On an application of dynamical systems theory to determine all the zeros of a vector function

Journal of Differential Equations

2001

This paper addresses stability properties of singular equilibria arising in quasilinear implicit ODEs. Under certain assumptions, local dynamics near a singular point may be described through a continuous or directionally continuous vector field. This fact motivates a classification of geometric singularities into weak and strong ones. Stability in the weak case is analyzed through certain linear matrix equations, a singular version of the Lyapunov equation being especially relevant in the study. Weak stable singularities include singular zeros having a spherical domain of attraction which contains other singular points. Regarding strong equilibria, stability is proved via a Lyapunov Schmidt approach under additional hypotheses. The results are shown to be relevant in singular root-finding problems. Academic Press

Section: Introductionmentioning

confidence: 99%

Stability of Singular Equilibria in Quasilinear Implicit Differential Equations

Riaza

Journal of Differential Equations

2001

“…For such a characterization, the next section brings together the global analysis developed by Zufiria and Guttalu [24] for the system (2) with the switch of sign algorithm of Branin.…”

Section: Branin's Algorithmmentioning

confidence: 99%

“…Zufiria and Guttalu [24) have elaborated on the relationship between the solutions of (1) and the trajectories of the dynamical system (2) (referred to as continuous Newton method) as well as on its typical global behavior. They also have shown that the system (2) can be integrated to obtain…”

Section: Branin's Algorithmmentioning

confidence: 99%

“…The role of this procedure in the algorithm becomes evident when a detailed study of the global behavior of the dynamical system composed of both the systems (2) and (3) is performed. A brief review of the global behavior of the system (2), taken from Zufiria and Guttalu [24], is first provided which is then accompanied by results for the system (3). The results for the combined dynamical system (2)-(3) follow later.…”

Section: Global Dynamics Of System (2)-(3)mentioning

confidence: 99%

“…For this purpose, a theory developed by Zufiria and Guttalu [24) for locating all the zeros of f will be employed.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Global Behavior of Branin’s Method

Guttalu

1994

Mechanics and Control

An analysis of Branin's method for finding all the real zeros of a vector function is carried out. It is based on a global study of this method perceived of as a dynamical system. Since Branin's algorithm is closely related to homotopy methods, this paper sheds some light on the global performance of these methods when employed for locating all the zeros of a vector function. We provide two examples for which the global behavior of Branin's algorithm is carried out using the cell-to-cell mapping computational technique.

Weak Singularities and the Continuous Newton Method

Riaza

Journal of Mathematical Analysis and Applications

1999

Self Cite

Some issues related to the determination of the singular roots of a nonlinear vector function f : ‫ޒ‬ n ª ‫ޒ‬ n are addressed in this paper. It is usually assumed that Newton-like fields are not defined at singularities; thus a particular treatment for these points is necessary. Nevertheless, in dimension 1 and in several higher dimensional instances it is possible to make a smooth extension of the field to singular points; when this is the case for a singular root, it can be treated in a way similar to that of regular ones. Necessary and sufficient conditions for this extension to be possible are given, under some structural assumptions, through the concept of weak singularity. The actual setting for this result is a general class of quotient functions which includes, in particular, the Newton field. For the specific case of the continuous-time Newton method, we enlarge some previous results concerning the relation between singular roots and equilibrium points of the extended field, as well as their asymptotic stability. Finally, a computational tool obtained from the extended continuous Newton method by means of the cell mapping technique is shown to be well behaved for the location of these singular roots. ᮊ