A global perspective for non-conservative dynamics

J. Palis

Annales de l'I.H.P. Analyse non linéaire (2005)

  • Volume: 22, Issue: 4, page 485-507
  • ISSN: 0294-1449

How to cite

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Palis, J.. "A global perspective for non-conservative dynamics." Annales de l'I.H.P. Analyse non linéaire 22.4 (2005): 485-507. <http://eudml.org/doc/78665>.

@article{Palis2005,
author = {Palis, J.},
journal = {Annales de l'I.H.P. Analyse non linéaire},
keywords = {attractor; stochastic stability; non-conservative dynamics; homoclinic bifurcation; hyperbolic decomposition},
language = {eng},
number = {4},
pages = {485-507},
publisher = {Elsevier},
title = {A global perspective for non-conservative dynamics},
url = {http://eudml.org/doc/78665},
volume = {22},
year = {2005},
}

TY - JOUR
AU - Palis, J.
TI - A global perspective for non-conservative dynamics
JO - Annales de l'I.H.P. Analyse non linéaire
PY - 2005
PB - Elsevier
VL - 22
IS - 4
SP - 485
EP - 507
LA - eng
KW - attractor; stochastic stability; non-conservative dynamics; homoclinic bifurcation; hyperbolic decomposition
UR - http://eudml.org/doc/78665
ER -

References

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  1. [1] Abraham R., Smale S., Nongenericity of Ω-stability, in: Global Analysis, Berkeley 1968, Proc. Sympos. Pure Math., vol. XIV, Amer. Math. Soc., 1970. Zbl0215.25102
  2. [2] Afraimovich V.S., Bykov V.V., Shil'nikov L.P., On the appearance and structure of the Lorenz attractor, Dokl. Acad. Sci. USSR234 (1977) 336-339. Zbl0451.76052MR462175
  3. [3] Afraimovich V.S., Shil'nikov L.P., The accessible transitions from Morse–Smale systems to systems with several periodic motions, Izv. Akad. Nauk SSSR38 (1974) 1248-1288. Zbl0322.58007MR423422
  4. [4] Afraimovich V.S., Shil'nikov L.P., On bifurcations of codimension 1, leading to the appearance of fixed points of a countable set of tori, Dokl. Akad. Nauk SSSR262 (1982) 777-780. Zbl0509.58033MR644867
  5. [5] Alves J.F., Bonatti C., Viana M., SRB measures for partially hyperbolic systems whose central direction is mostly expanding, Invent. Math.140 (2000) 351-398. Zbl0962.37012MR1757000
  6. [6] Andronov A., Leontovich E.A., Some cases of dependence of limit cycles on parameters, Uchen. Zap. Gor'kov. Univ.6 (1939) 3-24. 
  7. [7] Andronov A., Leontovich E.A., Gordon I.I., Majer A.G., Qualitative Theory of Dynamical Systems of Second Order, Nauka, Moscow, 1966, p. 569. MR199506
  8. [8] Andronov A., Leontovich E.A., Gordon I.I., Majer A.G., The Theory of Bifurcations of Dynamical Systems on the Plane, Nauka, Moscow, 1967, p. 487. Zbl0257.34001MR235228
  9. [9] Andronov A., Pontryagin L., Systèmes grossiers, Dokl. Akad. Nauk USSR14 (1937) 247-251. Zbl0016.11301
  10. [10] Anosov D.V., Geodesic flows on closed Riemannian manifolds of negative curvature, Proc. Steklov Math. Inst.90 (1967) 1-235. Zbl0176.19101MR224110
  11. [11] Arnéodo A., Coullet P., Tresser C., Possible new strange attractors with spiral structure, Comm. Math. Phys.79 (1981) 673-679. Zbl0485.58013MR623967
  12. [12] Arnold V.I., Small denominators I: On the mapping of a circle to itself, Izv. Akad. Nauk Math. Ser.25 (1961) 21-86. Zbl0152.41905MR140699
  13. [13] Arnold V.I., Lectures on bifurcations and versal families, Uspekhi Math. Nauk27 (1972) 119-184. Zbl0264.58005MR413191
  14. [14] Arnold V.I., Kolmogorov's hydrodynamics attractors. turbulence and stochastic processes: Kolmogorov's ideas 50 years on, Proc. Roy. Soc. London Ser. A434 (1991) 19-22. Zbl0726.76045MR1124924
  15. [15] Arnold V.I., Afraimovich V.S., Ilyasheko Yu., Shilnikov L.P., Bifurcation Theory and Catastrophe Theory, Springer, 1999. Zbl1038.37500MR1733750
  16. [16] Arroyo A., Rodriguez Hertz F., Homoclinic bifurcations and uniform hyperbolicity for three-dimensional flows, Ann. Inst. H. Poincaré20 (2003) 805-841. Zbl1045.37006MR1995503
  17. [17] Avila A., Lyubich M., de Melo W., Regular or stochastic dynamics in real analytic families of unimodal maps, Invent. Math.154 (2003) 451-550. Zbl1050.37018MR2018784
  18. [18] A. Avila, C.G. Moreira, Phase-parameter relation sharp statistical properties of unimodal maps, Contemp. Math., in press. Zbl1145.37022
  19. [19] A. Avila, C.G. Moreira, Statistical properties of unimodal maps: physical measures, periodic orbits and pathological laminations, Publ. Math. IHES, in press. Zbl1078.37030
  20. [20] A. Avila, C.G. Moreira, Statistical properties of unimodal maps: the quadratic family, Ann. of Math., in press. Zbl1078.37029
  21. [21] Avila A., Moreira C.G., Statistical properties of unimodal maps: smooth families with negative Schwarzian derivative, Astérisque286 (2003) 81-118. Zbl1046.37021MR2052298
  22. [22] Bamón R., Labarca R., Mañé R., Pacifico M.J., The explosion of singular cycles, Publ. Math. IHES78 (1993) 207-232. Zbl0801.58010MR1259432
  23. [23] Benedicks M., Carleson L., The dynamics of the Hénon map, Ann. of Math.133 (1991) 73-169. Zbl0724.58042MR1087346
  24. [24] M. Benedicks, M. Viana, Random perturbations and statistical properties of Hénon-like maps, Ann. Inst. H. Poincaré Anal. Non Linéaire. Zbl1131.37033MR2259614
  25. [25] Benedicks M., Viana M., Solution of the basin problem for Hénon-like attractors, Invent. Math.143 (2001) 375-434. Zbl0967.37023MR1835392
  26. [26] Benedicks M., Young L.-S., SBR-measures for certain Hénon maps, Invent. Math.112 (1993) 541-576. Zbl0796.58025MR1218323
  27. [27] Birkhoff G.D., Nouvelles recherches sur les systèmes dynamiques, Mem. Pont. Acad. Sci. Novi. Lyncaei1 (1935) 85-216. Zbl0016.23401
  28. [28] Bonatti C., Diaz L., Viana M., Dynamics Beyond Uniform Hyperbolicity, Encyclopaedia Math. Sci., vol. 102, Springer, 2004. Zbl1060.37020MR2105774
  29. [29] Bonatti C., Díaz L.J., Pujals E., A C 1 -generic dichotomy for diffeomorphisms: weak forms of hyperbolicity or infinitely many sinks or sources, Ann. of Math.158 (2003) 355-418. Zbl1049.37011MR2018925
  30. [30] Bonatti C., Díaz L.J., Pujals E., Rocha J., Robust transitivity and heterodimensional cycles, Astérisque286 (2003) 187-222. Zbl1056.37024
  31. [31] Bonatti C., Pumariño A., Viana M., Lorenz attractors with arbitrary expanding dimension, C. R. Acad. Sci. Paris, Sér. I Math.325 (1997) 883-888. Zbl0896.58043MR1485910
  32. [32] Bowen R., Ruelle D., The ergodic theory of Axiom A flows, Invent. Math.29 (1975) 181-202. Zbl0311.58010MR380889
  33. [33] Bunimovich L.A., Sinai Ya.G., Stochasticity of the attractor in the Lorenz model, in: Nonlinear Waves, Proc. Winter School, Nauka, Moscow, 1980, pp. 212-226. 
  34. [34] Cartwright M., Littlewood J., On non-linear differential equations of the second order, J. London Math. Soc.20 (1945) 127-153. Zbl0061.18903
  35. [35] Cartwright M., Littlewood J., On non-linear differential equations of the second order, Ann. of Math.48 (1947) 472-494. Zbl0029.12602MR21190
  36. [36] Colli E., Infinitely many coexisting strange attractors, Ann. Inst. H. Poincaré Anal. Non Linéaire15 (1998) 539-579. Zbl0932.37015MR1643393
  37. [37] Coullet P., Tresser C., Itérations d'endomorphims et groupe de renormalization, C. R. Acad. Sci. Paris, Sér. I287 (1978) 577-580. Zbl0402.54046MR512110
  38. [38] Dias Carneiro M.J., Palis J., Bifurcations and global stability of families of gradients, Publ. Math. IHES70 (1989) 70-163. Zbl0706.58042MR1067381
  39. [39] de Melo W., Structural stability of diffeomorphisms on two-manifolds, Invent. Math.21 (1973) 233-246. Zbl0291.58011MR339277
  40. [40] Van der Pol B., On relaxation oscillations, Philos. Mag. Ser. 72 (1926) 978-992. Zbl52.0450.05JFM52.0450.05
  41. [41] Díaz L.J., Robust nonhyperbolic dynamics and heterodimensional cycles, Ergodic Theory Dynam. Systems15 (1995) 291-315. Zbl0831.58035MR1332405
  42. [42] Díaz L.J., Pujals E., Ures R., Partial hyperbolicity and robust transitivity, Acta Math.183 (1999) 1-43. Zbl0987.37020MR1719547
  43. [43] Díaz L.J., Rocha J., Large measure of hyperbolic dynamics when unfolding heterodimensional cycles, Nonlinearity10 (1997) 857-884. Zbl0908.58042MR1457749
  44. [44] Dolgopyat D., On differentiability of SRB states for partially hyperbolic systems, Invent. Math.155 (2004) 359-449. Zbl1059.37021MR2031432
  45. [45] Feigenbaum M., Qualitative universality for a class of nonlinear transformations, J. Statist. Phys.19 (1978) 25-52. Zbl0509.58037MR501179
  46. [46] Franks J., Williams R., Anomalous Anosov Flows. Global Theory of Dynamical Systems, Lecture Notes in Math., vol. 819, Springer, 1980. Zbl0463.58021MR591182
  47. [47] Gonchenko S.V., Shil'nikov L.P., Turaev D.V., Dynamical phenomena in systems with structurally unstable Poincaré homoclinic orbits, Chaos6 (1996) 15-31. Zbl1055.37578MR1376892
  48. [48] A. Gorodetski, V. Kaloshin, How often surface diffeomorphisms have infinitely many sinks and hyperbolicity of periodic points near a homoclinic tangency, in press. Zbl1110.37034
  49. [49] Graczyk J., Swiatek G., Generic hyperbolicity in the logistic family, Ann. of Math.146 (1997) 1-52. Zbl0936.37015MR1469316
  50. [50] Guckenheimer J., Williams R.F., Structural stability of Lorenz attractors, Publ. Math. IHES50 (1979) 59-72. Zbl0436.58018MR556582
  51. [51] Hayashi S., Connecting invariant manifolds and the solution of the C 1 stability andΩ-stability conjectures for flows, Ann. of Math.145 (1997) 81-137. Zbl0871.58067
  52. [52] Hénon M., A two dimensional mapping with a strange attractor, Comm. Math. Phys.50 (1976) 69-77. Zbl0576.58018MR422932
  53. [53] Ilyashenko Yu., Li W., Nonlocal Bifurcations, Math. Surveys and Monographs, vol. 66, Amer. Math. Soc., 1999. Zbl1120.37308MR1650842
  54. [54] Jakobson M., On smooth mappings of the circle into itself, Math. USSR-Sb.14 (1971) 161-185. Zbl0241.58006MR290406
  55. [55] Jakobson M., Absolutely continuous invariant measures for one-parameter families of one-dimensional maps, Comm. Math. Phys.81 (1981) 39-88. Zbl0497.58017MR630331
  56. [56] V.Yu. Kaloshin, B.R. Hunt, Stretched exponential estimate on growth of the number of periodic points for prevalent diffeomorphisms, in press. Zbl1132.37011
  57. [57] Kifer Yu., Ergodic Theory of Random Perturbations, Birkhäuser, 1986. MR884892
  58. [58] Kifer Yu., Random Perturbations of Dynamical Systems, Birkhäuser, 1988. Zbl0659.58003MR1015933
  59. [59] Kozlovski O., Getting rid of the negative Schwarzian derivative condition, Ann. of Math.152 (2000) 743-762. Zbl0988.37044MR1815700
  60. [60] Kozlovski O., Axiom A maps are dense in the space of unimodal maps in the C k topology, Ann. of Math.157 (2003) 1-43. Zbl1215.37022MR1954263
  61. [61] O. Kozlovski, W. Shen, S. van Strien, Density of hyperbolicity in dimension one, Preprint, Warwick, 2004. Zbl1138.37013
  62. [62] O. Kozlovski, S. van Strien, W. Shen, Rigidity for real polynomials, Preprint, Warwick, 2003. Zbl1129.37020
  63. [63] Labarca R., Pacifico M.J., Stability of singular horseshoes, Topology25 (1986) 337-352. Zbl0611.58033MR842429
  64. [64] Levi M., Qualitative analysis of the periodically forced relaxation oscillations, Mem. Amer. Math. Soc.32 (244) (1981). Zbl0448.34032MR617687
  65. [65] Levinson N., A second order differential equations with singular solutions, Ann. of Math.50 (1949) 127-153. Zbl0045.36501MR30079
  66. [66] Liao S.-T., Hyperbolicity properties of the non-wandering sets of certain 3-dimensional systems, Acta Math. Sci.3 (1983) 361-368. Zbl0527.58031MR812542
  67. [67] Liao S.T., On the stability conjecture, Chinese Ann. of Math.1 (1980) 9-30. Zbl0449.58013MR591229
  68. [68] Littlewood J., On non-linear differential equations of the second order, III, Acta Math.97 (1957) 267-308. Zbl0081.08401MR88615
  69. [69] Littlewood J., On non-linear differential equations of the second order, IV, Acta Math.98 (1957) 1-110. Zbl0081.08401
  70. [70] Lorenz E.N., Deterministic nonperiodic flow, J. Atmosph. Sci.20 (1963) 130-141. 
  71. [71] Lyubich M., Almost every real quadratic map is either regular or stochastic, Ann. of Math.156 (2002) 1-78. Zbl1160.37356MR1935840
  72. [72] Mañé R., An ergodic closing lemma, Ann. of Math.116 (1982) 503-540. Zbl0511.58029MR678479
  73. [73] Mañé R., A proof of the C 1 stability conjecture, Publ. Math. IHES66 (1988) 161-210. Zbl0678.58022MR932138
  74. [74] Martens M., Nowicki T., Invariant measures for Lebesgue typical quadratic maps. Géométrie complexe et systèmes dynamiques (Orsay, 1995), Astérisque261 (2001) 239-252. Zbl0939.37020MR1755443
  75. [75] May R.M., Simple mathematical models with very complicated dynamics, Nature261 (1976) 459-467. 
  76. [76] McMullen C., Complex Dynamics and Renormalization, Ann. of Math. Stud., vol. 142, Princeton University Press, 1994. Zbl0822.30002MR1312365
  77. [77] Metzger R., Sinai–Ruelle–Bowen measures for contracting Lorenz maps and flows, Ann. Inst. H. Poincaré Anal. Non Linéaire17 (2000) 247-276. Zbl0983.37022MR1753089
  78. [78] Metzger R., Stochastic stability for contracting Lorenz maps and flows, Comm. Math. Phys.212 (2000) 277-296. Zbl1052.37018MR1772247
  79. [79] Mora L., Viana M., Abundance of strange attractors, Acta Math.171 (1993) 1-71. Zbl0815.58016MR1237897
  80. [80] Morales C., Pacifico M.J., A dichotomy for three-dimensional vector fields, Ergodic Theory Dynam. Systems23 (2003) 1575-1600. Zbl1040.37014MR2018613
  81. [81] Morales C., Pacifico M.J., Pujals E., Singular hyperbolic systems, Proc. Amer. Math. Soc.127 (1999) 3393-3401. Zbl0924.58068MR1610761
  82. [82] Morales C., Pacifico M.J., Pujals E., Robust transitive singular sets for 3-flows are partially hyperbolic attractors and repellers, Ann. of Math.160 (2004) 1-58. Zbl1071.37022MR2123928
  83. [83] Moreira C.G., Palis J., Viana M., Homoclinic tangencies and fractal invariants in arbitrary dimension, C. R. Acad. Sci. Paris, Sér. I Math.333 (2001) 475-480. Zbl1192.37032MR1859240
  84. [84] Moreira C.G., Yoccoz J.-C., Stable intersections of regular Cantor sets with large Hausdorff dimensions, Ann. of Math.154 (2001) 45-96. Zbl1195.37015MR1847588
  85. [85] Newhouse S., Diffeomorphisms with infinitely many sinks, Topology13 (1974) 9-18. Zbl0275.58016MR339291
  86. [86] Newhouse S., The abundance of wild hyperbolic sets and nonsmooth stable sets for diffeomorphisms, Publ. Math. IHES50 (1979) 101-151. Zbl0445.58022MR556584
  87. [87] Newhouse S., Palis J., Cycles and bifurcation theory, Astérisque31 (1976) 44-140. Zbl0322.58009MR516408
  88. [88] Newhouse S., Palis J., Takens F., Bifurcations and stability of families of diffeomorphisms, Publ. Math. IHES57 (1983) 5-71. Zbl0518.58031MR699057
  89. [89] Palis J., On Morse–Smale dynamical systems, Topology8 (1969) 385-405. Zbl0189.23902MR246316
  90. [90] Palis J., A global view of Dynamics and a conjecture on the denseness of finitude of attractors. Géométrie complexe et systèmes dynamiques (Orsay, 1995), Astérisque261 (1995) 335-347. Zbl1044.37014MR1755446
  91. [91] Palis J., Smale S., Structural stability theorems, in: Global Analysis, Berkeley 1968, Proc. Sympos. Pure Math., vol. XIV, Amer. Math. Soc., 1970, pp. 223-232. Zbl0214.50702MR267603
  92. [92] Palis J., Takens F., Stability of parametrized families of gradient vector fields, Ann. of Math.118 (1983) 383-421. Zbl0533.58018MR727698
  93. [93] Palis J., Takens F., Cycles and measure of bifurcation sets for two-dimensional diffeomorphisms, Invent. Math.82 (1985) 397-422. Zbl0579.58005MR811543
  94. [94] Palis J., Takens F., Hyperbolic and the creation of homoclinic orbits, Ann. of Math.1987 (1987) 337-374. Zbl0641.58029MR881272
  95. [95] Palis J., Takens F., Hyperbolicity and Sensitive-Chaotic Dynamics at Homoclinic Bifurcations, Cambridge University Press, 1993. Zbl0790.58014MR1237641
  96. [96] Palis J., Viana M., High dimension diffeomorphisms displaying infinitely many periodic attractors, Ann. of Math.140 (1994) 207-250. Zbl0817.58004MR1289496
  97. [97] Palis J., Yoccoz J.-C., Homoclinic tangencies for hyperbolic sets of large Hausdorff dimension, Acta Math.172 (1994) 91-136. Zbl0801.58035MR1263999
  98. [98] Palis J., Yoccoz J.-C., Fers à cheval non uniformément hyperboliques engendrés par une bifurcation homocline et densité nulle des attracteurs, C. R. Acad. Sci. Paris, Sér. I Math.333 (2001) 867-871. Zbl1015.37024MR1873226
  99. [99] Peixoto M., Structural stability on two-dimensional manifolds, Topology1 (1962) 101-120. Zbl0107.07103MR142859
  100. [100] Pesin Ya., Sinai Ya., Gibbs measures for partially hyperbolic attractors, Ergodic Theory Dynam. Systems2 (1982) 417-438. Zbl0519.58035MR721733
  101. [101] E. Pujals, Density of hyperbolicity and homoclinic bifurcation for 3d-diffeomorphisms in attracting regions, IMPA's preprint server, http://www.preprint.impa.br. MR2221742
  102. [102] Pujals E., Sambarino M., Homoclinic tangencies and hyperbolicity for surface diffeomorphisms, Ann. of Math.151 (2000) 961-1023. Zbl0959.37040MR1779562
  103. [103] Robbin J., A structural stability theorem, Ann. of Math.94 (1971) 447-493. Zbl0224.58005MR287580
  104. [104] Robinson C., Structural stability of vector fields, Ann. of Math.99 (1974) 154-175. Zbl0275.58012MR334283
  105. [105] Robinson C., Homoclinic bifurcation to a transitive attractor of Lorenz type, Nonlinearity2 (1989) 495-518. Zbl0704.58031MR1020439
  106. [106] Rovella A., The dynamics of perturbations of the contracting Lorenz attractor, Bull. Braz. Math. Soc.24 (1993) 233-259. Zbl0797.58051MR1254985
  107. [107] Ruelle D., A measure associated with Axiom A attractors, Amer. J. Math.98 (1976) 619-654. Zbl0355.58010MR415683
  108. [108] Rychlik M., Lorenz attractors through Shil'nikov-type bifurcation. Part 1, Ergodic Theory Dynam. Systems10 (1990) 793-821. Zbl0715.58027MR1091428
  109. [109] Sannami A., The stability theorems for discrete dynamical systems on two-dimensional manifolds, Nagoya Math. J.90 (1983) 1-55. Zbl0515.58022MR702250
  110. [110] Shil'nikov L.P., On the generation of periodic motion from a trajectory doubly asymptotic to an equilibrium state of saddle type, Math. USSR-Sb.6 (1968) 428-438. Zbl0188.15303
  111. [111] Simon R., A 3-dimensional Abraham–Smale example, Proc. Amer. Math. Soc.34 (1972) 629-630. Zbl0259.58006MR295391
  112. [112] Sinai Ya., Gibbs measures in ergodic theory, Russian Math. Surveys27 (1972) 21-69. Zbl0255.28016MR399421
  113. [113] Smale S., Diffeomorphisms with many periodic points, in: Differential and Combinatorial Topology, Princeton University Press, 1965. Zbl0142.41103MR182020
  114. [114] Smale S., Differentiable dynamical systems, Bull. Amer. Math. Soc.73 (1967) 747-817. Zbl0202.55202MR228014
  115. [115] Sullivan D., Bounds, quadratic differentials and renormalization conjectures, Ann. Math. Soc. Centennial Publ.2 (1992) 417-466. Zbl0936.37016MR1184622
  116. [116] Tedeschini-Lalli L., Yorke J., How often do simple dynamical systems have infinitely many coexisting sinks?, Comm. Math. Phys.106 (1986) 635-657. Zbl0602.58036MR860314
  117. [117] M. Tsujii, Physical measures for partially hyperbolic surface endomorphisms, Acta Math., in press. Zbl1105.37022MR2231338
  118. [118] Tucker W., A rigorous ODE solver and Smale's 14th problem, Found. Comput. Math.2 (2002) 53-117. Zbl1047.37012MR1870856
  119. [119] Ures R., Abundance of hyperbolicity in C 1 topology, Ann. Sci. École Norm. Sup.28 (1995) 747-760. Zbl0989.37013MR1355140
  120. [120] C. Vasquez, Statistical stability for diffeomorphisms with dominated splitting, in press. Zbl1147.37019
  121. [121] Viana M., Strange attractors in higher dimensions, Bull. Braz. Math. Soc.24 (1993) 13-62. Zbl0784.58044MR1224299
  122. [122] Viana M., Multidimensional nonhyperbolic attractors, Publ. Math. IHES85 (1997) 63-96. Zbl1037.37016MR1471866
  123. [123] Viana M., What's new on Lorenz strange attractors?, Math. Intelligencer22 (2000) 6-19. Zbl1052.37026MR1773551
  124. [124] Wen L., Homoclinic tangencies and dominated splittings, Nonlinearity15 (2002) 1445-1469. Zbl1011.37011MR1925423
  125. [125] Wen L., Generic diffeomorphisms away from homoclinic tangencies and heterodimensional cycles, Bull. Braz. Math. Soc.35 (2004) 419-452. Zbl1099.37024MR2106314
  126. [126] Young L.-S., Stochastic stability of hyperbolic attractors, Ergodic Theory Dynam. Systems6 (1986) 311-319. Zbl0633.58023MR857204

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