In this chapter we present a number of results from the theory of normal forms. The idea of normal forms consists in finding a polynomial change of variable which “improves” locally a nonlinear system, in order to more easily recognize its dynamics. As we shall see, normal form transformations apply to general classes of nonlinear systems in ℝ n near a fixed point, here the origin, by just assuming a certain smoothness of the vector field. In particular, this theory applies to the reduced systems provided by the center manifold theory given in the previous chapter.


Periodic Solution Normal Form Hopf Bifurcation Solvability Condition Center Manifold 
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© EDP Sciences 2011

Authors and Affiliations

  1. 1.Laboratoire de MathématiquesUniversité de Franche-ComtéBesançon CedexFrance
  2. 2.Laboratoire J.A.DieudonnéIUF, Université de NiceNice Cedex 02France

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