Algebra lineal y algunas de sus aplicaciones by L.I. GOLOVINA

By L.I. GOLOVINA

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Since We show that U ~ is a contraction in the supremum is not complete in the supremum norm, the con~ traction mapping principle does not imply that point in U, but it does imply that the completion of ii) U has a fixed ~ has a fixed point in with respect to the supremum norm. We show that the completion of U with respect to the supremum norm is contained in the set of functions from to y X th k with Lipschitz-continuous and with a second-order zero at the origin. point u* ~ of derivatives Thus, the fixed has the differentiability asserted in the theorem.

Associated with this orbit is a Poincar~ map. To define the Poincar~ map, let let N X be a point on the orbit, o be a codimension one manifold through transverse to the orbit. 15). The poincar~ map is a N U to V - P~(U) eN, with P~(xo) = Xo condition is spelled out below, and is reduced to a specific hypothesis on X in Section 4A. See also Section 4C. The case in which d Re )"(~)/d~ = 0 is discussed in Section 3A. In Section 3B it is shown that "vague attractor" can be replaced by "asymptotically stable".

L Assume further there is a real, simple, isolated eigenvalue such that O. 1) iR. 1) The curve is tangent to (Figure L 18). at IH These points and the are the only fixed points of in (0,0). 2). Hint: iH x iR Pick an eigenvector with eigenvalue 1. 1). tangent to (z,O) and Choose coordinates is the projection to the normalized Set in THE HOPF BIFURCATION AND ITS APPLICATIONS 26 these coordinates. l) _ 1 0: and we use the implicit function theorem to get a curve of zeros of C. g in (See Ruelle-Takens [1, p.

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