Versal deformations of D q -invariant 2-parameter families of planar vector fields

Grzegorz Świrszcz

Annales Polonici Mathematici (1995)

  • Volume: 62, Issue: 3, page 265-281
  • ISSN: 0066-2216

Abstract

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The paper deals with 2-parameter families of planar vector fields which are invariant under the group D q for q ≥ 3. The germs at z = 0 of such families are studied and versal families are found. We also give the phase portraits of the versal families.

How to cite

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Grzegorz Świrszcz. "Versal deformations of $D_q$-invariant 2-parameter families of planar vector fields." Annales Polonici Mathematici 62.3 (1995): 265-281. <http://eudml.org/doc/262832>.

@article{GrzegorzŚwirszcz1995,
abstract = {The paper deals with 2-parameter families of planar vector fields which are invariant under the group $D_q$ for q ≥ 3. The germs at z = 0 of such families are studied and versal families are found. We also give the phase portraits of the versal families.},
author = {Grzegorz Świrszcz},
journal = {Annales Polonici Mathematici},
keywords = {versal family; bifurcation; $D_q$-invariant; classification of families of planar vector fields; bifurcation diagrams},
language = {eng},
number = {3},
pages = {265-281},
title = {Versal deformations of $D_q$-invariant 2-parameter families of planar vector fields},
url = {http://eudml.org/doc/262832},
volume = {62},
year = {1995},
}

TY - JOUR
AU - Grzegorz Świrszcz
TI - Versal deformations of $D_q$-invariant 2-parameter families of planar vector fields
JO - Annales Polonici Mathematici
PY - 1995
VL - 62
IS - 3
SP - 265
EP - 281
AB - The paper deals with 2-parameter families of planar vector fields which are invariant under the group $D_q$ for q ≥ 3. The germs at z = 0 of such families are studied and versal families are found. We also give the phase portraits of the versal families.
LA - eng
KW - versal family; bifurcation; $D_q$-invariant; classification of families of planar vector fields; bifurcation diagrams
UR - http://eudml.org/doc/262832
ER -

References

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  1. [1] V. I. Arnold, On the loss of stability of oscillations near resonance and deformations of equivariant vector fields, Funktsional. Anal. i Prilozhen. 6 (2) (1977), 1-11 (in Russian). 
  2. [2] V. I. Arnold, Geometrical Methods in the Theory of Ordinary Differential Equations, Springer, New York, 1983. Zbl0507.34003
  3. [3] F. S. Berezovskaya and A. I. Khibnik, On bifurcations of separatrices in the problem of loss of stability of self-oscillations near the 1:4 resonance, Prikl. Mat. Mekh. 44 (1980), 938-943 (in Russian). Zbl0485.58015
  4. [4] R. I. Bogdanov, Versal deformations of singular points of vector fields on the plane in the case of zero eigenvalues, Trudy Sem. Petrovsk. 2 (1976), 37-65 (in Russian). 
  5. [5] F. Dumortier and R. Roussarie, On the saddle loop bifurcation, in: Bifurcations of Planar Vector Fields (Luminy 1989), Lecture Notes in Math. 1455, Springer, New York, 1990, 44-73. 
  6. [6] N. K. Gavrilov, On bifurcations of equilibrium state with one zero and one pair of pure imaginary roots, in: Methods of Qualitative Theory of Differential Equations, Gorki Univ., 1980, 33-40 (in Russian). 
  7. [7] M. Golubitsky, I. Stewart and D. Shaefer, Singularities and Groups in Bifurcation Theory, Vol. 2, Appl. Math. Sci. 69, Springer, New York, 1988. 
  8. [8] J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems and Bifurcations of Planar Vector Fields, Springer, New York, 1983. Zbl0515.34001
  9. [9] E. I. Khorozov, Versal deformations of equivariant vector fields in the case of symmetry of order 2 and 3, Trudy Sem. Petrovsk. 5 (1979), 163-192 (in Russian). Zbl0446.58010
  10. [10] A. I. Nieĭshtadt, Bifurcations of the phase portrait of a certain system of equations arising in the problem of loss of stability of self-oscillations near the 1:4 resonance, Prikl. Mat. Mekh. 42 (1978), 830-840 (in Russian). 
  11. [11] F. Takens, Forced oscillations and bifurcations, in: Applications of Global Analysis I, Comm. Math. Inst. Rijksuniv. Utrecht 3 (1974). 
  12. [12] A. Zegeling and R. E. Kooij, Uniqueness of limit cycles in polynomial systems with algebraic invariants, Bull. Austral. Math. Soc. 49 (1994), 7-20. Zbl0802.34030
  13. [13] A. Zegeling and R. E. Kooij, Equivariant unfoldings in the case of symmetry of order 4, preprint TU Delft, 1992. 
  14. [14] H. Żołądek, On versality of a certain family of symmetric vector fields on the plane, Mat. Sb. 120 (1983), 473-499 (in Russian). Zbl0516.58032
  15. [15] H. Żołądek, Bifurcations of a certain family of planar vector fields tangent to axes, J. Differential Equations 67 (1987), 1-55. Zbl0648.34068

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