Skip to main content
SpringerLink
Log in

Lectures on Dynamical Systems

  • Chapter
Dynamical Systems

Part of the book series: Progress in Mathematics ((PM,volume 8))

Abstract

A basic question in the theory of dynamical systems is to study the asymptotic behaviour of orbits. This has led to the development of many different subjects in mathematics. To name a few, we have ergodic theory, hamiltonian mechanics, and the qualitative theory of differential equations.

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

Access this chapter

Log in via an institution
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. R. Abraham and S. Smale, Non-generieity of Ω-stability, Proc. Symp. in Pure Math. 14 (1970), Amer. Math. Soc. 5–8.

    Article  Google Scholar 

  2. D. V. Anosov, Geodesic flows on closed Riemannian manifolds with negative curvature, Proc. Steklov Inst. Math. 90 (1967).

    Google Scholar 

  3. G. D. Birkhoff, Dynamical systems, Amer. Math. Soc. Colloq. Publ., vol. 9, Amer. Math. Soc., Providence, R. I., 1927.

    Google Scholar 

  4. R. Bowen, Markov partitions for Axiom A diffeoraorphisms, Amer. Jour. Math. 92 (1970), 725–747.

    Article  Google Scholar 

  5. R. Bowen, Markov partitions and minimal sets for Axiom A diffeo — morphisms, Amer. Jour. Math. 92, (1970), 907–918.

    Article  Google Scholar 

  6. R. Bowen, Some systems with unique equilibrium states; Math. Syp. Theory, vol. 8 (1974), 193–202.

    Article  Google Scholar 

  7. R. Bowen, Equilibrium states and the ergodic theory of Anosov-diffeomorphisms, Springer Lecture Notes in Math. 470 (1975).

    Google Scholar 

  8. R. Bowen, A horseshoe with positive measure, Inventiones Math. 29 (1975), 203–204.

    Article  Google Scholar 

  9. R. Bowen, On Axiom A diffeomorphisms, Notes of NSF Regional Conference, North Dakota State Univ., Fargo, N.D., June, 1977.

    Google Scholar 

  10. A. Dold, Lectures on algebraic topology, Springer-Verlag, New York, 1972.

    Book  Google Scholar 

  11. J. Franks, Anosov diffeomorphisms, Proc. Symp. Pure Math. 14 (1970), Amer. Math. Soc, 61–93.

    Article  Google Scholar 

  12. J. Gollub, T. Brunner, and B. Danly, Periodicity and choos in coupled non-linear oscillators, Science, Vol. 200, No. 4337, April 7, 1978, 48–50.

    Article  Google Scholar 

  13. O. Gurel and O. Rossler, Bifurcation Theory and its application to scientific disciplines, Proc. of New York Acad. of Science meeting, Nov., 1977.

    Google Scholar 

  14. J. Hale, Generic bifurcation with applications, lecture notes, Brown Univ., October, 1976.

    Google Scholar 

  15. P. Hartman, Ordinary differential equations, John Wiley and Sons, N. Y., 1964.

    Google Scholar 

  16. M. Hirsch, J. Palis, C. Pugh, and M. Shub, Neighborhoods of hyperbolic sets, Invent. Math. 9 (1970), 121–134.

    Article  Google Scholar 

  17. M. Hirsch and C. Pugh, Stable manifolds and hyperbolic sets, Proc. Symp. in Pure Math. 14 (1970), Amer. Math. Soc, Providence, R. I., 133–163.

    Article  Google Scholar 

  18. M. Hirsch, C. Pugh, and M. Shub, Invariant manifolds, Springer Lecture notes in Math. 583, 1977.

    Google Scholar 

  19. M. Irwin, On the stable manifold theorem, Bull. London Math. Soc. 2 (1970), 196–198.

    Article  Google Scholar 

  20. N. Levinson, A second order equation with singular solutions, Ann. of Math. 50 (1949), 127–153.

    Article  Google Scholar 

  21. R. Mane, The stability conjecture on two-dimensional manifolds, Topology, to appear.

    Google Scholar 

  22. R. Mane, Expansive homeomorphisms and topological dimension, preprint, IMPA, Rio de Janeiro, Brazil.

    Google Scholar 

  23. A. Manning, There are no new Anosov diffeomorphisms on tori, Amer. Jour. Math. 96 (1974), 422–429.

    Article  Google Scholar 

  24. J. Marsden and M. McCracken, The Hopf Bifurcation and its applications, Springer Lecture Notes in Appl. Math. Sci. 19 (1976).

    Book  Google Scholar 

  25. J. Moser, Stable and random motions in dynamical systems, Ann. of Math. Studies 77, Princeton Univ.-Press, Princeton, N. J., 1973.

    Google Scholar 

  26. S. Newhouse, Non-density of Axiom A (a) on S2, Proc. Symp. in Pure Math. 14 (1970), 191–202.

    Article  Google Scholar 

  27. S. Newhouse, On codimension one Anosov diffeomorphisms, Amer. Jour. Math. 92 (1970), 761–770.

    Article  Google Scholar 

  28. S. Newhouse, Hyperbolic limit sets, Trans. AMS 167 (1972), 125–150.

    Article  Google Scholar 

  29. S. Newhouse, Diffeomorphisms with infinitely many sinks, Topology 13 (1974), 9–18.

    Article  Google Scholar 

  30. S. Newhouse, Quasi-elliptic periodic points in conservative dynamical systems, Amer. Jour. Math. 99 (1977), 1061–1087.

    Article  Google Scholar 

  31. S. Newhouse, Topological entropy and Hausdorff dimension for area preserving diffeomorphisms of surfaces, Asterisque 51 (1978), 323–334.

    Google Scholar 

  32. S. Newhouse, The abundance of wild hyperbolic sets and non-smooth stable sets for diffeomorphisms, Publ. Math. IHES, to appear.

    Google Scholar 

  33. S. Newhouse, and J. Palis, Bifurcations, of Morse-Smale dynamical systems, Dynamical Systems, ed. M. Peixoto, Academic Press, 1973, 303–366.

    Google Scholar 

  34. S. Newhouse, and J. Palis, Cycles and Bifurcation Theory, Asterisque 31, (1976), 43–141.

    Google Scholar 

  35. S. Newhouse, J. Palis, and F. Takens, Stable arcs of diffeomorphisms.I, II, to appear.

    Google Scholar 

  36. S. Newhouse, D. Ruelle, and F. Takens, The occurrence of strange Axiom A attractors near constant vector fields on tori, to appear.

    Google Scholar 

  37. Z. Nitecki, Differentiable dynamics, MIT Press, 1971.

    Google Scholar 

  38. D. Ornstein, Ergodic theory, randomness, and dynamical systems, Yale Univ. Press, 1974.

    Google Scholar 

  39. J. Palis, On Morse-Smale dynamical systems, Topology 8 (1969), 385–405.

    Article  Google Scholar 

  40. J. Palis, Local structure of hyperbolic fixed points in Banach spaces, Anais da Acad. Bras. de Ciencias, 1968.

    Google Scholar 

  41. J. Palis, and W. Melo, Lectures on dynamical systems, IMPA, Rua Liuz de Camoes 68, Rio de Janeiro, Brazil, 1975.

    Google Scholar 

  42. V. Pliss, On a conjecture to due Smale, Diff. Equations, 7 and 8 (1971 and 1972).

    Google Scholar 

  43. R. Plykin, Sources and sinks for A-diffeomorphisms, Math. USSR Sbornik 23 (1974), 233–253.

    Article  Google Scholar 

  44. C. Pugh, An improved closing lemma and a general density theorem, Amer. Jour. Math. 89(1967), 1010–1021.

    Article  Google Scholar 

  45. C. Pugh, On a theorem of P. Hartman, Amer, Jour. of Math. 91 (1969), 363–367.

    Article  Google Scholar 

  46. C. Pugh, and R. C. Robinson, The C1 closing lemma including Hamiltonians, preprint, Univ. of Calif., Berkeley 1973.

    Google Scholar 

  47. D. Ruelle, Statistical mechanics on à compact set with Zu action satisfying expansiveness and specification, Trans. AMS 185 (1973), 237–251.

    Article  Google Scholar 

  48. D. Ruelle, A measure associated with Axiom A attractors, Amer. Jour. Math. 98 (1976), 619–654.

    Article  Google Scholar 

  49. D. Ruelle, Statistical mechanics and dynamical systems, Duke Univ. Math. Series III, Duke University, Durham, NC, 1977, 1–107.

    Google Scholar 

  50. J. Robbin, A structural stability theorem, Ann. Math. 94 (1971), 447–493.

    Article  Google Scholar 

  51. R. C. Robinson, Structural stability of C diffeomorphisms, Jour. Diff. Equations 22 (1976), 28–73.

    Article  Google Scholar 

  52. C. Simon, Springer Lecture Notes in Math. 206 (1971), 94–97.

    Article  Google Scholar 

  53. Ja. Sinai, Construction of Markov partitions, Funct. Anal. and its appl. 2 (1968), 70–80.

    Article  Google Scholar 

  54. M. Shub, Springer Lecture Notes in Math. 206 (1971), 28–29 and 39–40.

    Article  Google Scholar 

  55. S. Smale, Diffeormorphisms with many periodic points, Differential and Combinatorial Topology, Princeton Univ. Press, Princeton, N.J. 1975, 63–80.

    Google Scholar 

  56. S. Smale, Differentiable dynamical systems, Bull. AMS 73 (1967), 747–817.

    Article  Google Scholar 

  57. S. Smale, The Ω-stability theorem, Proc. Symp. in Pure Math. 14, (1970), Amer. Math. Soc., 289–298.

    Article  Google Scholar 

  58. S. Sternberg, On the structure of local homeomorphisms of Euclidean n-space II, Amer. Jour. Math. 80 (1958), 623–631.

    Article  Google Scholar 

  59. J. Stoker, Non-linear vibrations, Interscience Publishers, New York, 1950.

    Google Scholar 

  60. F. Takens, Partially hyperbolic fixed points, Topology 10 (1971) 133–147.

    Article  Google Scholar 

  61. R. Williams, One dimensional non-wandering sets, Topology 6 (1967), 473–487.

    Article  Google Scholar 

  62. R. Williams, Classification of one dimensional attractors, Proc. Symp. Pure Math 14 (1970), 341–363.

    Article  Google Scholar 

  63. R. Williams, Expanding attractors, Publ. Math. IHES 43 (1974), 169–203.

    Article  Google Scholar 

  64. J. Franks, Differentiably Ω-stable diffeomorphisms, Topology 11 (1972), 107–113.

    Article  Google Scholar 

  65. J. Franks, Absolutely structurally stable diffeomorphisms Proc. AMS 37 (1973), 293–296.

    Google Scholar 

  66. J. Franks, Time-dependent stable diffeomorphisms, Inventiones Math. 24 (1974), 163–172.

    Article  Google Scholar 

  67. J. Guckenheimer, Absolutely Ω-stable diffeomorphisms, Topology 11 (1972), 195–197.

    Article  Google Scholar 

  68. A. Manning, Axiom A diffeomorphisms have rational zeta functions, Bull. London Math. Soc. 3(1971), 215–220.

    Article  Google Scholar 

  69. J. Palis and S. Smale, Structural stability theorems, Proc. Symp. in Pure Math. 14 (1970), Amer. Math. Soc., 223–232.

    Article  Google Scholar 

  70. R. Bowen and D. Ruelle, The ergodic theory of Axiom A flows, Inventiones Math. 29 (1975), 181–202.

    Article  Google Scholar 

Download references

Authors
  1. Sheldon E. Newhouse
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

C. Marchioro

Rights and permissions

Reprints and permissions

Copyright information

© 1980 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Newhouse, S.E. (1980). Lectures on Dynamical Systems. In: Marchioro, C. (eds) Dynamical Systems. Progress in Mathematics, vol 8. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4899-3743-8_5

Download citation

  • DOI: https://doi.org/10.1007/978-1-4899-3743-8_5

  • Publisher Name: Birkhäuser, Boston, MA

  • Print ISBN: 978-0-8176-3024-9

  • Online ISBN: 978-1-4899-3743-8

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation