Further reduction of the Takens-Bogdanov normal form. (English) Zbl 0761.34027

Normal forms (analytic or formal) play an important role in the theory of systems of ODE, solving birfurcation or stability type problems. The idea of normalisation is the elimination of as many as possible terms. The authors formulate and prove a unique formal classification of analytic vector fields in \(R^ 2\) with nilpotent linear part. The investigation is divided in three subcases and except one subcase it is essentially complete. The paper has a high theoretical level, it applies the technics of Lie-algebras.


34C23 Bifurcation theory for ordinary differential equations
37G99 Local and nonlocal bifurcation theory for dynamical systems
17B70 Graded Lie (super)algebras
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[1] Baider, A, Unique normal forms for vector fields and Hamiltonians, J. differential equations, 77, 33-52, (1989) · Zbl 0689.70005
[2] Baider, A; Churchill, R.C, Unique normal forms for planar vector fields, Math. Z., 199, 303-310, (1988) · Zbl 0691.58012
[3] Baider, A; Sanders, J.A, Unique normal forms: the Hamiltonian nilpotent case, J. differential equations, 92, 282-304, (1991) · Zbl 0731.58060
[4] Bogdanov, R, Versal deformations of a singular point of a vector field on the plane in the case of zero eigenvalues, (), No. 4 · Zbl 0447.58009
[5] Bogdanov, R, Bifurcation of a limit cycle for a family of vector fields on the plane, (), No. 4 · Zbl 0518.58029
[6] Cushman, R; Sanders, J.A, Nilpotent normal forms and representation theory of sl(2, R), (), 31-51
[7] Cushman, R; Sanders, J.A; Cushman, R; Sanders, J.A, Invariant theory and normal form of Hamiltonian vectorfields with nilpotent linear part, (), 353-371 · Zbl 0712.58053
[8] Dumortier, F; Roussarie, R; Sotomayor, J, 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, 375-413, (1987) · Zbl 0608.58034
[9] Dumortier, F; Roussarie, R; Sotomayor, J, Generic 3-parameter families of planar vector fields, unfoldings of saddle, focus and elliptic singularities with nilpotent linear parts, (1988), Université de Bourgogne, preprint
[10] Sanders, J.A; van der Meer, J.C, Unique normal form of the Hamiltonian 1: 2-resonance, () · Zbl 0766.58051
[11] Takens, F, Singularities of vector fields, Ihes, 43, 47-100, (1974) · Zbl 0279.58009
[12] van der Meer, J.C, The Hamiltonian-Hopf bifurcation, () · Zbl 0853.58053
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