Sektion 6
Freitag, 22.09.2000, 17.00–17.20 Uhr, WIL C 129

An Extended System for Winged Cusp Points

Uwe Schnabel, TU Dresden, Institut für Numerische Mathematik

Gerd Pönisch, TU Dresden, Institut für Numerische Mathematik

A point  (x*, c*, a*)  is called a winged cusp point of the nonlinear system  F (x, c, a) = 0 , F : Rn × R1 × R3 --> Rn, if

rank@xF (x*,c*, a*) = rank@(x,c)F (x*,c*,a*) = n -  1,
          *  *  *
rank@F  (x  ,c ,a )  = n,

and if the Ljapunov-Schmidt reduced function has the normal form g(q, c) = ±q3 ± c2,  g : R1 × R1 --> R1. A minimally extended system F (x, c, a) = 0, f(x, c, a) = 0 is proposed for defining winged cusp points, where f : Rn × R1 × R3 --> R4 are 4 scalar functions characterizing the winged cusp. The functions fi : Rn × R1 × R3 --> R1, i = 1, ..., 4, depend on the partial derivatives @ qg, @cg, @q2g, and @q@cg. For the description of the fi an evaluable type of Ljapunov-Schmidt reduced function is introduced. The regular solution (x*, c*, a*) of the (n + 4)-dimensional extended system delivers the desired winged cusp point (x*, c*) as first part. For numerically solving these systems, a two-stage Newton-type method are proposed. Computational differentiation is applied for computing the several partial derivatives needed. A numerical example is given.

The talk is based on the papers:

[1] Schnabel, U., Pönisch, G., Janovský, V. Reduced functions characterizing singular points and their relations. Preprint IOKOMO–04–1999, TU Dresden, 1999.
[2] Schnabel, U., Walther, A. Berechnung singulärer Punkte: Newtonverfahren und Automatische Differentiation. Preprint IOKOMO–05–1999, TU Dresden, 1999.