# Versal deformations of ${D}_{q}$-invariant 2-parameter families of planar vector fields

Annales Polonici Mathematici (1995)

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

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topGrzegorz 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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- [2] V. I. Arnold, Geometrical Methods in the Theory of Ordinary Differential Equations, Springer, New York, 1983. Zbl0507.34003
- [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
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- [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.
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- [8] J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems and Bifurcations of Planar Vector Fields, Springer, New York, 1983. Zbl0515.34001
- [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] 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] F. Takens, Forced oscillations and bifurcations, in: Applications of Global Analysis I, Comm. Math. Inst. Rijksuniv. Utrecht 3 (1974).
- [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] A. Zegeling and R. E. Kooij, Equivariant unfoldings in the case of symmetry of order 4, preprint TU Delft, 1992.
- [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] 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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