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 Funktsional. Anal. i Prilozhen.: Year: Volume: Issue: Page: Find

 Funktsional. Anal. i Prilozhen., 2005, Volume 39, Issue 3, Pages 37–53 (Mi faa73)

On the Number of Unbounded Solution Branches in a Neighborhood of an Asymptotic Bifurcation Point

A. M. Krasnosel'skii, D. I. Rachinskii

Institute for Information Transmission Problems, Russian Academy of Sciences

Abstract: We suggest a method for studying asymptotically linear vector fields with a parameter. The method permits one to prove theorems on asymptotic bifurcation points (bifurcation points at infinity) for the case of double degeneration of the principal linear part. We single out a class of fields that have more than two unbounded branches of singular points in a neighborhood of a bifurcation point. Some applications of the general theorems to bifurcations of periodic solutions and subharmonics as well as to the two-point boundary value problem are given.

Keywords: asymptotic bifurcation point, solution branch, asymptotically homogeneous operator, periodic oscillations, subharmonic

DOI: https://doi.org/10.4213/faa73

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English version:
Functional Analysis and Its Applications, 2005, 39:3, 194–206

Bibliographic databases:

UDC: 517.988.67

Citation: A. M. Krasnosel'skii, D. I. Rachinskii, “On the Number of Unbounded Solution Branches in a Neighborhood of an Asymptotic Bifurcation Point”, Funktsional. Anal. i Prilozhen., 39:3 (2005), 37–53; Funct. Anal. Appl., 39:3 (2005), 194–206

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/faa73
• https://doi.org/10.4213/faa73
• http://mi.mathnet.ru/eng/faa/v39/i3/p37

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This publication is cited in the following articles:
1. Ya. M. Dymarskii, “Manifold Method in Eigenvector Theory of Nonlinear Operators”, Journal of Mathematical Sciences, 154:5 (2008), 655–815
2. Krasnosel'skii, AM, “On Disconnected Unbounded Sets of Forced Oscillations”, Doklady Mathematics, 78:2 (2008), 660
3. A. M. Krasnosel'skii, D. I. Rachinskii, “Criteria of resonance origin in a single-circuit control system with saturation”, Autom. Remote Control, 69:8 (2008), 1297–1310
4. Krasnosel'skii, AM, “Double degeneracy in the problem on unbounded branches of forced oscillations”, Doklady Mathematics, 77:2 (2008), 170
5. Krasnosel'skii A.M., “Unbounded branches of periodic solutions”, Differ. Equ., 45:3 (2009), 344–364
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