Bifurcations and stability of families of diffeomorphisms

Sheldon E. Newhouse; Jacob Palis; Floris Takens

Publications Mathématiques de l'IHÉS (1983)

  • Volume: 57, page 5-71
  • ISSN: 0073-8301

How to cite

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Newhouse, Sheldon E., Palis, Jacob, and Takens, Floris. "Bifurcations and stability of families of diffeomorphisms." Publications Mathématiques de l'IHÉS 57 (1983): 5-71. <http://eudml.org/doc/103989>.

@article{Newhouse1983,
author = {Newhouse, Sheldon E., Palis, Jacob, Takens, Floris},
journal = {Publications Mathématiques de l'IHÉS},
keywords = {center manifolds; Hopf point; quasi-transversal intersection; Hopf bifurcations; quasi-hyperbolic periodic orbits; mild conjugacy; saddle- node; strong unstable foliation; flip},
language = {eng},
pages = {5-71},
publisher = {Institut des Hautes Études Scientifiques},
title = {Bifurcations and stability of families of diffeomorphisms},
url = {http://eudml.org/doc/103989},
volume = {57},
year = {1983},
}

TY - JOUR
AU - Newhouse, Sheldon E.
AU - Palis, Jacob
AU - Takens, Floris
TI - Bifurcations and stability of families of diffeomorphisms
JO - Publications Mathématiques de l'IHÉS
PY - 1983
PB - Institut des Hautes Études Scientifiques
VL - 57
SP - 5
EP - 71
LA - eng
KW - center manifolds; Hopf point; quasi-transversal intersection; Hopf bifurcations; quasi-hyperbolic periodic orbits; mild conjugacy; saddle- node; strong unstable foliation; flip
UR - http://eudml.org/doc/103989
ER -

References

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  1. [1] A. A. ANDRONOV, E. A. LEONTOVICH, I. I. GORDON, A. G. MAIER, Theory of bifurcations of dynamic systems on a plane, Isr. Program Scientific Transl., Jerusalem, 1971. 
  2. [2] L. BLOCK, J. FRANKE, Existence of periodic points of maps of S1, Invent. Math., 22 (1973), 69-73. Zbl0272.58005MR50 #11326
  3. [3] Th. BRÖCKER, L. C. LANDER, Differentiable germs and catastrophes, London Math. Soc. Lecture Notes, 17, Cambridge Univ. Press, 1975. Zbl0302.58006MR58 #13132
  4. [4] P. BRUNOVSKI, On one-parameter families of diffeomorphisms, I and II, Comment. Mat. Univ. Carolinae, 11 (3) (1970), 559-582 ; 12 (4) (1971), 765-784. Zbl0202.23104
  5. [5] P. BRUNOVSKI, Generic properties of the rotation number of one-parameter diffeomorphisms of the circle, Czech. Math. J., 24 (1974), 74-90. Zbl0308.58007MR49 #11567
  6. [6] A. DENJOY, Les trajectoires à la surface du tore, C. r. Acad. Sci., 223 (1946), 5-8. Zbl0063.01085MR8,34e
  7. [7] J. GUCKENHEIMER, One-parameter families of vector fields on two-manifolds : another nondensity theorem, Dynamical Systems, Academic Press, 1973, 111-128. Zbl0285.58007MR49 #11565
  8. [8] P. HARTMAN, On local homeomorphisms of Euclidean spaces, Bol. Soc. Mat. Mexicana, 5 (2) (1960), 220-241. Zbl0127.30202MR25 #5253
  9. [9] M. HERMAN, Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations, Publ. Math. I.H.E.S., 49 (1979), 5-234. Zbl0448.58019MR81h:58039
  10. [10] M. HERMAN, Mesure de Lebesgue et nombre de rotation, Geometry and Topology, Springer Lecture Notes in Math., 597 (1977), 271-293. Zbl0366.57007MR56 #16682
  11. [11] M. W. HIRSCH, C. C. PUGH, Stable manifolds and hyperbolic sets, Proc. Symp. Pure Math. A.M.S., 14 (1970), 133-164. Zbl0215.53001MR42 #6872
  12. [12] M. W. HIRSCH, C. C. PUGH, M. SHUB, Invariant manifolds, Springer Lecture Notes in Math., 583 (1977). Zbl0355.58009MR58 #18595
  13. [13] I. KUPKA, Contributions à la théorie des champs génériques, Contributions to differential equations, vol. 2 (1963), 475-484 and vol. 3 (1964), 411-420. Zbl0149.41003MR29 #2818a
  14. [14] D. C. LEWIS, Formal power series transformations, Duke Math. J., 5 (1939), 794-805. Zbl0022.32703MR1,123cJFM65.1242.03
  15. [15] I. P. MALTA, Hyperbolic Birkhoff centers, Trans. A.M.S., 262 (1980), 181-193 ; announced in An. Acad. Brasil. Ciências, 51 (1979), (I), 27-29. Zbl0458.58017
  16. [16] A. MANNING, There are no new Anosov diffeomorphisms on tori, Amer. J. Math., 96 (1974), 422-429. Zbl0242.58003MR50 #11324
  17. [17] J. E. MARSDEN, M. MCCRACKEN, The Hopf bifurcation and its applications, Appl. Math. Sci., 19, Springer-Verlag, 1976. Zbl0346.58007MR58 #13209
  18. [18] W. C. de MELO, Moduli and stability of two-dimensional diffeomorphisms, Topology, 19 (1980), 9-21. Zbl0447.58025MR81c:58047
  19. [18a] W. de MELO, J. PALIS and S. VAN STRIEN, Characterizing diffeomorphisms with modulus of stability one, Dynamical Systems and Turbulence, Warwick 1980, Springer Lecture Notes in Math., 898 (1981), 266-286. Zbl0482.58022
  20. [19] S. NEWHOUSE, J. PALIS, Bifurcations of Morse-Smale dynamical systems, Dynamical Systems, Academic Press, 1973, 303-366. Zbl0279.58011MR48 #12600
  21. [20] S. NEWHOUSE, J. PALIS, Cycles and bifurcation theory, Astérisque, 31 (1976), 43-140. Zbl0322.58009MR58 #24366
  22. [21] S. NEWHOUSE, J. PALIS, F. TAKENS, Stable arcs of diffeomorphisms, Bull. A.M.S., 82 (1976), 499-502. Zbl0339.58008MR53 #6640
  23. [22] S. NEWHOUSE, The abundance of wild hyperbolic sets and non-smooth stable sets, Publ. Math. I.H.E.S., 50 (1979), 101-152. Zbl0445.58022MR82e:58067
  24. [23] R. PALAIS, Local triviality of the restriction map for embeddings, Comm. Math. Helvet., 34 (1960), 305-312. Zbl0207.22501MR23 #A666
  25. [24] J. PALIS, On Morse-Smale dynamical systems, Topology, 8 (1969), 385-405. Zbl0189.23902MR39 #7620
  26. [25] J. PALIS, S. SMALE, Structural stability theorems, Proc. Symp. Pure Math. A.M.S., 14 (1970), 223-232. Zbl0214.50702MR42 #2505
  27. [26] J. PALIS, F. TAKENS, Topological equivalence of normally hyperbolic dynamical systems, Topology, 16 (1977), 335-345. Zbl0391.58015MR57 #14049
  28. [26a] J. PALIS and F. TAKENS, Stability of parametrized families of gradient vector fields, Preprint IMPA, to appear. Zbl0533.58018
  29. [27] J. PALIS, A differentiable invariant of topological conjugacies and moduli of stability, Astérisque, 51 (1978), 335-346. Zbl0396.58015MR58 #13189
  30. [28] J. PALIS, Moduli of stability and bifurcation theory, Proc. Int. Congres of Math. Helsinki (1978), 835-839. Zbl0467.34032MR81c:58046
  31. [29] H. POINCARÉ, Œuvres complètes, t. 1, Gauthier-Villars, 1952, 137-158. 
  32. [30] C. C. PUGH, M. SHUB, Linearization of normally hyperbolic diffeomorphisms and flows, Invent. Math., 10 (1970), 187-198. Zbl0206.25802MR44 #1055
  33. [31] C. ROBINSON, Global structural stability of a saddle-node bifurcation, Trans. A.M.S., 236 (1978), 155-172. Zbl0406.58022MR57 #7683
  34. [32] D. RUELLE, F. TAKENS, On the nature of turbulence, Comm. Math. Phys., 20 (1971), 167-192 ; A note concerning our paper on the nature of turbulence, Comm. Math. Phys., 23 (1971), 343-344. Zbl0223.76041MR44 #1297
  35. [33] S. SMALE, Stable manifolds for differential equations and diffeomorphisms, Ann. Scuola Norm. Sup. Pisa, 18 (1963), 97-116. Zbl0113.29702MR29 #2818b
  36. [34] S. SMALE, Differentiable dynamical systems, Bull. A.M.S., 73 (1967), 747-817. Zbl0202.55202MR37 #3598
  37. [35] J. SOTOMAYOR, Generic one parameter families of vector fields, Publ. Math. I.H.E.S., 43 (1974), 4-56. Zbl0279.58008
  38. [36] J. SOTOMAYOR, Generic bifurcations of dynamical systems, Dynamical Systems, Academic Press 1973, 549-560. 
  39. [37] S. J. VAN STRIEN, Center manifolds are not C∞, Math. Z., 166 (1979), 143-145. Zbl0403.58021MR80j:58049
  40. [38] F. TAKENS, Partially hyperbolic fixed points, Topology, 10 (1971), 133-147. Zbl0214.22901MR46 #6399
  41. [39] F. TAKENS, Normal forms for certain singularities of vector fields, Ann. Inst. Fourier, 23 (1973), (2), 163-195. Zbl0266.34046MR51 #1872
  42. [40] F. TAKENS, Forced oscillations and bifurcations, Applications of Global Analysis I, Comm. of the Math. Inst., R.U. Utrecht (Holland), 1974. Zbl1156.37315MR57 #17720
  43. [41] F. TAKENS, Global phenomena in bifurcations of dynamical systems with simple recurrence, Jber. d. Dt. Math.-Verein, 81 (1979), 87-96. Zbl0419.58012MR80f:58034
  44. [42] R. WILLIAMS, The D.A. map of Smale and structural stability, Proc. Symp. Pure Math. A.M.S., 14 (1970), 329-334. Zbl0213.50303MR41 #9296

Citations in EuDML Documents

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  1. Hammouda Annabi, Un invariant de conjugaison pour certaines familles à un paramètre de champs de vecteurs ou de difféomorphismes
  2. Mohamed Larbi Annabi, Dépendance continue par rapport aux paramètres dans la bifurcation de Hopf-Takens de codimension 3
  3. Pedro Duarte, Plenty of elliptic islands for the standard family of area preserving maps
  4. Eduardo Colli, Infinitely many coexisting strange attractors
  5. Lorenzo J. Díaz, Raúl Ures, Persistent homoclinic tangencies and the unfolding of cycles
  6. Lluis Alsedà, Antonio Falcó, A characterization of the kneading pair for bimodal degree one circle maps
  7. Eleonora Catsigeras, Heber Enrich, Homoclinic tangencies near cascades of period doubling bifurcations
  8. S. Sergio Plaza, Global stability of saddle-node bifurcation of a periodic orbit for vector fields
  9. Rodrigo Bamon, Rafael Labarca, Ricardo Mañé, Maria-José Pacífico, The explosion of singular cycles
  10. M. J. Dias Carneiro, Jacob Palis, Bifurcations and global stability of families of gradients

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