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Bifurcation from multiple complex eigenvalues

Cohen, Donald S. (1977) Bifurcation from multiple complex eigenvalues. Journal of Mathematical Analysis and Applictions, 57 (3). pp. 505-521. ISSN 0022-247X. https://resolver.caltech.edu/CaltechAUTHORS:20200113-142201121

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Abstract

We shall study bifurcation and stability for nonlinear ordinary differential systems of arbitrary dimension when an equilibrium solution loses its stability by virtue of two pairs (α(λ) ± iβ(λ) ± iδ(λ)) complex conjugate eigenvalues of the linearized system simultaneously crossing the imaginary axis. Such a situation is not at all uncommon, and we shall give applications where this situation is indeed the usual phenomenon encountered. In these situations considerably different and more diverse behavior can occur than in the simpler bifurcation at a simple complex eigenvalue (Hopf bifurcation). As we shall see, the complexity of bifurcating solutions will result principally from the simple fact that superpositions such as βt + sin δt are not periodic if β and δ are incommensurate. The extension of our theory to account for arbitrary numbers of pairs of complex conjugate eigenvalues will be clear.


Item Type:Article
Related URLs:
URLURL TypeDescription
https://doi.org/10.1016/0022-247x(77)90243-8DOIArticle
Additional Information:© 1977 Published by Elsevier Inc. Submitted by W. F. Ames. This work was supported in part by the U. S. Army Research Office under Contract DAHC-04-68-C-0006 and the National Science Foundation under Grant GP-32157X2.
Funders:
Funding AgencyGrant Number
Army Research Office (ARO)DAHC-04-68-C-0006
NSFGP-32157X2
Issue or Number:3
Record Number:CaltechAUTHORS:20200113-142201121
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20200113-142201121
Official Citation:Donald S Cohen, Bifurcation from multiple complex eigenvalues, Journal of Mathematical Analysis and Applications, Volume 57, Issue 3, 1977, Pages 505-521, ISSN 0022-247X, https://doi.org/10.1016/0022-247X(77)90243-8. (http://www.sciencedirect.com/science/article/pii/0022247X77902438)
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:100690
Collection:CaltechAUTHORS
Deposited By: Tony Diaz
Deposited On:13 Jan 2020 23:42
Last Modified:13 Jan 2020 23:42

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