Skip to main content

Towards the General Theory of Global Planar Bifurcations

  • Conference paper
  • First Online:
Book cover Mathematical Sciences with Multidisciplinary Applications

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 157))

Abstract

This is an outline of a theory to be created, as it was seen in April 2015. An addendnum to the proofs at the end of the chapter describes the recent developments.

To Christiane Rousseau, a wonderful mathematician and organizer of scientific life, and a dear friend.

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

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 169.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 219.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 219.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Arnold, V.I, Afrajmovich, V.S., Ilyashenko, Y.S., Shilnikov, L.P.: Bifurcation Theory and Catastrophe Theory. Translated from the 1986 Russian original by N. D. Kazarinoff, Reprint of the 1994 English edition from the series Encyclopedia of Mathematical Sciences [Dynamical systems. V, Encyclopedia Mathematical Science, vol. 5, viii+271 pp. Springer, Berlin (1994); Springer, Berlin (1999)

    Google Scholar 

  2. Dumortier, F., Roussarie, R., Rousseau, C.: Hilbert’s 16th problem for quadratic vector fields. J. Differ. Equ. 110 (1), 86–133 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  3. Écalle, J.: Introduction aux fonctions analysables et preuve constructive de la conjecture de Dulac (French). Hermann, Paris (1992)

    Google Scholar 

  4. Fedorov, R.M.: Upper bounds for the number of orbital topological types of polynomial vector fields on the plane “modulo limit cycles” (Russian). Uspekhi Mat. Nauk 59 (3)(357), 183–184 (2004). Translation in Russian Math. Surveys 59 (3), 569–570 (2004)

    Google Scholar 

  5. Ilyashenko, Y.S.: Finiteness Theorems for Limit Cycles. Translated from the Russian by H. H. McFaden. Translations of Mathematical Monographs, vol. 94, x+288 pp. American Mathematical Society, Providence, RI (1991)

    Google Scholar 

  6. Ilyashenko, Y.S.: Local dynamics and nonlocal bifurcations. In: Bifurcations and Periodic Orbits of Vector Fields (Montreal, PQ, 1992). NATO Advanced Science Institute Series C: Mathematical Physical Sciences, vol. 408, pp. 279–319. Kluwer Academic Publisher, Dordrecht (1993)

    Google Scholar 

  7. Ilyashenko, Y., Weigu, L.: Nonlocal Bifurcations. Mathematical Surveys and Monographs, vol. 66, xiv+286 pp. American Mathematical Society, Providence, RI (1999)

    Google Scholar 

  8. Ilyashenko, Y., Yakovenko, S.: Smooth normal forms for local families of diffeomorphisms and vector fields. Russ. Math. Surv. 46 (1), 3–39 (1991)

    MathSciNet  Google Scholar 

  9. Ilyashenko, Y., Yakovenko, S.: Nonlinear Stokes Phenomena in smooth classification problems. In: Nonlinear Stokes Phenomena. Advances in Soviet Mathematics, vol. 14, pp. 235–287. American Mathematical Society, Providence, RI (1993)

    Google Scholar 

  10. Ilyashenko, Y., Yakovenko, S.: Finite cyclicity of elementary polycycles in generic families. In: Concerning the Hilbert 16th Problem. American Mathematical Society Translation Series 2, vol. 165, pp. 21–95. American Mathematical Society, Providence, RI (1995)

    Google Scholar 

  11. Kaleda, P.I., Schurov, I.V.: Cyclicity of elementary polycycles with a fixed number of singular points in generic k-parametric families (Russian). Algebra i Analiz 22 (4), 57–75 (2010); Translation in St. Petersburg Math. J. 22 (4), 557–571 (2011)

    Google Scholar 

  12. Kaloshin, V.: The existential Hilbert 16-th problem and an estimate for cyclicity of elementary polycycles. Invent. Math. 151 (3), 451–512 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  13. Kaloshin, V.: Around the Hilbert-Arnold problem. In: On Finiteness in Differential Equations and Diophantine Geometry. CRM Monograph Series, vol. 24, pp. 111–162. American Mathematical Society, Providence, RI (2005)

    Google Scholar 

  14. Kotova, A., Stanzo, V.: On few-parameter generic families of vector fields on the two-dimensional sphere. In: Concerning the Hilbert 16th Problem, pp. 155–201. American Mathematical Society Translation Series 2, vol. 165. American Mathematical Society, Providence, RI (1995)

    Google Scholar 

  15. Leontovich, E.: On the generation of limit cycles from separatrices (Russian). Doklady Akad. Nauk SSSR (N.S.) 78, 641–644 (1951)

    Google Scholar 

  16. Malta, I.P., Palis, J.: Families of vector fields with finite modulus of stability. In: Dynamical systems and turbulence, Lecture Notes in Mathematics, vol. 898, pp. 212–229 (1980)

    MathSciNet  MATH  Google Scholar 

  17. Roussarie, R.: On the number of limit cycles which appear by perturbation of separatrix loop of planar vector fields. Bol. Soc. Brasil Mat. 17 (2), 67–101 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  18. Roussarie, R.: Weak and continuous equivalences for families on line diffeomorphisms. In: Dynamical Systems and Bifurcation Theory (Rio de Janeiro, 1985). Pitman Research Notes in Mathematics Series, vol. 160, pp. 377–385. Longman Sci. Tech, Harlow (1987)

    Google Scholar 

  19. Roussarie, R.: A note on finite cyclicity property and Hilbert’s 16th problem. In: Dynamical Systems Valparaiso 1986. Lecture Notes in Mathematics, vol. 1331, pp. 161–168. Springer, Berlin (1988)

    Google Scholar 

  20. Stantzo, V.: Bifurcations of the polycycle “Saddle Lip”. Tr. Mat. Inst. Steklova 213 (1997), Differ. Uravn. s Veshchestv. i Kompleks. Vrem., 152–212. Translation in Proc. Steklov Inst. Math. 213 (2), 141–199 (1996)

    Google Scholar 

  21. Trifonov, S.: Desingularization in families of analytic differential equations. In: Concerning the Hilbert 16th Problem. American Mathematical Society Translation Series 2, vol. 165, pp. 97–129. American Mathematical Society, Providence, RI (1995)

    Google Scholar 

  22. Trifonov, S.I.: Cyclicity of elementary polycycles of generic smooth vector fields (Russian). Tr. Mat. Inst. Steklova 213 (1997). Differ. Uravn. s Veshchestv. i Kompleks. Vrem, 152–212; Translation in Proc. Steklov Inst. Math. 213 (2), 141–199 (1996)

    Google Scholar 

Download references

Acknowledgements

The article was prepared within the framework of the Academic Fund Program at the National Research University Higher School of Economic (HSE) in (2016–17) (grant # 16-05-0066) and supported within the framework of a subsidy granted to the HSE by the Government of Russian Federation for the implementation of the Global Competitiveness program.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Y. Ilyashenko .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2016 Springer International Publishing Switzerland

About this paper

Cite this paper

Ilyashenko, Y. (2016). Towards the General Theory of Global Planar Bifurcations. In: Toni, B. (eds) Mathematical Sciences with Multidisciplinary Applications. Springer Proceedings in Mathematics & Statistics, vol 157. Springer, Cham. https://doi.org/10.1007/978-3-319-31323-8_13

Download citation

Publish with us

Policies and ethics