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

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