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Bifurcations of Dynamical Systems

  • John Guckenheimer
Chapter
Part of the Progress in Mathematics book series (PM, volume 8)

Abstract

The subject of these lectures is the bifurcation theory of dynamical systems. They are not comprehensive, as we take up some facets of bifurcation theory and largely ignore others. In particular, we focus our attention on finite dimensional systems of difference and differential equations and say almost nothing about infinite dimensional systems. The reader interested in the infinite dimensional theory and its applications should consult the recent survey of Marsden [66] and the conference proceedings edited by Rabinowitz [89]. We also neglect much of the multidimensional bifurcation theory of singular points of differential equations. The systematic exposition of this theory is much more algebraic than the more geometric questions considered here, and Arnold [7,9] provides a good survey of work in this area. We confine our interest to questions which involve the geometric orbit structure of dynamical systems. We do make an effort to consider applications of the mathematical phenomena illustrated. For general background about the theory of dynamical systems consult [102]. Our style is informal and our intent is pedagogic. The current state of bifurcation theory is a mixture of mathematical fact and conjecture. The demarcation between the proved and unproved is small [11]. Rather than attempting to sort out this confused state of affairs for the reader, we hope to provide the geometric insight which will allow him to explore further.

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Bibliography

  1. 1.
    V.S. Afraimovic, V.V. Bykov, L.P. Siĺnikov, On the appearance and structure of the Lorenz attractor, Doklady Acad. Nauk, 237, 336–339, 1977.Google Scholar
  2. 2.
    V.S. Afraimovic and L.P. Siĺnikov, On small periodic perturbations of autonomous systems, Soviet Math Dokl. 15, 206–211, 1976.Google Scholar
  3. 3.
    A. Andronov and E. Leontovich, Sur la theorie de la variation de la structure qualitative de la division du plan en trajectores, Dokl. Akad. Nauk. 21, 427–430, 1938.Google Scholar
  4. 4.
    A. Andronov, E. Leontovich, I. Gordon, and A. Maier, The Theory of Bifurcation of Plane Dynamical Systems, 1971.Google Scholar
  5. 5.
    V.I. Arnold, Small denominators I, Mappings of the circle onto itself., Izv. Akad. Nauk. SSSR Ser. Mat. 25, 21–86, 1961.Google Scholar
  6. 6.
    V.I. Arnold, Singularities of smooth mappings, Russian Math. Surveys 23, 1–43, 1968.Google Scholar
  7. 7.
    V.I. Arnold, On Matrices Depending upon a Parameter, Russian Math Surveys 26 (1971), No. 2, 29–43.Google Scholar
  8. 8.
    V.I. Arnold, On the algebraic unsolvability of Lyapounov stability and the problem of classification of singular points of an analytic system of differential equations, Funct. Anal. Appl. 4, 173–180, 1970.Google Scholar
  9. 9.
    V.I. Arnold, Lectures on Bifurcations in Versal Families, Russian Math Surveys 27 (1972) 54–123.Google Scholar
  10. 10.
    V.I. Arnold, Critical Points of Smooth Functions, Proceedings International Congress of Mathematicians, v. 1, Vancouver, B.C., 1974, 19–40.Google Scholar
  11. 11.
    V.I. Arnold, Loss of stability of self oscillations close to resonance and versai deformations of equivariant vector fields, Funct. Anal. Appl. 11, 85–92, 1977.Google Scholar
  12. 12.
    R. Aris, Chemical Reactors and Some Bifurcation Phenomena, preprint, Minneapolis, 1977.Google Scholar
  13. 13.
    J. Beddington, C. Free, and J. Lawton , Dynamic Complexity in Predator-Prey Models Framed in Difference Equations, Nature 255, 1975, 58–60.Google Scholar
  14. 14.
    L. Block and J. Franke, Existence of periodic points for maps of S1 Inventiones Math. 22, 69–73, 1973.Google Scholar
  15. 15.
    R.I. Bogdanov, Orbital equivalence of singular points of vector fields on the plane, Funct. Anal. Appl. 10, 316–317, 1976.Google Scholar
  16. 16.
    R.I. Bogdanov, Versal deformations of a singular point of a vector field on a plane in the case of zero eigenvalues, Proceedings of the I.G. Petrovskii Seminar, 2, 37–65, 1976.Google Scholar
  17. 17.
    H. Brolin, Invariant sets under iteration of rational functions, Arkiv för Mathematik, 6, 103–144, 1965.Google Scholar
  18. 18.
    P. Brunovsky, On one parameter families of diffeomorphisms, Comment. Math. Univ. Carolinae 11, 559–582, 1970.Google Scholar
  19. 19.
    F.H. Busse, Magnetohydrodynamics of the Earth’s Dynamo, Annnal Review of Fluid Mechanics, 10, 1978, 435–462.Google Scholar
  20. 20.
    M.L. Cartwright, and J.E. Littlewood, On Non-linear Differential Equations of the Second Order: I. The Equation ÿ + k(l-y2) ẏ + y = bλkcos(λt+a), k large. J. London Math. Soc. 20, 1945, 180–189f.Google Scholar
  21. 21.
    S. Chow, J. Hale, and M. Mallet-Paret, Applications of Generic Bifurcation II, Arch. Rational Mech. Anal., 62, 1976, 209–235.Google Scholar
  22. 22.
    A.E. Cook and P.H. Roberts, The Rikitake two-disc dynamo system. Proc. Camb. Phil. Soc. 68, 1970, 547–569.Google Scholar
  23. 23.
    A. Denjoy, Sur les courbes définies par les equations différentielles à la surface du tore, J. Math. Pures. Appl. [9], 11, 333–375, 1932.Google Scholar
  24. 24.
    F. Dumortier, Singularities of vector fields in the plane, Journal of Differential Equations, 23, 53–106, 1977.Google Scholar
  25. 25.
    M.P. Fatou, Sur les équations fonctionelles, Bull. Soc. Math. France, 47, 161–271, 1919 and 48, 33–94 and 208–314, 1920.Google Scholar
  26. 26.
    M.J. Feigenbaum, The formal development of recursive universality, preprint, Los Alamos, 1977.Google Scholar
  27. 27.
    M.J. Feigenbaum, Quantitative universality for a class of nonlinear transformations, mimeographed, Los Alamos.Google Scholar
  28. 28.
    S.D. Feit, Characteristic exponents and strange attractors, preprint. New Haven, 1978.Google Scholar
  29. 29.
    P.R. Fenstermacher, H.L. Swinney, S.V. Benson, and J.R. Gollub, Bifurcations to Periodic, Quasiperiodic, and Chaotic Regimes in Rotating and Convecting Fluids, preprint, 1977.Google Scholar
  30. 30.
    J.E. Flaherty and F.C. Hoppensteadt, Frequency entrainment of a forced van der Pol oscillator, Studies in Appl. Math, 58, 5–15, 1978.Google Scholar
  31. 31.
    Gantmacher, Theory of Matrices.Google Scholar
  32. 32.
    N.K. Gavrilov and L.P. Siĺnikov, On three dimensional dynamical systems close to systems with a structurally unstable homoclinic curve, Math. USSR, Sb. 17, 467–485, 1972 and 19, 139–156, 1973.Google Scholar
  33. 33.
    A.W. Gillies, On the transformation of singularities and limit cycles of the variational equations of van der Pol, Quart. J. Mech and Applied Math. VII (2), 152–167, 1954.Google Scholar
  34. 34.
    J. Gollub, and H. Swinney, Onset of Turbulence in a Rotating Fluid, Physical Review Letters, 35, 1975, 927–930.Google Scholar
  35. 35.
    M. Golubitsky and D. Schaeffer, A theory for imperfect bifurcation via singularity theory, preprint, 1978.Google Scholar
  36. 36.
    J. Grasman, E.J.M. Veling, and G.M. Willems, Relaxation oscillations governed by a van der Pol equation with periodic forcing term, Siam J. Appl. Math 31 (1976) 667–676.Google Scholar
  37. 37.
    J. Guckenheimer, Bifurcation and catastrophe, Dynamical Systems, ed. M. Peixoto, Academic Press, 95–110, 1973.Google Scholar
  38. 38.
    J. Guckenheimer, A Strange, Strange Attractor, in the Hopf Bifurcation, ed. J. Marsden, M. McCracken, Springer Verlag 1976.Google Scholar
  39. 39.
    J. Guckenheimer, On the bifurcation of maps of the interval, Inventiones Mathematicae, 39 (1977), 165–178.Google Scholar
  40. 40.
    J. Guckenheimer, Bifurcation of quadratic functions, to appear in Proceedings of Conference on Bifurcation Theory and its Applications, November, 1977, N.Y. Academy of Sciences.Google Scholar
  41. 41.
    J. Guckenheimer, G. Oster, and A. Ipaktchi, Dynamics of Density Dependent Population Models, J. of Mathematical Biology, 4, 101–147, 1977.Google Scholar
  42. 42.
    J. Guckenheimer and R.F. Williams, Structural Stability of Lorenz Attractors, to appear.Google Scholar
  43. 43.
    H. Haken, Analogy between higher instabilities in fluids and lasers, Physics Letters 53A, 77–78, 1975.Google Scholar
  44. 44.
    P. Hartman , Ordinary Differential Equations, Wiley, 1964.Google Scholar
  45. 45.
    M. Hassell, J. Lawton, and R. May, Patterns of Dynamical Behavior in Single-species Populations, J. Animal Ecology, 45, 1976, 471–86.Google Scholar
  46. 46.
    C. Hayashi, Nonlinear Oscillations in Physical Systems, McGraw-Hill, New York, 1964.Google Scholar
  47. 47.
    M. Henon, A two dimensional mapping with a strange attractor, Comm. Math. Phys. 50, 69–78, 1976.Google Scholar
  48. 48.
    M.R. Herman, Measure de Lebesque et nombre de rotation, Geometry and Topology, Springer Lecture Notes in Mathematics 597, ed. J. Palis and M. doCarmo, 1977, 271–293.Google Scholar
  49. 49.
    M.R. Herman, Sur la conjugaison différentiable des difféomorphismes du cercle á des rotations, Thesis, Orsay, 1976.Google Scholar
  50. 50.
    M. Hirsch, C. Pugh, and M. Snub, Invariant Manifolds, Springer Lecture Notes in Mathematics 583, 1977.Google Scholar
  51. 50a.
    P. Holmes, A nonlinear oscillator with a strange attractor, preprint, Ithaca, 1977.Google Scholar
  52. 51.
    P.J. Holmes, and D.A. Rand, Bifurcations of the Forced van der Pol Oscillator, mimeographed, Southampton, 1976.Google Scholar
  53. 52.
    G. Iooss and D. Joseph, Bifurcation and stability of n T-periodic solutions branching from T-periodic solutions at points of resonance, preprint, 1978.Google Scholar
  54. 53.
    M.V. Jakobson, On smooth mappings of the circle into itself, Math USSR Sb., 14, 161–185, 1971.Google Scholar
  55. 54.
    D. Joseph, Stability of Fluid Motions, 2 vols.. Springer-Verlag, 1976.Google Scholar
  56. 55.
    J.L. Kaplan and J.A. Yorke, Preturbulence: a regime observed in a fluid flow of Lorenz, preprint. College Park, 1977.Google Scholar
  57. 56.
    N. Keyfitz, Introduction to the mathematics of populations. Addison-Wesley, 1968.Google Scholar
  58. 57.
    N. Kopell and L. Howard, Bifurcations under nongeneric conditions, Advances in Mathematics, 13, 274–283, 1974.Google Scholar
  59. 58.
    S. Lefschetz, On a theorem of Bendixson, J. Diff. Eq. 4, 66–101, 1968.Google Scholar
  60. 59.
    M. Levi, Thesis, Courant Institute of New York University, 1978.Google Scholar
  61. 60.
    N. Levinson, A Second Order Differential Equation with Singular Solutions, Annals of Math, 50, 1949, 127, 153.Google Scholar
  62. 61.
    T. Li and J.A. Yorke, Period three implies chaos. Am. Math. Monthly, 82, 1975, 985–992.Google Scholar
  63. 62.
    J.E. Littlewood, On nonlinear differential equations of the second order: III. The equation ÿ-k(l-y2) ẏ+y = bµkcos(µt+α) for large k, and its generalizations, Acta Mathematica, 97 (1957), 268–308.Google Scholar
  64. 63.
    J.E. Littlewood, On nonlinear differential equations of the second order: IV. The general equation ÿ + kf(y)ẏ+g(y) = bk(p(φ)), φ=t+α Acta Mathematica 98, 1–110, 1957.Google Scholar
  65. 64.
    E.N. Lorenz, Deterministic nonperiodic flow, J. Atmospheric Sciences, 20, 1963, 130–141.Google Scholar
  66. 65.
    M.C. Mackey and L. Glass, Oscillation and chaos in physiological control systems, Science 197, 1977, 287–289.Google Scholar
  67. 66.
    J.E. Marsden, Qualitative methods in bifurcation theory, preprint, Berkeley, 1978.Google Scholar
  68. 67.
    J. Marsden and M. McCracken, The Hopf Bifurcation Theorem and Its Applications, Springer Verlag, 1976.Google Scholar
  69. 68.
    R.M. May, Simple Mathematical Models with very Complicated Dynamics, Nature 261, 1976, 459–466.Google Scholar
  70. 69.
    W. de Melo, Moduli of stability of two dimensional diffeomorphisms, preprint, Rio de Janeiro, 1978.Google Scholar
  71. 70.
    J. Milnor and W. Thurston, On iterated maps of the interval I and II, mimeographed, Princeton, 1977.Google Scholar
  72. 71.
    P.J. Myrberg, Iteration der reelen polynome zweiten grades III, Annales Acad. Scient. Fennicae, 336/33, 1–18, 1963.Google Scholar
  73. 72.
    S. Newhouse, Diffeomorphisms with Infinitely Many Sinks, Topology, 12, 1974, 9–18.Google Scholar
  74. 73.
    S. Newhouse, On simple arcs between structurally stable flows, Dynamical Systems — Warwick 1974, Springer Lecture Notes in Mathematics 468, 1975, 209–233.Google Scholar
  75. 74.
    S.E. Newhouse, The abundance of wild hyperbolic sets and non-smooth stable sets for diffeomorphisms, mimeographed, IHES, 1977.Google Scholar
  76. 75.
    S. Newhouse and J. Palis, Bifurcations of Morse-Smale dynamical systems, Dynamical Systems, ed. M. Peixoto, Academic Press, New York, 1973, 303–366.Google Scholar
  77. 76.
    S. Newhouse and J. Palis, Cycles and Bifurcation Theory, Asterisque 31, 43–140, 1976.Google Scholar
  78. 77.
    S. Newhouse, J. Palis, and F. Takens, Stable Arcs of Diffeomorphisms Bull. Am. Math. Soc. 82, 499–502, 1976 and to appear.Google Scholar
  79. 78.
    S. Newhouse and M. Peixoto, There is a simple arc between any two Morse-Smale flows. Asterisque 31, 15–42, 1976.Google Scholar
  80. 79.
    A.J. Nicholson, The Self-adjustment of Populations to Change. Cold Spring Harbor Symp. Quant. Biol. XXII, 1957, 153–173.Google Scholar
  81. 80.
    G. Oster, Internal Variables in Population Dynamics, Some Questions in Mathematical Biology VII, 1976, 37–68.Google Scholar
  82. 81.
    G. Oster, Lectures in Population Dynamics, Modern Modelling of Continuum Phenomena, SIAM, 1977.Google Scholar
  83. 82.
    J. Palis, On Morse-Smale dynamical systems. Topology, 8, 385–405, 1969.Google Scholar
  84. 83.
    J. Palis, Some developments on stability and bifurcations of dynamical systems, Geometry and Topology, Springer Lecture Notes in Mathematics 597, ed. J. Palis and M. doCarmo, 1977, 495–506.Google Scholar
  85. 84.
    J. Palis, A differentiable invariant of topological conjugacies and moduli of stability. Asterisque, 49–50, 1978.Google Scholar
  86. 85.
    J. Palis and F. Takens, Topological equivalence of normally hyperbolic dynamical systems, Topology, 16, 335–345, 1977.Google Scholar
  87. 86.
    M. Peixoto, Structural Stability on Two Dimensional Manifolds, Topology 1, (1962) 101–120.Google Scholar
  88. 87.
    P. Plikin, Sources and Sinks in Axiom A Diffeomorphisms on Surfaces. Mat. Sbornik 94, 1974, 243–264.Google Scholar
  89. 88.
    H. Poincaré, Les méthodes nonvelles de la mécanique céleste, vols. I, II, III, Paris 1892, 1893, 1899; repreint, Dover, New York, 1957.Google Scholar
  90. 89.
    P. Rabinowitz, ed., Applications of Bifurcation Theory, Academic Press, 1977.Google Scholar
  91. 90.
    D. Ruelle, The Lorenz attractor and the problem of turbulence, conference report Bielefeld, 1975.Google Scholar
  92. 91.
    D. Ruelle, Sensitive dependence on initial condition and turbulent behavior of dynamical systems, conference on Bifurcation Theory and its Applications, N.Y. Academy of Sciences, 1977.Google Scholar
  93. 92.
    D. Ruelle and F. Takens, On the nature of turbulence, Comm. Math. Phys. 20, 167–192 (1971).Google Scholar
  94. 93.
    A.N. Šarkovskii, Coexistence of cycles of a continuous map of the line into itself, Ukr. Math 2.1, 61–71, 1964.Google Scholar
  95. 94.
    A. Seidenberg, Reduction of singularities of the differential equation Ady = Bdx, Am. J. Math, 90, 248–269, 1968.Google Scholar
  96. 95.
    R. Shaw, Strange attractors, chaotic behavior, and information flow, preprint, Santa Cruz, 1978.Google Scholar
  97. 96.
    A.N. Shoshitaishvili, Bifurcations of topological type at singular points of parametrized vector fields, Funct. anal. appl. 6, 169–170, 1972.Google Scholar
  98. 97.
    C.L. Siegel, Über die normalform analytischer Differential-gleichungen in der Nähe einer Gleichgewichtslosung, Göttinger Nachrichten der Akad. der Wissenschaften, 1952, 21–30.Google Scholar
  99. 98.
    L. Silnikov, On a new type of bifurcation of multidimensional dynamical systems, Sov. Math. Dokl, 10, 1368–1371, 1969.Google Scholar
  100. 99.
    L.P. Sil’nikov, A contribution to the probem of the structure of an extended neighborhood of a structurally stable equilibrium of saddle-focus type, Math USSR Sb. 10, 91–102, 1970.Google Scholar
  101. 100.
    D. Singer, Stable orbits and bifurcation of maps of the interval, preprint, 1977.Google Scholar
  102. 101.
    S. Smale, Diffeomorphisms with many periodic proints, in Differential and Combinatorial Topology, Princeton, 1965, 63–80.Google Scholar
  103. 102.
    S. Smale, Differentiable Dynamical Systems, Bull. Am. Math. Soc. 73, 1967, 747–817.Google Scholar
  104. 103.
    S. Smale, On the Differential Equations of Species in Competition, J. Math. Biol. 3, 5–7 (1976).Google Scholar
  105. 104.
    J. Sotomayor, Bifurcations of vector fields on two dimensional manifolds, Publ. I.H.E.S. #43, 1–46, 1973.Google Scholar
  106. 105.
    J. Sotomayor, Generic bifurcations of dynamical systems, Dynamical Systems, ed. M. Peixoto, Academic Press, 1973, 561–582.Google Scholar
  107. 106.
    J. Sotomayor, Structural stability and bifurcation theory, Dynamical Systems, ed. M. Peixoto, Academic Press, 1973, 549–560.Google Scholar
  108. 107.
    S. Sternberg, Local contractions and a theorem of Poincaré, Am. J. Math 79, 787–789, 1957.Google Scholar
  109. 108.
    S. Sternberg, On the structure of local homeomorphisms of Euclidean n-space II and III, Am. J. Math., 80, 623–631, 1958 and 81, 578–604, 1959.Google Scholar
  110. 109.
    J.J. Stoker, Nonlinear vibrations in mechanical and electrical systems, Wiley-Interscience.Google Scholar
  111. 110.
    F. Takens, Partially Hyperbolic Fixed Points, Topology, 10, 133–147, 1971.Google Scholar
  112. 111.
    F. Takens, Singularities of Vector Fields, Publ. I.H.E.S. #43, 47–100, 1973.Google Scholar
  113. 112.
    F. Takens, Integral curves near mildly degenerate singular points of vector fields, Dynamical Systems, ed. M. Peixoto, Academic Press, 1973, 599–617.Google Scholar
  114. 113.
    F. Takens, A nonstabilizable jet of a singularity of a vector field, Dynamical Systems, ed. M. Peixoto, Academic Press, 1973, 583–598.Google Scholar
  115. 114.
    F. Takens, Forced Oscillations and Bifurcations, Applications of Global Analysis, Communications of Maths. Institute, Rijksuniversiteit, Utrecht, 3 1974, 1–59.Google Scholar
  116. 115.
    F. Takens, Constrained equations, a study of implicit differential equations and their discontinuous solutions, Mathematisch Instituut Rijksuniversiteit Groningen, Report ZW-75–03, (1975).Google Scholar
  117. 116.
    R. Thorn, Structural Stability and Morphogenesis, W.A. Benjamin, Inc., Reading, Mass. 1975.Google Scholar
  118. 117.
    A. Uppal, W.H. Ray, A.B. Poore, On the dynamic behavior of continuous stirred tank reactors, Chemical Engineering Science, 29, 967–985, 1974.Google Scholar
  119. 118.
    R.F. Williams, The Structure of Lorenz Attractors, to appear,Google Scholar

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© Springer-Verlag Berlin Heidelberg 1980

Authors and Affiliations

  • John Guckenheimer
    • 1
  1. 1.Santa CruzUSA

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