Hostname: page-component-89b8bd64d-46n74 Total loading time: 0 Render date: 2026-05-09T05:08:37.738Z Has data issue: false hasContentIssue false
  • English
  • Français

An example derived from the Lorenz attractor

Part of: Smooth dynamical systems: general theory Dynamical systems with hyperbolic behavior

Published online by Cambridge University Press:  24 February 2026

MING LI ,
FAN YANG ,
JIAGANG YANG  and
RUSONG ZHENG
MING LI
Affiliation:
School of Mathematical Sciences, Nankai University, Tianjin, PR China (e-mail: limingmath@nankai.edu.cn)
FAN YANG*
Affiliation:
Department of Mathematics, Wake Forest University, Winston-Salem, USA
JIAGANG YANG
Affiliation:
Departamento de Geometria, UFF, Niteroi, Brazil (e-mail: yangjg@impa.br)
RUSONG ZHENG
Affiliation:
Department of Computational Mathematics and Control, Shenzhen MSU-BIT University, Shenzhen, PR China (e-mail: zhengrs@smbu.edu.cn)
*
e-mail: yangf@wfu.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider a DA-type surgery of the famous Lorenz attractor in dimension 4. This kind of surgery was first used by Smale [Differentiable dynamical systems. Bull. Amer. Math. Soc. (N.S.) 73(6) (1967), 747–817] and Mañé [Contributions to the stability conjecture. Topology 17(4) (1978), 383–396] to give important examples in the study of partially hyperbolic systems. Our construction gives the first example of a singular chain recurrence class which is Lyapunov stable, away from homoclinic tangencies, and exhibits robustly heterodimensional cycles. Moreover, the chain recurrence class has the following interesting property: there exists robustly a two-dimensional sectionally expanding subbundle (containing the flow direction) of the tangent bundle such that it is properly included in a subbundle of the finest dominated splitting for the tangent flow.

Keywords

smooth dynamicsvector fields with singularitieschain recurrence classes

MSC classification

Primary: 37C10: Vector fields, flows, ordinary differential equations
Secondary: 37D30: Partially hyperbolic systems and dominated splittings 37D45: Strange attractors, chaotic dynamics

Information

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 46 , Issue 5 , May 2026 , pp. 1189 - 1218
Check for updates
Copyright
© The Author(s), 2026. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Abraham, R. and Smale, S.. Nongenericity of $\varOmega$ -stability. Global Analysis (Proceedings of a Symposium in Pure Mathematics, 14). Ed. S.-S. Chern and S. Smale. American Mathematical Society, Providence, RI, 1970, pp. 5–8.CrossRefGoogle Scholar
Afraimovich, V., Bykov, V. and Shil’nikov, L.. On the origin and structure of the Lorenz attractor. Dokl. Akad. Nauk 234 (1977), 336–339.Google Scholar
Araújo, V. and Melbourne, I.. Existence and smoothness of the stable foliation for sectional hyperbolic attractors. Bull. Lond. Math. Soc. 49(2) (2017), 351–367.CrossRefGoogle Scholar
Araújo, V. and Pacifico, M. J.. Three-Dimensional Flows (Ergebnisse der Mathematik Und Ihrer Grenzgebiete. 3. Folge/a Series of Modern Surveys in Mathematics, 53). Springer, Berlin–Heidelberg, 2010.CrossRefGoogle Scholar
Arbieto, A. and Morales, C.. A dichotomy for higher-dimensional flows. Proc. Amer. Math. Soc. 141(8) (2013), 2817–2827.CrossRefGoogle Scholar
Arroyo, A. and Rodriguez Hertz, F.. Homoclinic bifurcations and uniform hyperbolicity for three-dimensional flows. Ann. Inst. Henri Poincaré C Anal. Non Linéaire 20(5) (2003), 805–841.Google Scholar
Barros, D., Bonatti, C. and Pacifico, M. J.. Upper, down, two-sided Lorenz attractor, collisions, merging, and switching. Ergod. Th. & Dynam. Sys. 44(10) (2024), 2737–2781.CrossRefGoogle Scholar
Bautista, S.. The geometric Lorenz attractor is a homoclinic class. Bol. Mat. 11(1) (2004), 69–78.Google Scholar
Bonatti, C.. SURVEY towards a global view of dynamical systems, for the ${C}^1$ -topology. Ergod. Th. & Dynam. Sys. 31(4) (2011), 959–993.CrossRefGoogle Scholar
Bonatti, C. and da Luz, A.. Star fows and multisingular hyperbolicity. J. Eur. Math. Soc. (JEMS) 23(8) (2021), 2649–2705.CrossRefGoogle Scholar
Bonatti, C. and Díaz, L. J.. Robust heterodimensional cycles and ${C}^1$ -generic dynamics. J. Inst. Math. Jussieu 7(3) (2008), 469–525.CrossRefGoogle Scholar
Bonatti, C., Díaz, L. J., Pujals, E. R. and Rocha, J.. Robustly transitive sets and heterodimensional cycles. Astérisque 286 (2003), 187–222.Google Scholar
Bonatti, C., Díaz, L. J. and Viana, M.. Dynamics beyond Uniform Hyperbolicity (Encyclopedia of Mathematical Sciences (Mathematical Physics), 102). Springer, Berlin, 2005.Google Scholar
Bonatti, C., Gan, S. and Wen, L.. On the existence of non-trivial homoclinic classes. Ergod. Th. & Dynam. Sys. 27(5) (2007), 1473–1508.CrossRefGoogle Scholar
Bonatti, C., Gan, S. and Yang, D.. Dominated chain recurrent class with singularities. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 14(1) (2015), 83–99.Google Scholar
Bonatti, C., Li, M. and Yang, D.. A robustly chain transitive attractor with singularities of different indices. J. Inst. Math. Jussieu 12(3) (2013), 449–501.CrossRefGoogle Scholar
Bonatti, C., Pumariño, A. and Viana, M.. Lorenz attractors with arbitrary expanding dimension. C. R. Acad. Sci. 325(8) (1997), 883–888.CrossRefGoogle Scholar
Bonatti, C. and Viana, M.. SRB measures for partially hyperbolic systems whose central direction is mostly contracting. Israel J. Math. 115(1) (2000), 157–193.CrossRefGoogle Scholar
Crovisier, S.. Birth of homoclinic intersections: a model for the central dynamics of partially hyperbolic systems. Ann. of Math. (2) 172(3) (2010), 1641–1677.CrossRefGoogle Scholar
Crovisier, S.. Partial hyperbolicity far from homoclinic bifurcations. Adv. Math. 226(1) (2011), 673–726.CrossRefGoogle Scholar
Crovisier, S. and Pujals, E. R.. Essential hyperbolicity and homoclinic bifurcations: a dichotomy phenomenon/mechanism for diffeomorphisms. Invent. Math. 201(2) (2015), 385–517.CrossRefGoogle Scholar
Crovisier, S., Sambarino, M. and Yang, D.. Partial hyperbolicity and homoclinic tangencies. J. Eur. Math. Soc. (JEMS) 17(1) (2015), 1–49.CrossRefGoogle Scholar
Crovisier, S. and Yang, D.. Homoclinic tangencies and singular hyperbolicity for three-dimensional vector fields. Preprint. 2018, arXiv:1702.05994v2.Google Scholar
Crovisier, S. and Yang, D.. Robust transitivity of singular hyperbolic attractors. Math. Z. 298(1) (2021), 469–488.CrossRefGoogle Scholar
da Luz, A.. Star flows with singularities of different indices. Preprint. 2020, arXiv:1806.09011v2.Google Scholar
Díaz, L. J.. Persistence of cycles and nonhyperbolic dynamics at heteroclinic bifurcations. Nonlinearity 8(5) (1995), 693–713.CrossRefGoogle Scholar
Gan, S. and Yang, D.. Morse-Smale systems and horseshoes for three dimensional singular flows. Ann. Sci. Éc. Norm. Supér. (4) 51(1) (2018), 39–112.Google Scholar
Gan, S., Yang, J. and Zheng, R.. Lyapunov stable chain recurrence classes for singular flows. Preprint, 2024, arXiv:2202.09742.Google Scholar
Gourmelon, N.. Generation of homoclinic tangencies by ${C}^1$ -perturbations. Discrete Contin. Dyn. Syst. 26(1) (2009), 1–42.Google Scholar
Guckenheimer, J. and Holmes, P.. Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields (Applied Mathematical Sciences, 12). Springer-Verlag, New York, 1983.CrossRefGoogle Scholar
Guckenheimer, J. and Williams, R. F.. Structural stability of Lorenz attractors. Publ. Math. Inst. Hautes Études Sci. 50 (1979), 59–72.CrossRefGoogle Scholar
Hirsch, M. W., Pugh, C. C. and Shub, M.. Invariant Manifolds (Lecture Notes in Mathematics, 583). Springer-Verlag, Berlin–New York, 1977.CrossRefGoogle Scholar
Li, D. and Turaev, D.. Persistence of heterodimensional cycles. Invent. Math. 236(3) (2024), 1413–1504.CrossRefGoogle Scholar
Li, M., Gan, S. and Wen, L.. Robustly transitive singular sets via approach of an extended linear Poincaré flow. Discrete Contin. Dyn. Syst. 13(2) (2005), 239–269.CrossRefGoogle Scholar
Liao, G., Viana, M. and Yang, J.. The entropy conjecture for diffeomorphisms away from tangencies. J. Eur. Math. Soc. (JEMS) 15(6) (2013), 2043–2060.CrossRefGoogle Scholar
Lorenz, E. N.. Deterministic nonperiodic flow. J. Atmos. Sci. 20(2) (1963), 130–141.2.0.CO;2>CrossRefGoogle Scholar
Mañé, R.. Contributions to the stability conjecture. Topology 17(4) (1978), 383–396.CrossRefGoogle Scholar
Mañé, R.. An ergodic closing lemma. Ann. of Math. (2) 116(3) (1982), 503–540.CrossRefGoogle Scholar
Metzger, R. and Morales, C.. Sectional-hyperbolic systems. Ergod. Th. & Dynam. Sys. 28(5) (2008), 1587–1597.CrossRefGoogle Scholar
Morales, C. A., Pacifico, M. J. and Pujals, E. R.. Robust transitive singular sets for 3-flows are partially hyperbolic attractors or repellers. Ann. of Math. (2) 160(2) (2004), 375–432.CrossRefGoogle Scholar
Newhouse, S. E.. Nondensity of axiom A(a) on ${S}^2$ . Global Analysis (Proceedings of a Symposium in Pure Mathematics, 1). Ed. S.-S. Chern and S. Smale. American Mathematical Society, Providence, RI, 1970, pp. 191–202.CrossRefGoogle Scholar
Newhouse, S. E.. Diffeomorphisms with infinitely many sinks. Topology 13(1) (1974), 9–18.CrossRefGoogle Scholar
Pacifico, M. J., Yang, F. and Yang, J.. Entropy theory for sectional hyperbolic flows. Ann. Inst. Henri Poincaré C Anal. Non Linéaire 38(4) (2021), 1001–1030.CrossRefGoogle Scholar
Palis, J.. A global view of dynamics and a conjecture on the denseness of finitude of attractors. Astérisque 261 (2000), 335–347.Google Scholar
Palis, J.. A global perspective for non-conservative dynamics. Ann. Inst. Henri Poincaré C Anal. Non Linéaire 22(4) (2005), 485–507.CrossRefGoogle Scholar
Palis, J.. Open questions leading to a global perspective in dynamics. Nonlinearity 21(4) (2008), 37–43.CrossRefGoogle Scholar
Palis, J.. Homoclinic bifurcations, sensitive-chaotic dynamics and strange attractors. Dynamical Systems and Related Topics (Advanced Series in Dynamical Systems, 9). Ed. K. Shiraiwa. World Scientific Publishing Co., River Edge, NJ, 1991, pp. 466–472.Google Scholar
Palis, J. and Takens, F.. Hyperbolicity and Sensitive Chaotic Dynamics at Homoclinic Bifurcations. Cambridge University Press, Cambridge, 1995.Google Scholar
Pujals, E. R. and Sambarino, M.. Homoclinic tangencies and hyperbolicity for surface diffeomorphisms. Ann. of Math. (2) 151(3) (2000), 961–1023.CrossRefGoogle Scholar
Salgado, L.. Singular hyperbolicity and sectional Lyapunov exponents of various orders. Proc. Amer. Math. Soc. 147(2) (2019), 735–749.CrossRefGoogle Scholar
Salgado, L.. Star flows: a characterization via Lyapunov functions. Bull. Braz. Math. Soc. (N.S.) 55(3) (2024), 31.CrossRefGoogle Scholar
Shi, Y., Gan, S. and Wen, L.. On the singular-hyperbolicity of star flows. J. Mod. Dyn. 8(2) (2014), 191–219.CrossRefGoogle Scholar
Shi, Y., Yang, F. and Yang, J.. A countable partition for singular flows, and its application on the entropy theory. Israel J. Math. 249(1) (2022), 375–429.CrossRefGoogle Scholar
Shil’nikov., L. P. A case of the existence of a denumerable set of periodic motions. Dokl. Akad. Nauk 160(3) (1965), 558–561.Google Scholar
Smale, S.. Differentiable dynamical systems. Bull. Amer. Math. Soc. (N.S.) 73(6) (1967), 747–817.CrossRefGoogle Scholar
Tucker, W.. The Lorenz attractor exists. C. R. Acad. Sci. Ser. Math. 328(12) (1999), 1197–1202.Google Scholar
Turaev, D. V. and Shil’nikov, L. P.. An example of a wild strange attractor. Sb. Math. 189(2) (1998), 291–314.CrossRefGoogle Scholar
Wen, L.. Homoclinic tangencies and dominated splittings. Nonlinearity 15(5) (2002), 1445.CrossRefGoogle Scholar
Williams, R.. The ‘DA’ maps of Smale and structural stability. Global Analysis (Proceedings of a Symposium in Pure Mathematics, 14). Ed. S.-S. Chern and S. Smale. American Mathematical Society, Providence, RI, 1970, pp. 329–334.CrossRefGoogle Scholar