Contemporary MathematicsMathematics
Turning Points and Bifurcations for Homotopies of Analytic Maps
Eugene Allgower, Stefan-Gicu Cruceanu, and Simon J. Tavener
ABSTRACT. Curves that are implicitly defined by equations H(t, z) = 0, where the
mapping H : [0, 1] ×
Cn

Cn
is analytic in the z-variables and smooth in the t-
variable are studied under arclength parametrization. Turning points at which
˙
t = 0 and
¨
t = 0 are seen to be bifurcation points (where the dot notation refers to differentiation with
respect to the arclength parameter). At the bifurcation point the bifurcating curves have
orthogonal tangents and the same absolute curvature, but of opposite sign. In particular,
numerical homotopy methods for polynomial systems of equations can be facilitated when
a predictor-corrector continuation algorithm is used. By switching branches at turning
points, the homotopy paths can be monotonically traced in the t-variable.
1. Introduction
Numerical continuation methods have found successful use in the approximation of
solution curves of ordinary and partial differential equations involving a so-called bifur-
cation parameter. In this setting, a single connected component is traced and points of
special interest, such as turning points and bifurcation points may be sought. Another
frequent application of numerical continuation involves the approximation of all of the
complex solutions of systems of n complex polynomials in n complex variables. In this
setting, if the system is nondegenerate, there are theoretical results concerning the number
of solutions the system has. A considerable amount of effort has gone into formulating
efficient homotopies so that all of the complex solutions of a polynomial system may be
found by tracing homotopy paths, see for example, [5, 6, 8].
This has led to attempts to approximate all of the solutions to certain types of bound-
ary value problems by means of putting finite difference approximations into a complex
setting, and applying the homotopy method to the resulting system, see [1, 2, 4]. One of
the challenges to this approach is that the number of solutions to the system generally be-
comes very large if the number of mesh points becomes large. For example, for polynomial
systems the number of solutions is the product of the degrees of the individual equations
(counting multiplicities and points at infinity). Yet on the other hand, only the (usually
much smaller) number of purely real solutions is of interest. The remedy taken in [1, 2, 4]
has been to start with a very coarse finite difference mesh and to refine it by introducing
a new mesh point in a continuous way. After the homotopy paths have been traced to
conclusion the process may be repeated.
1991 Mathematics Subject Classification. Primary, 65H20 ; Secondary, 65H10, 14Q05.
c 0000 (copyright holder)
1
Contemporary
Volume 496, 2009
c 2009 American Mathematical Society
1
http://dx.doi.org/10.1090/conm/496/09715
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