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On a codimension 3 bifurcation of plane vector fields with \({\mathbb{Z}}_ 2\) symmetry. (English) Zbl 0724.34014

The paper deals with a 3-parameter family of plane vector fields \((1)\quad \dot x=f(x,\lambda):=A(\lambda)x+h(x,\lambda),\) \(x\in {\mathbb{R}}^ 2\), \(\lambda \in {\mathbb{R}}^ 3\), \(f=(f_ 1,f_ 2)\) in \(C^{\infty}\), with \(h(0,\lambda)=0\) for all \(\lambda\), \(-h(- x,\lambda)=h(x,\lambda)\) for all \(\lambda\),x. Under suitable assumptions on A(\(\lambda\)) and f, (0,0) is a singularity of codimension 3. Using normal forms (1) is reduced to a simple system. Then, it is shown the existence of a homoclinic trajectory as well as periodic trajectories surrounding three equilibrium points and periodic trajectories surrounding single equilibrium points.

MSC:

34C23 Bifurcation theory for ordinary differential equations
34C20 Transformation and reduction of ordinary differential equations and systems, normal forms
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[1] V. I. Arnold: Geometric Methods in the Theory of Ordinary Differential Equations. Springer-Verlag, New York 1983.
[2] N. N. Bautin, E. A. Leontovich: Methods and Examples of Qualitative Study of Dynamical Systems in the Plane. Nauka, Moscow 1978
[3] R. I. Bogdanov: Versal deformations of a singular point of vector fields in the plane in the case of zero eigenvalues. Selecta Math. Soviet 1 (1981), 389-421 (Proc. of Petrovski Seminar, 2 (1976), 37-65) · Zbl 0518.58030
[4] R. I. Bogdanov: Bifurcations of limit cycles of a certain family of vector fields in the plane. Selecta Math. Soviet 1 (1981), 373-387 (Proc. of Petrovski Seminar 2 (1976), 23-36) · Zbl 0518.58029
[5] J. Carr: Application of Center Manifold Theory. Springer-Verlag, New York 1981. · Zbl 0464.58001
[6] J. Carr S. N. Chow, J. K. Hale: Abelian integrals and bifurcation theory. J. Differential Equations 59 (1985), 413-437. · Zbl 0587.34033
[7] S. N. Chow, J. K. Hale: Methods of Bifurcation Theory. Springer-Verlag, New York 1982. · Zbl 0487.47039
[8] S. N. Chow, J. A. Sanders: On the number of critical points of the period. J. Differential Equations 64 (1986), 51-66. · Zbl 0594.34028
[9] R. Cushman, J. A. Sanders: A codimension two bifurcation with a third-order Picard-Fuchs equation. J. Differential Equations 59 (1985), 243 - 256. · Zbl 0571.34021
[10] G. Dangelmayer, J. Guckenheimer: On a four parameter family of plane vector fields. Archive for Rational Mechanics and Analysis, 97 (1987), 321 - 352. · Zbl 0654.34025
[11] B. Drachman S. A. Van Gils, Zhang Zhi-Fen: Abelian integrals for quadratic vector fields. J. Reine Angew. Math. 382 (1987), 165-180. · Zbl 0621.58033
[12] F. Dumortier R. Roussarie, J Sotomayor: Generic 3-parameter families of vector fields on the plane. Unfolding a singularity with nilpotent linear part. The cusp-case of codimension 3, Ergodic Theory Dynamical Systems 7 (1987), No. 3, 375-413. · Zbl 0608.58034
[13] C. Elpic E. Tirapegni M. Brachet P. Coullet, G. Iooss: A simple global characterization of normal forms of singular vector fields. Preprint No. 109, University of Nice, 1986, Physica 29 D (1987), 95-127. · Zbl 0633.58020
[14] J. Guckenheimer: Multiple bifurcation problems for chemical reactors. Physica 20 D (1986), 1-20. · Zbl 0593.34043
[15] J. Guckenheimer: A condimension two bifurcation with circular symmetry. in ”Multiparameter Bifurcation Theory”, AMS series: Contemporary Math. 56 (1986), 175-184.
[16] J. Guckenheimer, P. Holmes: Nonlinear Oscillations. Dynamical Systems and Bifurcations of Vector Fields, Springer-Verlag, New York 1983. · Zbl 0515.34001
[17] E. I. Horozov: Versal deformation of equivariant vector fields with \(Z_2\) or \(Z_3\) symmetry. Proc. of Petrovski Seminar 5 (1979), 163-192
[18] J. K. Hale: Introduction to dynamic bifurcation. in ”Bifurcation Theory and Applications” (L. Salvadoei, pp. 106-151, LNM 1057, Springer-Verlag 1984. · Zbl 0544.58016
[19] Yu. S. Ilyashenko: Multiplicity of limit cycles arising from perturbations of the form \(w' = = P_2/Q_1\) of a Hamiltonian equation in the real and complex domain. Amer. Math. Soc. Transl. Vol. 118, No. 2, pp. 191-202, AMS, Providence, R. I., 1982.
[20] Yu. S. Ilyashenko: Zeros of special abelian integrals in a real domain. Funct. Anal. Appl. 11 (1977), 309-311. · Zbl 0413.58011
[21] M. Medved: Generic bifurcations of vector fields with a singularity of codimension 2. in ”Equadiff 5, Bratislava 1981”, Proceedings, Teubner-Texte zur Math., Band 47, Teubner-Verlag 1982, pp. 260-263.
[22] M. Medved: The unfoldings of a germ of vector fields in the plane with a singularity of codimension 3. Czechosl. Math. J. 35 (110), 1 (1985), 1-42. · Zbl 0591.58022
[23] M. Medved: Normal forms and bifurcations of some equivariant vector fields. to appear in Mathematica Slovaca 1990. · Zbl 0735.58024
[24] M. Medved: On a codimension three bifurcations. Časopis pro pěstování matem., 109 (1984), 3-26.
[25] J. A. Sanders, R. Cushman: Limit cycles in the Josephson equation. SIAM J. Math. Anal., Vol. 17, No. 3 (1986), 495-511. · Zbl 0604.58041
[26] F. Tokens: Unfoldings of certain singularities of vector fields: Generalized Hopf bifurcation. J. Differential Equations 14 (1973), 476-493. · Zbl 0273.35009
[27] F. Takens: Forced oscillations and bifurcations. in ”Applications of Global Analysis”, Comment, of Math. Inst. Rijksuniversiteit Ultrecht 1974.
[28] H. Žoladek: Bifurcation of certain families of planar vector fields tangent to the axes. Differential Equations 67 (1987), 1-55. · Zbl 0648.34068
[29] H. Žoladek: On versality of certain families of symmetric vector fields on the plane. Math. Sb. 120 (1983), 473-499. · Zbl 0516.58032
[30] H. Žoladek: Abelian integrals in unfolding of codimension 3 singular planar vector fields I. The saddle and elliptic cases, II. The focus case. To appear in Lecture Notes in Math.
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