Get 20M+ Full-Text Papers For Less Than $1.50/day. Start a 14-Day Trial for You or Your Team.

Learn More →

On the Andronov–Hopf Bifurcation Theorem

On the Andronov–Hopf Bifurcation Theorem Di erential Equations, Vol. 37, No. 5, 2001, pp. 640{646. Translated from Di erentsial'nye Uravneniya, Vol. 37, No. 5, 2001, pp. 609{615. Original Russian Text Copyright c 2001 by Izmailov. ORDINARY DIFFERENTIAL EQUATIONS A. F. Izmailov Computing Center, Russian Academy of Sciences, Moscow, Russia Received September 13, 1999 1. INTRODUCTION. THE ANDRONOV{HOPF BIFURCATION THEOREM The theorem [1] on the level set of a nonlinear mapping near a singular point claims that if the mapping is 2-regular at the singular point, then its level set is locally di eomorphic to the set of zeros of its second di erential. (The de nition of 2-regularity and the rigorous statement of this theorem are given in Section 2 below.) In the present paper, as an application of this theorem, we suggest a simple proof of the Andronov{Hopf bifurcation theorem (for brevity, referred to as the bifurcation theorem in what follows). We obtain the bifurcation theorem as an immediate corollary to the theorem in [1]; the only diculty is in verifying the 2-regularity of the corresponding mapping. The Andronov{Hopf bifurcation (the bifurcation of a cycle) is one of the most comprehen- sively studied examples of branching solutions of nonlinear problems, which has numerous http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Differential Equations Springer Journals

On the Andronov–Hopf Bifurcation Theorem

Izmailov, A.
Differential Equations , Volume 37 (5) – Oct 12, 2004
Download PDF
7 pages

Loading next page...
 
/lp/springer-journals/on-the-andronov-hopf-bifurcation-theorem-7HQvzjFgoF

References (3)

Publisher
Springer Journals
Copyright
Copyright © 2001 by MAIK “Nauka/Interperiodica”
Subject
Mathematics; Difference and Functional Equations; Ordinary Differential Equations; Partial Differential Equations
ISSN
0012-2661
eISSN
1608-3083
DOI
10.1023/A:1019260414080
Publisher site
See Article on Publisher Site

Abstract

Di erential Equations, Vol. 37, No. 5, 2001, pp. 640{646. Translated from Di erentsial'nye Uravneniya, Vol. 37, No. 5, 2001, pp. 609{615. Original Russian Text Copyright c 2001 by Izmailov. ORDINARY DIFFERENTIAL EQUATIONS A. F. Izmailov Computing Center, Russian Academy of Sciences, Moscow, Russia Received September 13, 1999 1. INTRODUCTION. THE ANDRONOV{HOPF BIFURCATION THEOREM The theorem [1] on the level set of a nonlinear mapping near a singular point claims that if the mapping is 2-regular at the singular point, then its level set is locally di eomorphic to the set of zeros of its second di erential. (The de nition of 2-regularity and the rigorous statement of this theorem are given in Section 2 below.) In the present paper, as an application of this theorem, we suggest a simple proof of the Andronov{Hopf bifurcation theorem (for brevity, referred to as the bifurcation theorem in what follows). We obtain the bifurcation theorem as an immediate corollary to the theorem in [1]; the only diculty is in verifying the 2-regularity of the corresponding mapping. The Andronov{Hopf bifurcation (the bifurcation of a cycle) is one of the most comprehen- sively studied examples of branching solutions of nonlinear problems, which has numerous

Journal

Differential Equations – Springer Journals

Published: Oct 12, 2004

There are no references for this article.