Newhouse, S.; Palis, J.; Takens, F. Bifurcations and stability of families of diffeomorphisms. (English) Zbl 0518.58031 Publ. Math., Inst. Hautes Étud. Sci. 57, 5-71 (1983). Page: −5 −4 −3 −2 −1 ±0 +1 +2 +3 +4 +5 Show Scanned Page Cited in 11 ReviewsCited in 176 Documents MSC: 37G99 Local and nonlocal bifurcation theory for dynamical systems 37D15 Morse-Smale systems 34D30 Structural stability and analogous concepts of solutions to ordinary differential equations 37C85 Dynamics induced by group actions other than \(\mathbb{Z}\) and \(\mathbb{R}\), and \(\mathbb{C}\) 37C75 Stability theory for smooth dynamical systems Keywords:center manifolds; Hopf point; quasi-transversal intersection; Hopf bifurcations; quasi-hyperbolic periodic orbits; mild conjugacy; saddle- node; strong unstable foliation; flip Citations:Zbl 0339.58008; Zbl 0355.58009 × Cite Format Result Cite Review PDF Full Text: DOI Numdam EuDML References: [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] L. Block, J. Franke, Existence of periodic points of maps of S1,Invent. Math.,22 (1973), 69–73. · Zbl 0272.58005 · doi:10.1007/BF01425575 [3] Th. Bröcker, L. C. Lander, Differentiable germs and catastrophes,London Math. Soc. Lecture Notes,17, Cambridge Univ. Press, 1975. · Zbl 0302.58006 [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. [5] P. Brunovski, Generic properties of the rotation number of one-parameter diffeomorphisms of the circle,Czech. Math. J.,24 (1974), 74–90. · Zbl 0308.58007 [6] A. Denjoy, Les trajectoires à la surface du tore,C. r. Acad. Sci.,223 (1946), 5–8. · Zbl 0063.01085 [7] J. Guckenheimer, One-parameter families of vector fields on two-manifolds: another nondensity theorem,Dynamical Systems, Academic Press, 1973, 111–128. · Zbl 0285.58007 [8] P. Hartman, On local homeomorphisms of Euclidean spaces,Bol. Soc. Mat. Mexicana,5 (2) (1960), 220–241. · Zbl 0127.30202 [9] M. Herman, Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations,Publ. Math. I.H.E.S.,49 (1979), 5–234. [10] M. Herman, Mesure de Lebesgue et nombre de rotation,Geometry and Topology, Springer Lecture Notes in Math.,597 (1977), 271–293. · Zbl 0366.57007 [11] M. W. Hirsch, C. C. Pugh, Stable manifolds and hyperbolic sets,Proc. Symp. Pure Math. A.M.S.,14 (1970), 133–164. · Zbl 0215.53001 [12] M. W. Hirsch, C. C. Pugh, M. Shub, Invariant manifolds,Springer Lecture Notes in Math.,583 (1977). [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. [14] D. C. Lewis, Formal power series transformations,Duke Math. J.,5 (1939), 794–805. · Zbl 0022.32703 · doi:10.1215/S0012-7094-39-00565-X [15] I. P. Malta, Hyperbolic Birkhoff centers,Trans. A.M.S.,262 (1980), 181–193; announced inAn. Acad. Brasil. Ciências,51 (1979), (1), 27–29. · Zbl 0396.58022 · doi:10.1090/S0002-9947-1980-0583851-4 [16] A. Manning, There are no new Anosov diffeomorphisms on tori,Amer. J. Math.,96 (1974), 422–429. · Zbl 0242.58003 · doi:10.2307/2373551 [17] J. E. Marsden, M. McCracken, The Hopf bifurcation and its applications,Appl. Math. Sci.,19, Springer-Verlag, 1976. · Zbl 0346.58007 [18] W. C. de Melo, Moduli and stability of two-dimensional diffeomorphisms,Topology,19 (1980), 9–21. · Zbl 0447.58025 · doi:10.1016/0040-9383(80)90028-2 [19] W. de Melo, J. Palis andS. Van Strien, Characterizing diffeomorphisms with modulus of stability one,Dynamical Systems and Turbulence, Warwick 1980,Springer Lecture Notes in Math.,898 (1981), 266–286. · doi:10.1007/BFb0091919 [20] S. Newhouse, J. Palis, Bifurcations of Morse-Smale dynamical systems,Dynamical Systems, Academic Press, 1973, 303–366. · Zbl 0279.58011 [21] S. Newhouse, J. Palis, Cycles and bifurcation theory,Astérisque,31 (1976), 43–140. · Zbl 0322.58009 [22] S. Newhouse, J. Palis, F. Takens, Stable arcs of diffeomorphisms,Bull. A.M.S.,82 (1976), 499–502. · Zbl 0339.58008 · doi:10.1090/S0002-9904-1976-14073-6 [23] S. Newhouse, The abundance of wild hyperbolic sets and non-smooth stable sets,Publ. Math. I.H.E.S.,50 (1979), 101–152. · Zbl 0445.58022 [24] R. Palais, Local triviality of the restriction map for embeddings,Comm. Math. Helvet.,34 (1960), 305–312. · Zbl 0207.22501 · doi:10.1007/BF02565942 [25] J. Palis, On Morse-Smale dynamical systems,Topology,8 (1969), 385–405. · Zbl 0189.23902 · doi:10.1016/0040-9383(69)90024-X [26] J. Palis, S. Smale, Structural stability theorems,Proc. Symp. Pure Math. A.M.S.,14 (1970), 223–232. · Zbl 0214.50702 [27] J. Palis, F. Takens, Topological equivalence of normally hyperbolic dynamical systems,Topology,16 (1977), 335–345. · Zbl 0391.58015 · doi:10.1016/0040-9383(77)90040-4 [28] J. Palis andF. Takens,Stability of parametrized families of gradient vector fields, Preprint IMPA, to appear. [29] J. Palis, A differentiable invariant of topological conjugacies and moduli of stability,Astérisque,51 (1978), 335–346. · Zbl 0396.58015 [30] J. Palis, Moduli of stability and bifurcation theory,Proc. Int. Congres of Math. Helsinki (1978), 835–839. [31] H. Poincaré,OEuvres complètes, t. 1, Gauthier-Villars, 1952, 137–158. [32] C. C. Pugh, M. Shub, Linearization of normally hyperbolic diffeomorphisms and flows,Invent. Math.,10 (1970), 187–198. · Zbl 0206.25802 · doi:10.1007/BF01403247 [33] C. Robinson, Global structural stability of a saddle-node bifurcation,Trans. A.M.S.,236 (1978), 155–172. · Zbl 0406.58022 · doi:10.1090/S0002-9947-1978-0467832-8 [34] 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. · Zbl 0223.76041 · doi:10.1007/BF01646553 [35] S. Smale, Stable manifolds for differential equations and diffeomorphisms,Ann. Scuola Norm. Sup. Pisa,18 (1963), 97–116. · Zbl 0113.29702 [36] S. Smale, Differentiable dynamical systems,Bull. A.M.S.,73 (1967), 747–817. · Zbl 0202.55202 · doi:10.1090/S0002-9904-1967-11798-1 [37] J. Sotomayor, Generic one parameter families of vector fields,Publ. Math. I.H.E.S.,43 (1974), 4–56. [38] J. Sotomayor, Generic bifurcations of dynamical systems,Dynamical Systems, Academic Press 1973, 549–560. · Zbl 0296.58007 [39] S. J. Van Strien, Center manifolds are not CMath. Z.,166 (1979), 143–145. · Zbl 0403.58021 · doi:10.1007/BF01214040 [40] F. Takens, Partially hyperbolic fixed points,Topology,10 (1971), 133–147. · Zbl 0214.22901 · doi:10.1016/0040-9383(71)90035-8 [41] F. Takens, Normal forms for certain singularities of vector fields,Ann. Inst. Fourier,23 (1973), (2), 163–195. · Zbl 0266.34046 [42] F. Takens, Forced oscillations and bifurcations,Applications of Global Analysis I,Comm. of the Math. Inst., R.U. Utrecht (Holland), 1974. [43] F. Takens, Global phenomena in bifurcations of dynamical systems with simple recurrence,Jber. d. Dt. Math.-Verein,81 (1979), 87–96. · Zbl 0419.58012 [44] R. Williams, The D.A. map of Smale and structural stability,Proc. Symp. Pure Math. A.M.S.,14 (1970), 329–334. This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.