Hostname: page-component-745bb68f8f-b95js Total loading time: 0 Render date: 2025-01-30T21:54:20.153Z Has data issue: false hasContentIssue false
  • English
  • Français

Newton's diagram method for nonlinear equations with several small parameters

Published online by Cambridge University Press:  17 February 2009

Peter Aizengendler
Peter Aizengendler
Affiliation:
Professor Peter Aizengendler, late of Pscov University, Russia, died in November 2000. This paper, his last mathematical testament, is published with the kind consent of his son, Dr Mark Aizengendler, 11 Varram Way, West Lakes Shore, SA 5020; e-mail:ark15jan@yahoo.com.au.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this article, we generalise Newton's diagram method for finding small solutions ξ(λ) of equations f (ξ,λ) = 0 (0,0) = 0 with f analytic (see [1, 2, 4, 6]) to the case of a multi-dimensional function f, unknown variable ζ and small parameter λ. This method was briefly described in [1]. The method has many different applications and allows one to solve some inflexible problems. In particular, the method can be used in very difficult bifurcation problems, for example, for systems with small imperfections.

Type
Research Article
Information
The ANZIAM Journal , Volume 44 , Issue 2 , October 2002 , pp. 247 - 259
Copyright
Copyright © Australian Mathematical Society 2002

References

[1]Aizengendler, P. G., “Newton's method in the theory of implicit functions”, Izv. Vysš. Učebn. Zaved. Matematika 7 (110) (1971) 3–8, (Russian) MR 46.Google Scholar
[2]Aizengendler, P. G., “On asymptotics and number of small solutions of nonlinear equations”, in Functional analysis, No. 8 (Russian), (Ulyanovsk. Gos. Ped. Inst., Ul'yanovsk, 1977) 3–14.Google Scholar
[3]Aizengendler, P. G., “Methods of post-bifurcation analysis in systems with a small vector parameter”, in Functional analysis, No. 26 (Russian), (Ul'yanovsk. Gos. Ped. Inst., Ul'yanovsk, 1986) 3–15.Google Scholar
[4]Aizengendler, P. G. and Valnberg, M. M., “A theory of branching of solutions of nonlinear equations in the many-dimensional case”, Dokl. Akad. Nauk SSSR 163 (1965) 543–546Google Scholar
Translated in Soviet Math. Dokl. 6 (1965) 936–939.Google Scholar
[5]Chow, S. N., Hale, J. K. and Mallet-Paret, J., “Applications of generic bifurcations. I, II”, Arch. Rational Mech. Anal. 59 (1975) 159–188; 62 (1976) 209–235.CrossRefGoogle Scholar
[6]Vaĭnberg, M. M. and Trenogin, V. A., Theory of branching of solutions of non-linear equations, Transl. from Russian (Noordhoff, Leyden, 1974).Google Scholar