Skip to main content
SpringerLink
Log in

The Dynamics of Vector Fields with Singularities

  • Published:
Acta Mathematica Sinica, English Series Aims and scope Submit manuscript

Abstract

We give a brief survey on the dynamics of vector fields with singularities. The aim of this survey is not to list all results in this field, but only to introduce some results from several viewpoints and some technics.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Abdenur, F., Avila, A., Bochi, J.: Robust transitivity and topological mixing for C1-flows. Proc. Amer. Math. Soc., 132, 699–705 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  2. Abdenur, F., Bonatti, C., Crovisier, S.: Nonuniform hyperbolicity for C1-generic diffeomorphisms. Israel J. Math., 183, 1–60 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  3. Abraham, R., Smale, S.: Nongenericity of Ω-stability, In: Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968), Amer. Math. Soc., Providence, RI, 1970, 5–8

    Chapter  Google Scholar 

  4. Afraĭmovič, V., Bykov, V., Shilnikov, L.: The origin and structure of the Lorenz attractor. Dokl. Akad. Nauk SSSR, 234, 336–339 (1977)

    MathSciNet  Google Scholar 

  5. Aoki, N.: The set of Axiom A diffeomorphisms with no cycles. Bol. Soc. Bras. Mat., 23, 21–65 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  6. Arbieto, A., Lopez, A., Morales, C.: Homoclinic classes for sectional-hyperbolic sets. Kyoto J. Math., 56, 531–538 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  7. Araújo, V., Melbourne, I.: Exponential decay of correlations for nonuniformly hyperbolic flows with a C1+α stable foliation, including the classical Lorenz attractor. Ann. Henri Poincaré, 17, 2975–3004 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  8. Araújo, V., Melbourne, I.: Mixing properties and statistical limit theorems for singular hyperbolic flows without a smooth stable foliation. Adv. Math., 349, 212–245 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  9. Araújo, V., Pacifico, M. J.: Three-dimensional flows. Ergebnisse der Mathematik und ihrer Grenzgebiete, Vol. 53, Springer, Heidelberg, 2010

    Book  MATH  Google Scholar 

  10. Araújo, V., Pacifico, M., Pujals, E., et al.: Singular-hyperbolic attractors are chaotic. Trans. Amer. Math. Soc., 361(5), 2431–2485 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  11. Arbieto, A., Morales, C., Santiago, B.: Lyapunov stability and sectional-hyperbolicity for higher-dimensional flows. Math. Ann., 361, 67–75 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  12. Arroyo, A., Rodriguez Hertz, F.: Homoclinic bifurcations and uniform hyperbolicity for three-dimensional flows. Ann. Inst. H. Poincaré Anal. Non Linéaire, 20, 805–841 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  13. Bonatti, C., Crovisier, S.: Récurrence et généricité. Invent. Math., 158, 33–104 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  14. Bonatti, C., da Luz, A.: Star flows and multisingular hyperbolicity, accepted by J. Eur. Math. Soc., 2020, arXiv:1705.05799

  15. Bonatti, C., Díaz, L. J., Viana, M.: Dynamics Beyond Uniform Hyperbolicity. A Global Geometric and Probabilistic Perspective. Encyclopaedia of Mathematical Sciences, Vol. 102. Mathematical Physics, III, Springer-Verlag, Berlin, 2005

    MATH  Google Scholar 

  16. Bonatti, C., Gan, S. B., Wen, L.: On the existence of non-trivial homoclinic classes. Ergodic Theory Dynam. Systems, 27, 1473–1508 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  17. Bonatti, C., Gan, S. B., Yang, D. W.: Dominated chain recurrent class with singularities. Ann. Sc. Norm. Super. Pisa Cl. Sci., 14, 83–99 (2015)

    MathSciNet  MATH  Google Scholar 

  18. Bonatti, C., Pumariño, A., Viana, M.: Lorenz attractors with arbitrary expanding dimension. C. R. Acad. Sci. Paris Sér. I Math., 325, 883–888 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  19. Bautista, S., Morales, C.: Recent progress on sectional-hyperbolic systems. Dynamical Systems, 30, 369–382 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  20. Bowen, R.: Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms. Lecture Notes in Mathematics, Vol. 470, Springer-Verlag, Berlin, 1977

    MATH  Google Scholar 

  21. Bowen, R., Ruelle, D.: The ergodic theory of Axiom A fows. Invent. Math., 29, 181–202 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  22. Buzzi, J., Yang, D. W.: On the mixing for partially hyperbolic flows. in preparation, 2020

  23. Crovisier, S.: Birth of homoclinic intersections: a model for the central dynamics of partially hyperbolic systems. Ann. of Math., 172, 1641–1677 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  24. Crovisier, S., da Luz, A., Yang, D. W., et al.: On the notions of singular domination and (multi-)singular hyperbolicity, Sci. China Math., 63, 1721–1744 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  25. Crovisier, S., Pujals, E.: Essential hyperbolicity and homoclinic bifurcations: a dichotomy phenomenon/mechanism for diffeomorphisms. Invent. Math., 201, 385–517 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  26. Crovisier, S., Wang, X., Yang, D., et al.: The ergodic theory of multi-singular hyperbolic vector fields, in preparation, 2020

  27. Crovisier, S., Yang, D. W.: On the density of singular hyperbolic three-dimensional vector fields: a conjecture of Palis. C. R. Math. Acad. Sci. Paris, 353, 85–88 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  28. Crovisier, S., Yang, D. W.: Homoclinic tangencies and singular hyperbolicity for three-dimensional vector fields, preprint, 2017, arXiv:1702.05994

  29. Crovisier, S., Yang, D. W.: Robust transitivity of singular hyperbolic attractors. Mathematische Zeitschrift, 298, 469–488 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  30. Crovisier, S., Yang, D. W., Zhang, J. H.: Empirical measures of partially hyperbolic attractors. accepted by Comm. Math. Phys., 375, 725–764 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  31. Da Luz, A. Star flows with singularities of different indices, preprint, 2018, arXiv:1806.09011

  32. Doering, C. I.: Persistently transitive vector fields on three-dimensional manifolds. In: Dynamical Systems and Bifurcation Theory (Rio de Janeiro, 1985), 59–89. Pitman Res. Notes Math. Ser., Vol. 160, Longman Sci. Tech., Harlow (1987)

    Google Scholar 

  33. Gan, S. B., Wen, L.: Nonsingular star flows satisfy Axiom A and the no-cycle condition. Invent. Math., 164, 279–315 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  34. Gan, S. B., Yang, D. W.: Morse-Smale systems and horseshoes for three dimensional singular flows. Ann. Sci. Ec. Norm. Supér., 51(1), 39–112 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  35. Ghys, E.: The Lorenz attractor, a paradigm for chaos. In: Chaos, Prog. Math. Phys., Vol. 66, Birkhäuser/Springer, Basel, 2013, 1–54

    MATH  Google Scholar 

  36. Guchenheimer, J.: A strange, strange attractor, In: The Hopf Bifurcation Theorems and Its Applications (Applied Mathematical Series), Vol. 19, Springer-Verlag, 1976, 368–381

  37. Guchenheimer, J., Williams, R.: Structural stability of Lorenz attractors. Inst. Hautes Etudes Sci. Publ. Math., 50, 59–72 (1979)

    Article  MathSciNet  Google Scholar 

  38. Hayashi, S.: Diffeomorphisms in \({{\cal F}^1}(M)\) satisfy Axiom A. Ergodic Theory Dyn. Syst., 12, 233–253 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  39. Hayashi, S.: Connecting invariant manifolds and the solution of the C1-stability and Ω-stability conjectures for flows. Ann. Math., 145, 81–137 (1997). Correction: Ann. Math., 150, 353–356 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  40. Hirsch, M., Pugh, C., Shub, M.: Invariant manifolds. Lecture Notes in Mathematics, Vol. 583, Springer-Verlag, 1977

  41. Katok, A.: Lyapunov exponents, entropy and periodic orbits for diffeomorphisms. Publ. Math. I.H.E.S., 51, 137–173 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  42. Kiriki, S., Li, M., Soma, T.: Geometric Lorenz flows with historic behavior. Discrete Contin. Dyn. Syst., 36, 7021–7028 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  43. Ledrappier, F., Young, L. S.: The metric entropy of diffeomorphisms. I. Characterization of measures satisfying Pesin’s entropy formula. Ann. of Math., 122, 509–539 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  44. Leplaideur, R., Yang, D. W.: SRB measure for higher dimensional singular partially hyperbolic flows. Ann. Inst. Fourier, 67, 2703–2717 (2017)

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  46. Li, M., Gan, S. B., Wen, L.: Robustly transitive singular sets via approach of an extended linear Poincare flow. Discrete Contin. Dyn. Syst., 13, 239–269 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  47. Liao, S. T.: A basic property of a certain class of differential systems (in Chinese). Acta Math. Sin., 22, 316–343 (1979)

    MATH  Google Scholar 

  48. Liao, S. T.: On the stability conjecture. Chinese Ann. Math., 1(1), 9–30 (1980)

    MathSciNet  MATH  Google Scholar 

  49. Liao, S. T.: Obstruction sets (I) (in Chinese). Acta Math. Sin., 23, 411–453 (1980)

    MATH  Google Scholar 

  50. Liao, S. T.: On (η, d)-contractible orbits of vector fields. Systems Science and Mathematical Sciences, 2, 193–227 (1989)

    MathSciNet  MATH  Google Scholar 

  51. Lorenz, E. N.: Deterministic nonperiodic flow. J. Atmosph. Sci., 20, 130–141 (1963)

    Article  MathSciNet  MATH  Google Scholar 

  52. Luzzatto, S., Melbourne, I., Paccaut, F.: The Lorenz attractor is mixing. Comm. Math. Phys., 260, 393–401 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  53. Mañé, R.: An ergodic closing lemma. Ann. of Math., 116, 503–540 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  54. Mañé, R.: A proof of the C1 stability conjecture. Inst. Hautes Etudes Sci. Publ. Math., 66, 161–210 (1988)

    Article  MATH  Google Scholar 

  55. Metzger, R., Morales, C.: Sectional-hyperbolic systems. Ergodic Theory Dynam. Systems, 28, 1587–1597 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  56. Morales, C., Pacifico, M.: A dichotomy for three-dimensional vector fields. Ergodic Theory and Dynamical Systems, 23, 1575–1600 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  57. Morales, C., Pacifico, M., Pujals, E.: Robust transitive singular sets for 3-flows are partially hyperbolic attractors or repellers. Ann. of Math., 160, 375–432 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  58. Newhouse, S.: Nondensity of axiom A(a) on S2. In: Global Analysis Proc. Symp. Pure Math. AMS, Vol. 14, 1970, 191–202

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

    Article  MathSciNet  MATH  Google Scholar 

  60. Newhouse, S.: The abundance of wild hyperbolic sets and nonsmooth stable sets for diffeomorphisms. Inst. Hautes Etudes Sci. Publ. Math., 50, 101–151 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  61. Pacifico, M., Yang, F., Yang, J. G.: Entropy theory for sectional hyperbolic flows, accepted by Ann. Inst. H. Poincaré Anal. Non Linéaire, 2020, arXiv:1901.07436

  62. Palis, J.: Homoclinic bifurcations, sensitive-chaotic dynamics and strange attractors. In: Dynamical Systems and Related Topics (Nagoya, 1990). Adv. Ser. Dynam. Systems, Vol. 9, World Sci. Publishing, River Edge, NJ, 1991, 466–472

    Google Scholar 

  63. Palis, J.: A global view of dynamics and a conjecture of the denseness of finitude of attractors. Astérisque, 261, 335–347 (2000)

    MathSciNet  MATH  Google Scholar 

  64. Palis, J.: A global perspective for non-conservative dynamics. Ann. Inst. H. Poincare Anal. Non Lineaire, 22, 485–507 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  65. Palis, J.: Open questions leading to a global perspective in dynamics. Nonlinearity, 21, 37–43 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  66. Palis, J., Smale, S.: Structural stability theorems, Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968), Amer. Math. Soc., Providence, RI, 1970, 223–231

    MATH  Google Scholar 

  67. Pliss, V.: On a conjecture due to Smale. Diff. Uravenenija, 8, 262–268 (1972)

    Google Scholar 

  68. Peixoto, M. M.: Structural stability on two-dimensional manifolds. Topology, 1, 101–120 (1962)

    Article  MathSciNet  MATH  Google Scholar 

  69. Pugh, C.: The closing lemma. Amer. J. Math., 89, 956–1009 (1967)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  71. Robinson, C.: Structural stability of C1 flows. Lecture Notes in Mathematics, Vol. 468, Springer-Verlag, Berlin/New York, 1974

    Google Scholar 

  72. Pujals, E., Sambarino, M.: Homoclinic tangencies and hyperbolicity for surface diffeomorphisms. Annals of Math., 151, 961–1023 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  73. Ruelle, D.: A measure associated with axiom-A attractors. Amer. J. Math., 98, 619–654 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  74. Ruelle, D.: Positivity of entropy production in non equilibrium statistical mechanics. Journal of Statistical Physics, 85, 1–23 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  75. Ruelle, D.: Historical behaviour in smooth dynamical systems. In: Global Analysis of Dynamical Systems, Inst. Phys., Bristol, 2001, 63–66

  76. Shi, Y., Gan, S. B., Wen, L.: On the singular-hyperbolicity of star flows. J. Mod. Dyn., 8(2), 191–219 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  77. Simon, R. C.: A 3-dimensional Abraham-Smale example. Proc. Amer. Math. Soc., 34, 629–630 (1972)

    MathSciNet  MATH  Google Scholar 

  78. Sinai, Y.: Gibbs measures in ergodic theory. Uspehi Mat. Nauk, 27, 21–64 (1972)

    MathSciNet  MATH  Google Scholar 

  79. Smale, S.: Diffeomorphisms with many periodic points, In: Differential and Combinatorial Topology (A Symposium in Honor of Marston Morse), Princeton Univ. Press, Princeton, NJ, 1965, 63–80

    Chapter  Google Scholar 

  80. Smale, S.: Dynamical Systems on n-dimensional Manifolds, Symposium on Differential Equations and Dynamical Systems (Puerto Rico), Academic Press, New York, 1967

    Google Scholar 

  81. Smale, S.: Differentiable dynamical systems. Bull. Amer. Math. Soc., 73, 747–817 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  82. Viana, M.: What’s new on Lorenz strange attractors? Math. Intelligencer, 22, 6–19 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  83. Wen, L.: On the C1 stability conjecture for flows. J. Differential Equations, 129, 334–357 (1996)

    Article  MathSciNet  Google Scholar 

  84. Xiao, Q. Y., Zheng, Z. H.: C1 weak Palis conjecture for nonsingular flows. Discrete Contin. Dyn. Syst., 38, 1809–1832 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  85. Yang, D., Zhang, Y.: On the finiteness of uniform sinks. J. Differential Equations, 257, 2102–2114 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  86. Yang, J. G.: Cherry flow: physical measures and perturbation theory. Ergodic Theory Dynam. Systems, 37, 2671–2688 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  87. Yoccoz, J. C.: Recent developments in dynamics. In: Proceedings of the International Congress of Mathematicians, Vols. 1, 2 (Zuärich, 1994), Birkhaäuser, Basel, 1995, 246–265

    MATH  Google Scholar 

  88. Zhu, S., Gan, S. B., Wen, L.: Indices of singularities of robustly transitive sets. Disc. Cont. Dynam. Syst., 21, 945–957 (2008)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

I would like to thank Y. Zang for providing some pictures.

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, Soochow University, Suzhou, 215006, P. R. China

    Da Wei Yang

Authors
  1. Da Wei Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Da Wei Yang.

Additional information

Supported by NSFC (Grant Nos. 11671288, 11822109, 11790274)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Yang, D.W. The Dynamics of Vector Fields with Singularities. Acta. Math. Sin.-English Ser. 38, 225–248 (2022). https://doi.org/10.1007/s10114-022-0471-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10114-022-0471-5

Keywords

MR(2010) Subject Classification

Search

Navigation