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A simple global characterization for normal forms of singular vector fields. (English) Zbl 0633.58020
We derive a new global characterization of the normal forms of amplitude equations describing the dynamics of competing order parameters in degenerate bifurcation problems. Using an appropriate scalar product in the space of homogeneous vector polynomials, we show that the nonlinearities in the normal form commute with the one parameter Lie group generated by the adjoint of the original critical linear operator. This leads to a very efficient constructive method to compute both the nonlinear coefficients and the unfolding of the normal form. Explicit examples, and results obtained when there are additional symmetries, are also presented. All the examples presented also include the proofs of completeness and minimality of the corresponding normal forms.
Recently some of the authors have generalized these techniques to include periodic perturbations [C. Elphick, G. Iooss and E. Tirapegui, Phys. Lett. A 120, 459 (1987)], dynamical systems with Grassmann variables [C. Elphick, J. Math. Phys. 28, 1243-1249 (1987), J. Phys. A 20, 4371-4382 (1987)] and to treat Hamiltonian systems [C. Elphick, “Global aspects of Hamiltonian normal forms”, Phys. Lett. A 127, 418-424 (1988)].
Reviewer: C.Elphick

MSC:
37G99 Local and nonlocal bifurcation theory for dynamical systems
34A34 Nonlinear ordinary differential equations and systems
37D99 Dynamical systems with hyperbolic behavior
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