Skip to main content
SpringerLink
Log in

On a class of semilocal bifurcations of lorenz type

  • Published:
Acta Mathematica Sinica Aims and scope Submit manuscript

Abstract

In this paper, we study a two-parameter family of systemsE c in whichE 0 as a contour consisting of a saddle point and two periodic motions of saddle type, i.e., the situation is similar to that described by Lorenz equations for parametersb=8/3, σ=10,r=r l =24.06, and get some results concerning bifurcation phenomenon and dynamical behavior of the orbits ofE c in a small neighborhood of the contour for |ε| near zero. Thus, under a few natural assumptions which are verified numerically, we can explain some numerical results of Lorenz equations for parameters near the above values in a mathematically precise way, which is different from the methods of J. Guckenheimer et al.([3], [4]), by considering Lorenz equation as a one—or two—dimensional map.

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. Lorenz, E.,Deterministic nonperiodic flow, J. Atmos. Sci., 20, 130–141.

  2. Kaplan, J. L. & Yorke, J. A.,Perturbulence, a regime observed in a fluid flow model of Lorenz, Comm. Math. Phys.,67 (1979), 93–108.

    Article  MATH  Google Scholar 

  3. Guckenheimer, J., A strange attractor, in Marsden J. and McCracken, J., The Hopf Bifurcation and Applications, Applied Mathematical Sciences No. 19: Berlin, Springer-Verlag, 1976.

    Google Scholar 

  4. Williams, R.F., The structure of Lorenz attractors Publ. Math. IHES,50, 101–152.

  5. Afraimovich, V.S., Bykov, V.V. & Shil'nikov, L.P.,On structurally unstable attraction limit sets of Lorenz attractor type, Trans. Moscow Math. Soc. 1983, Issue2, 153–216.

    Google Scholar 

  6. Sparrow, C., The Lorenz Equations, Springer-Verlag, New York, Heidelberg, Berlin.

  7. Deng Bo, Shil' nikov problem, exponential expansion strong λ-lemma,C 1-linearization and homoclinic bifurcation, to appear.

  8. Guan Ke-ying, A geometric interpretation to the Lorenz attractor (preprint).

  9. Arnol'd, V.I., Afraimovich, V.S., Ilyashenko, Yu. S., Shil'nikov, L.P., Bifurcation Theory. Itogi Nauki i Tekhniki. Sovremennye Problemy Mathematiki. Fundamentalnye Napravlenia, 5, VINITI, Moscow (1986): 5–217. (English translation: Encyclopaedia of Mathematical Sciences, Dynamical Systems 5, Springer (1988).)

    Google Scholar 

  10. Afraimovich, V. S. & shil'nikov, L.P.,On critical sets of Morse-Smale systems, Trans. Moscow Math. Soc.,28 (1973), 179–212.

    Google Scholar 

  11. Li Weigu, A class of semilocal bifurcations. PhD. thesis, Department of Mathematics, Peking University.

  12. Shil'nikov, L.P.,On a Poincare-Birkhoff problem, Math. USSR-Sbornik,3(1967), 353–371.

    Article  Google Scholar 

  13. Shil'nikov, L.P., Turaev, D.V.,On bifurcations of a homoclinic “figure eight” for a saddle with a negative saddle value Soviet Math. Dokl.,34 (1987), 397–401.

    Google Scholar 

  14. Chow, S.N. & Hale, J. K., Methods of Bifurcation Theory, Springer-Verlag, New York, Heidelberg, Berlin.

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Peking University, 100871, Beijing, P.R. China

    Weigu Li

Authors
  1. Weigu Li
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Li, W. On a class of semilocal bifurcations of lorenz type. Acta Mathematica Sinica 8, 158–176 (1992). https://doi.org/10.1007/BF02629936

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02629936

Keywords

Search

Navigation