The Poincare-Dulac theorem for nonlinear representations of nilpotent lie algebras

Arnal, Didier; Ammar, Mabrouk Ben; Pinczon, Georges

doi:10.1007/bf00400976

Cited by 19 publications

(22 citation statements)

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“…The introduction of normal forms with respect to a non-semisimple matrix requires a more difficult Ž w x. procedure and will not be considered here cf. 10,11 . Ž Up to a linear change of coordinates possibly after complexification of .…”

Section: Basic Assumptions and Preliminariesmentioning

confidence: 97%

Resonant Bifurcations

Cicogna¹

2000

Journal of Mathematical Analysis and Applications

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We consider dynamical systems depending on one or more real parameters, and assuming that, for some ''critical'' value of the parameters, the eigenvalues of the linear part are resonant, we discuss the existenceᎏunder suitable hypothesesᎏof a general class of bifurcating solutions in correspondence with this resonance. These bifurcating solutions include, as particular cases, the usual stationary and Hopf bifurcations. The main idea is to transform the given dynamical system into Ž . normal form in the sense of Poincare and Dulac and to impose that thé normalizing transformation is convergent, using the convergence conditions in the form given by A. Bruno. Some specifically interesting situations, including the cases of multiple-periodic solutions and of degenerate eigenvalues in the presence of symmetry, are also discussed in some detail. ᮊ

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Section: Basic Assumptions and Preliminariesmentioning

confidence: 97%

Resonant Bifurcations

Cicogna¹

2000

Journal of Mathematical Analysis and Applications

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“…(7) 8 Equivalently, this could be defined in terms of a (set of) differential form, the contact forms, which should vanish identically. 9 Note that this projection could take smooth manifolds into non-smooth ones, as can be seen e.g. by considering solutions of the ordinary differential equation U2 -X.…”

Section: Geometrical Settingmentioning

confidence: 99%

“…This function defines a graph r(f) in M r(f)-f(x,u)EM: u=f(X)JCM (8) and once again we identify an analytic object (the function f) and the corresponding geometrical one (the manifold F(f)). Quite obviously, once we have u =: f (x), we also have all of its partial derivatives, so r(f) can be lifted, as discussed above, to a manifold in MW: (9) this is automatically compatible with the contact structure.…”

Section: Geometrical Settingmentioning

confidence: 99%

“…Let us first consider the action of Diff (M) on an algebraic equation :A, which we write in the form F(x, u) --0. As it was discussed before, A identifies a solution manifold SF in M; notice that this corresponds to a level surface of F. If we are given a vector field X E diff (M), this will generate a one-parameter subgroup To of Diff (M); in general To : (x, u) --> (g.0 (x, u), h,9 (x, u)), so that T,9 : SF --T, 9 (SF) = f (x, u) c M : F(g-,g (x, u), h-,o (x, u)) = 01 (39) The condition needed in order to have SF invariant under T,9, i.e. T,9 : SF --+ SF, is, in terms of the vector field, X : SF --> TSF; we define the symmetry group of F as GF = Ig E Diff(M): 9: SF ' SFJ (40) and-its algebra is…”

Section: Invariance Of Equations and Of Solutionsmentioning

confidence: 99%

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Symmetry and Perturbation Theory in Nonlinear Dynamics

1999

Lecture Notes in Physics

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“…Normaliser le champ T formellement, (1) car alors on peut utiliser la structure de l'algèbre de Lie en question pour préciser les formes normales possibles.…”

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