Broer, H. W.; Takens, F. Formally symmetric normal forms and genericity. (English) Zbl 0704.58047 Dynamics reported, Vol. 2, 39-59 (1989). [For the entire collection see Zbl 0659.00009.] The paper is devoted to generic properties of several classes of dynamical systems on manifolds, like Hamiltonian systems, volume preserving systems, and so on. Generic properties are known which imply a certain local simplicity of the system, but global dynamics can be very complicated. In this paper, the authors deal with such global complexity of some generic properties. Some extensions of theorems of the second author about normal form problems on diffeomorphisms and vector fields are discussed. Whenever one of their normal form theorems is applicable, authors speak of integrable approximations. Three examples of such locally integrable approximations are discussed in the paper from the point of view of symmetry and genericity: area preserving diffeomorphisms of the plane, unfolding of a Hopf saddle-node and period doubling node. These are examples of locally integrable approximation not generic due to symmetry. Since non structural stability arises even in the generic case, authors deal with the dynamics of this “generic case” in the presence of a “non generic integrable approximation”, independent from possible perturbations to dynamical system which are topologically different. Reviewer: B.Valino Alonso Cited in 1 ReviewCited in 11 Documents MSC: 37G05 Normal forms for dynamical systems 37C80 Symmetries, equivariant dynamical systems (MSC2010) Keywords:generic properties of dynamical systems; normal forms; symmetry groups; Hamiltonian systems; volume preserving systems Citations:Zbl 0659.00009 PDFBibTeX XML