in

    Google Scholar 

  • H. Haken: Z. Naturforsch. 8A, 228 (1954) where the case of a unitary representation of T was treated and the form (3.1.20) proven. For further results see

    Google Scholar 

  • H. Haken: In Dynamics of Synergetic Systems, Springer Ser. Synergetics, Vol. 6, ed. by H. Haken (Springer, Berlin, Heidelberg, New York 1980 ) p. 16

    Google Scholar 

  • For the proof of Theorem 3.8.2 I used auxiliary theorems represented in

    Google Scholar 

  • N. Dunford, J. T. Schwartz: Linear Operators, Pure and Applied Mathematics, Vol. VII, Parts I — III (Wiley, Interscience, New York 1957 ) For attempts of other authors at this problem cf.

    Google Scholar 

  • N. N. Bogoliubov, I. A. Mitropolskii, A. M. Samoilento: Methods of Accelerated Convergence in Nonlinear Mechanics (Springer, Berlin, Heidelberg, New York 1976), where further references are given.

    Google Scholar 

  • The results of Sect. 3.9 are taken from N. N. Bogoliubov, I. A. Mitropolskii, A. M. Samoilento 1. c.

    Google Scholar 

4. Stochastic Nonlinear Differential Equations

  • For the original papers on the ItO and Stratonovich calculus see

    Google Scholar 

  • Ito: Lectures on Stochastic Processes (Tata Institute of Fundamental Research, Bombay 1961) K. Ito: Stochastic Processes ( Universitet Matematisk Institut, Aarhus 1969 )

    Google Scholar 

  • K. Ito, H.P. McKean: Diffusion Processes and Their Sample Paths (Springer, Berlin, Heidelberg, New York 1965 )

    Google Scholar 

  • K. Ito: Nagoya Math. J. 1, 35 (1950)

    MathSciNet  MATH  Google Scholar 

  • K. Ito: Nagoya Math. J. 3, 55 (1951)

    MathSciNet  MATH  Google Scholar 

  • K.Ito: On Stochastic Differential Equations (Am. Math. Soc. New York, 1951 )

    Google Scholar 

  • P. Langevin: Sur la theorie du mouvement brownien. C. R. Acad. Sci. Paris 146, 530 (1908)

    MATH  Google Scholar 

  • R. L. Stratonovich: SIAM J. Control 4, 362 (1966) Recent texts and monographs include

    Google Scholar 

  • I. I. Gihmann, A. V. Skorohod: Stochastic Differential Equations (Springer, Berlin, Heidelberg, New York 1972 )

    Google Scholar 

  • L. Arnold: Stochastic Differential Equations ( Oldenbourg, Munchen 1973 )

    Google Scholar 

  • N. G. van Kampen: Stochastic Processes in Physics and Chemistry ( North-Holland, Amsterdam 1981 )

    MATH  Google Scholar 

  • C. W. Gardiner: Handbook of Stochastic Methods. Springer Ser. Synergetics, Vol. 13 ( Springer, Berlin, Heidelberg, New York 1983 )

    Google Scholar 

5. The World of Coupled Nonlinear Oscillators

  1. This section gives only a sketch, for detailed treatments of nonlinear oscillators see

    Google Scholar 

  2. N. N. Bogoliubov, Y. A. Mitropolsky: Asymptotic Methods in the Theory of Nonlinear Oscillations (Hindustan Publ. Corp., New Delhi 1961 )

    Google Scholar 

  3. N. Minorski: Nonlinear Oscillations ( Van Nostrand, Toronto 1962 )

    Google Scholar 

  4. A. Andronov, A. Vitt, S. E. Khaikin: Theory of Oscillators ( Pergamon, London 1966 )

    MATH  Google Scholar 

  5. Perturbations of Quasiperiodic Motion for Time-Independent Amplitudes (Persistence of Quasiperiodic Motion) Our presentation is based on results reported in

    Google Scholar 

  6. N. N. Bogoliubov, I. A. Mitropolskii, A. M. Samoilento: Methods of Accelerated Convergence in Nonlinear Mechanics (Springer, Berlin, Heidelberg, New York 1976 )

    Google Scholar 

6. Nonlinear Coupling of Oscillators: The Case of Persistence of Quasiperiodic Motion

  • A. N. Kolmogorov: Dokl. Akad. Nauk. USSR 98, 527 (1954)

    MathSciNet  MATH  Google Scholar 

  • V. I. Arnol’d: Russ. Math. Surv. 18, 9 (1963)

    MATH  Google Scholar 

  • Moser: Math. Ann. 169, 136 (1967)

    MathSciNet  MATH  Google Scholar 

  • In this chapter we essentially follow Moser’s paper, but we use a somewhat different representation Further reading

    Google Scholar 

  • J. Moser: “Nearly Integrable and Integrable Systems”, in Topics in Nonlinear Dynamics, ed. by S. Jorna (AIP Conf. Proc. 46, 1 1978 )

    Google Scholar 

  • M. V. Berry: “Regular and Irregular Motion”, in Topics in Nonlinear Dynamics, ed. by S. Jorna (AIP Conf. Proc. 46, 16 1978 )

    Google Scholar 

7. Nonlinear Equations. The Slaving Principle

  • This chapter is based on a slight generalization of

    Google Scholar 

  • H. Haken, A. Wunderlin: Z. Phys. B47, 179 (1982)

    MathSciNet  Google Scholar 

  • An early version, applied to quasiperiodic motion (in the case of laser theory) was developed by H. H.ken: Talk at the International Conference on Optical Pumping, Heidelberg (1962); also H. Haken, H. Sauermann: Z. Phys. 176, 47 (1963)

    Google Scholar 

  • Here, by an appropriate decomposition of the variables into rapidly oscillating parts and slowly varying amplitudes, the atomic variables were expressed by the field modes (order parameters) Other procedures are given in

    Google Scholar 

  • H. Haken: Z. Phys. B20, 413 (1975); B21, 105 (1975); B22, 69 (1975); B23, 388 (1975) and H. Haken: Z. Phys. B29, 61 (1978); B30, 423 (1978)

    Google Scholar 

  • The latter procedures are based on rapidly converging continued fractions, at the expense that the slaved variables depend on the order parameters (unstable modes) at previous times (in higher order approximation). These papers included fluctuations of the Langevin type.

    Google Scholar 

  • In a number of special cases (in particular, if the fluctuations are absent), relations can be established to other theorems and procedures, developed in mathematics, theoretical physics, or other disciplines.

    Google Scholar 

  • Relations between the slaving principle and the center manifold theorem (and related theorems) are studied by

    Google Scholar 

  • A. Wunderlin, H. Haken: Z. Phys. B44, 135 (1981) For the center manifold theorem, see

    Google Scholar 

  • V. A. Pliss: Izv. Akad. Nauk SSSR., Mat. Ser. 28, 1297 (1964)

    MathSciNet  MATH  Google Scholar 

  • A. Kelley: In Transversal Mappings and Flows, ed. by R. Abraham, J. Robbin ( Benjamin, New York 1967 )

    Google Scholar 

  • In contrast to the center manifold theorem, the slaving principle contains fluctuations, includes the surrounding of the center manifold, and provides a construction of s(u,q), t).

    Google Scholar 

8. Nonlinear Equations. Qualitative Macroscopic Changes

  • In this chapter I present an approach initiated in 1962 (H. Haken: Talk at the International Conference on Optical Pumping, Heidelberg 1962), and applied to laser theory including quasi-periodic motion, e.g. bifurcation to tori (see, e.g.

    Google Scholar 

  • H. Haken, H. Sauermann: Z. Phys. 176, 47 (1963)

    ADS  Google Scholar 

  • H. Haken: Laser Theory, in Encylopedia of Physics, Vol. XXV, 2c, Light and Matter lc (Springer, Berlin, Heidelberg, New York 1970) and reprint edition Laser Theory (Springer, Berlin, Heidelberg, New York 1983 )

    Google Scholar 

  • This author’s approach is based on the slaving principle and represents, in modern language, “dynamic bifurcation theory” (which allows one to cope with transients and fluctuations). “Static” bifurcation theory was initiated in the classical papers by

    Google Scholar 

  • H. Poincare: Les methodes nouvelles de la mecanique celeste T. I (Gauthier-Villars, Paris 1892) H. Poincare: Acta Math. 7, 1 (1885)

    MATH  Google Scholar 

  • A. M. Lyapunov: Sur le masse liquide homogene donnee d’un mouvement de rotation. Zap. Acad. Nauk, St. Petersburg 1, 1 (1906)

    Google Scholar 

  • E. Schmidt: Zur Theorie der linearen and nichtlinearen Integralgleichungen, 3. Teil, Math. Annalen 65, 370 (1908)

    MATH  Google Scholar 

  • While this field seems to have been more or less dormant for a while (with the exception of bifurcation theory in fluid dynamics), the past decade has seen a considerable increase of interest as reflected by recent texts. We mention in particular

    Google Scholar 

  • D. H. Sattinger: Topics in Stability and Bifurcation Theory, Lecture Notes Math., Vol. 309 ( Springer, Berlin, Heidelberg, New York 1972 )

    Google Scholar 

  • D. H. Sattinger: Group Theoretic Methods in Bifurcation Theory, Lecture Notes Math., Vol. 762 ( Springer, Berlin, Heidelberg, New York 1980 )

    Google Scholar 

  • G. looss: Bifurcation of Maps and Applications, Lecture Notes, Mathematical Studies ( North-Holland, Amsterdam 1979 )

    Google Scholar 

  • Looss, D. D. Joseph: Elementary Stability and Bifurcation Theory (Springer, Berlin, Heidelberg, New York 1980 )

    Google Scholar 

  • These authors deal in an elegant fashion with “static” bifurcation theory.

    Google Scholar 

  • Haken: Synergetics, Springer Ser. Synergetics, Vol. 1, 3rd. ed. (Springer, Berlin, Heidelberg, New York 1983 ) Here we follow

    Google Scholar 

  • H. Haken: Unpublished material References on catastrophe theory are

    Google Scholar 

  • R. Thom: Structural Stability and Morphogenesis ( Benjamin, Reading, MA 1975 )

    MATH  Google Scholar 

  • H. Haken: Synergetics, Springer Ser. Synergetics, Vol. 1, 3rd. ed. ( Springer, Berlin, Heidelberg, New York 1983 )

    Google Scholar 

  • The branching of oscillatory solutions was first treated in the classical paper

    Google Scholar 

  • E. Hopf: Abzweigung einer periodischen Losung eines Differentialsystems. Berichie der Mathematisch-Physikalischen Klasse der Sachsischen Akademie der Wissenschaften zu Leipzig XCIV, 1 (1942) For recent treatments see

    Google Scholar 

  • J. Marsden, M. McCracken: The Hopf Bifurcation and Its Applications. Lecture Notes Appl. Math. Sci., Vol. 18 ( Springer, Berlin, Heidelberg, New York 1976 )

    Google Scholar 

  • D. D. Joseph: Stability of Fluids Motion. Springer Tracts Natural Philos., Vols. 27, 28 ( Springer, Berlin, Heidelberg, New York 1976 )

    Google Scholar 

  • A. S. Monin, A. M. Yaglom: Statistical Fluid Mechanics, Vol. I ( MIT Press, Cambridge, MA 1971 )

    Google Scholar 

  • H. Haken: Synergetics, Springer Ser. Synergetics, Vol. 1, 3rd. ed. ( Springer, Berlin, Heidelberg, New York 1983 )

    Google Scholar 

  • H. Haken: Laser Theory, in Encylopedia of Physics, Vol. XXV, 2c, Light and Matter lc (Springer, Berlin, Heidelberg, New York 1970) and reprint edition Laser Theory (Springer, Berlin, Heidelberg, New York 1983 )

    Google Scholar 

  • R. L. Stratonovich: Topics in the Theory of Random Noise, Vols. 1, 2 (Gordon and Breach, Ncw York 1963, 1967 )

    Google Scholar 

  • H. Haken: Z. Phys. B29, 61 (1978)

    Google Scholar 

  • H. Haken: Unpublished material

    Google Scholar 

  • H. Haken: Z. Phys. B30, 423 (1978)

    MathSciNet  Google Scholar 

  • A. Chenciner, G, Iooss: Arch. Ration. Mech. Anal. 69, 109 (1979)

    MathSciNet  MATH  Google Scholar 

  • G. R. Sell: Arch. Ration. Mech. Anal. 69, 199 (1979)

    MathSciNet  MATH  Google Scholar 

  • G. R. Sell: In Chaos and Order in Nature, Springer Ser. Synergetics, Vol. 11, ed. by H. Haken (Springer, Berlin, Heidelberg, New York 1980 ) p. 84

    Google Scholar 

  • L. D. Landau, E. M. Lifshitz: In Course of Theoretical Physics Vol. 6, Fluid Mechanics (Pergamon, London, New York 1959)

    Google Scholar 

  • E. Hopf: Commun. Pure Appl. Math. 1, 303 (1948)

    MathSciNet  MATH  Google Scholar 

  • D. Ruelle, F. Takens: Commun. Math. Phys. 20, 167 (1971)

    MathSciNet  ADS  MATH  Google Scholar 

  • S. Newhouse, D. Ruelle, F. Takens: Commun. Math. Phys. 64, 35 (1978)

    MathSciNet  ADS  MATH  Google Scholar 

  • S. Grossmann, S. Thomae: Z. Naturforsch. 32A, 1353 (1977)

    MathSciNet  ADS  Google Scholar 

  • M. J. Feigenbaum: J. Stat. Phys. 19, 25 (1978); Phys. Lett. 74A, 375 (1979)

    Google Scholar 

  • P. Collet, J. P. Eckmann: Iterated Maps on the Interval as Dynamical Systems ( Birkhauser, Boston 1980 )

    MATH  Google Scholar 

  • T. Geisel, J. Nierwetberg: In Evolution of Order and Chaos, Springer Ser. Synergetics, Vol. 17, ed. by H. Haken (Springer, Berlin, Heidelberg, New York 1982 ) p. 187

    Google Scholar 

  • Y. Pomeau, P. Manneville: Commun. Math. Phys. 77, 189 (1980)

    MathSciNet  ADS  Google Scholar 

  • Mayer-Kress, H. Haken: Phys. Lett. 82A, 151 (1981)

    MathSciNet  Google Scholar 

9. Spatial Patterns

  • Examples are provided by the Navier-Stokes Equations, e.g.

    Google Scholar 

  • Haken: Synergetics Springer Ser. Synergetics, Vol. 1, 3rd. ed. (Springer, Berlin, Heidelberg, New York 1983) and

    Google Scholar 

  • O. A. Ladyzhenskaya: The Mathematical Theory of Viscous Incompressible Flow ( Gordon and Breach, New York 1963 )

    MATH  Google Scholar 

  • D. D. Joseph: Stability of Fluid Motions I and II, Springer Tracts Natural Philos., Vols. 27, 28 (Springer, Berlin, Heidelberg, New York 1976 ) Reaction Diffusion equations are treated, e. g., in

    Google Scholar 

  • P. C. Fife: In Dynamics of Synergetic Systems Springer Ser. Synergetics, Vol. 6, ed. by H. Haken (Springer, Berlin, Heidelberg, New York 1980) p. 97, with further references

    Google Scholar 

  • J. S. Turner: Adv. Chem. Phys. 29, 63 (1975)

    Google Scholar 

  • J. W. Turner: Trans. NY Acad. Sci. 36, 800 (1974), Bull. Cl. Sci. Acad. Belg. 61, 293 (1975)

    MATH  Google Scholar 

  • Y. Schiffmann: Phys. Rep. 64, 87 (1980) Compare also Sect. 1. 5. 2

    Google Scholar 

  • H. Haken: Synergetics, Springer Ser. Synergetics, Vol. 1, 3rd. ed. ( Springer, Berlin, Heidelberg, New York 1983 )

    Google Scholar 

  • H. Haken: Synergetics, Springer Ser. Synergetics, Vol. 1, 3rd. ed. (Springer, Berlin, Heidelberg, New York 1983) See also the references cited in Sect. 9. 1

    Google Scholar 

  • H. Haken: Z. Phys. B21, 105 (1975)

    Google Scholar 

  • H. Haken: Z. Phys. B22, 69 (1975); B23, 388 (1975)

    Google Scholar 

  • For a different approach (for a more restricted class of problems) based on scaling see Y. Kuramoto, T. Tsusuki: Prog. Theor. Phys. 52, 1399 (1974)

    Google Scholar 

  • A. Wunderlin, H. Haken: Z. Phys. B21, 393 (1975)

    Google Scholar 

  • H. Haken: Unpublished material Equation (9.5.15) with A = 0 was derived differently by

    Google Scholar 

  • J. Swift, P. C. Hohenberg: Phys. Rev. A15, 319 (1977)

    ADS  Google Scholar 

  • H. Haken: Synergetics, Springer Ser. Synergetics, Vol. 1, 3rd. ed. ( Springer, Berlin, Heidelberg, New York 1983 )

    Google Scholar 

10. The Inclusion of Noise

  • W. Horsthemke, R. Lefever: Noise-Induced Transitions, Springer Ser. Synergetics, Vol. 15 ( Springer, Berlin, Heidelberg, New York 1983 )

    Google Scholar 

  • H. Haken: Synergetics, Springer Ser. Synergetics, Vol. 1, 3rd. ed. ( Springer, Berlin, Heidelberg, New York 1983 )

    Google Scholar 

  • H. Risken: Z. Phys. 186, 85 (1965); 191, 302 (1966)

    MathSciNet  Google Scholar 

  • H. Risken, H. D. Vollmer: Z. Phys. 201, 323 (1967); 204, 240 (1967)

    Google Scholar 

  • H. Risken: In Progress in Optics, Vol. VIII, ed by E. Wolf ( North-Holland, Amsterdam 1970 ) p. 239

    Google Scholar 

  • E. Uhlenbeck, L. S. Ornstein: Phys. Rev. 36, 823 (1930)

    ADS  MATH  Google Scholar 

  • N. Wax (ed.): Selected Papers on Noise and Statistical Processes ( Dover, New York 1954 )

    Google Scholar 

  • R. Graham, H. Haken: Z. Phys. 248, 289 (1971)

    MathSciNet  ADS  Google Scholar 

  • Risken: Z. Phys. 251, 231 (1972) For related work see

    Google Scholar 

  • H. Haken: Z. Phys. 219, 246 (1969)

    ADS  Google Scholar 

  • H. Haken: Rev. Mod. Phys. 47, 67 (1975)

    MathSciNet  ADS  Google Scholar 

  • R. Graham: Z. Phys. B40, 149 (1980)

    Google Scholar 

  • H. Haken: Z. Phys. 219, 246 (1969)

    ADS  Google Scholar 

  • Compare (H. Haken: Synergetics Springer Ser. Synergetics, Vol. 1, 3rd. ed. (Springer, Berlin, Heidelberg, New York 1983)), where also another approach is outlined. That approach starts right away from the master equation or Fokker-Planck equation and eliminates the slaved variables from these equations.

    Google Scholar 

11. Discrete Noisy Maps

  • Basic works on discrete maps have been cited in Sect. 8.11.2. Scaling properties of discrete noisy maps have been analized in

    Google Scholar 

  • J. P. Crutchfield, B. A. Huberman: Phys. Lett. 77 A, 407 (1980)

    Google Scholar 

  • J. P. Crutchfield, M. Nauenberg, J. Rudnick: Phys. Rev. Lett. 46, 935 (1981)

    MathSciNet  ADS  Google Scholar 

  • B. Shraiman, C. E. Wayne, P. C. Martin: Phys. Rev. Lett. 46, 933 (1981) We shall follow essentially

    Google Scholar 

  • Mayer-Kress, H. Haken: J. Stat. Phys. 26, 149 (1981)

    MathSciNet  ADS  MATH  Google Scholar 

  • Haken, G. Mayer-Kress: Z. Phys. B43, 185 (1981)

    MathSciNet  Google Scholar 

  • H. Haken: In Chaos and Order in Nature. Springer Ser. Synergetics, Vol. 11, ed. by H. Haken (Springer, Berlin, Heidelberg, New York 1981 ) p. 2

    Google Scholar 

  • H. Haken, A. Wunderlin: Z. Phys. B46, 181 (1982)

    MathSciNet  Google Scholar 

12. Example of an Unsolvable Problem in Dynamics

Appendix

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    Hermann Haken

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Haken, H. (2004). Advanced Topics. In: Synergetics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-10184-1_2

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