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1995 | 62 | 3 | 265-281
Tytuł artykułu

Versal deformations of $D_q$-invariant 2-parameter families of planar vector fields

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
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.
Słowa kluczowe
Rocznik
Tom
62
Numer
3
Strony
265-281
Opis fizyczny
Daty
wydano
1995
otrzymano
1994-10-27
poprawiono
1994-12-15
Twórcy
  • Institute of Mathematics, University of Warsaw, Banacha 2, 02-097 Warszawa, Poland
Bibliografia
  • [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] V. I. Arnold, Geometrical Methods in the Theory of Ordinary Differential Equations, Springer, New York, 1983.
  • [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).
  • [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] 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] 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] M. Golubitsky, I. Stewart and D. Shaefer, Singularities and Groups in Bifurcation Theory, Vol. 2, Appl. Math. Sci. 69, Springer, New York, 1988.
  • [8] J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems and Bifurcations of Planar Vector Fields, Springer, New York, 1983.
  • [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).
  • [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.
  • [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).
  • [15] H. Żołądek, Bifurcations of a certain family of planar vector fields tangent to axes, J. Differential Equations 67 (1987), 1-55.
Typ dokumentu
Bibliografia
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Identyfikator YADDA
bwmeta1.element.bwnjournal-article-apmv62z3p265bwm
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