Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

Request Permissions   Purchase Content 
 

 

Spectral flow is a complete invariant for detecting bifurcation of critical points


Authors: James C. Alexander and Patrick M. Fitzpatrick
Journal: Trans. Amer. Math. Soc. 368 (2016), 4439-4459
MSC (2010): Primary 46T99, 47J15, 58E05, 58E07
DOI: https://doi.org/10.1090/tran/6474
Published electronically: January 13, 2016
MathSciNet review: 3453376
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Given a one-parameter path of equations for which there is a trivial branch of solutions, to determine the points on the branch from which there bifurcate nontrivial solutions, there is the heuristic principle of linearization. That is to say, at each point on the branch, linearize the equation, and justify the inference that points on the branch that are bifurcation points for the path of linearized equations are also bifurcation points for the original path of equations. In quite general circumstances, for the bifurcation of critical points, we show that, at isolated singular points of the path of linearizations, a property of the path that is known to be sufficient to force bifurcation of nontrivial critical points is also necessary.

To be more precise, let $ I$ be an open interval of real numbers that contains the point $ \lambda _0$ and $ B$ an open ball about the origin of a real, separable Hilbert space $ H.$ Let $ \psi \colon I\times B\to \ R$ be a family of $ C^2$ functions. For $ \lambda \in I,$ assume $ \nabla _x\psi (\lambda ,0)=0,$ and set $ \mathop {\rm Hessian}\nolimits _x\psi (\lambda ,\,0)\equiv L_\lambda .$ Assume $ L_\lambda $ is invertible if $ \lambda \ne \lambda _0$ and $ L_{\lambda _0}$ is Fredholm. It is known that if the spectral flow of $ L\colon I\to {\mathcal L}(H)$ across $ \lambda _0$ is nonzero, then in each neighborhood of $ (\lambda _0,\,0)$ there are pairs $ (\lambda ,\,x),$ $ x\ne 0,$ for which $ \nabla _x\psi (\lambda ,x)=0. $ We prove that if $ L\colon I\to {\mathcal L}(H)$ is a continuous path of symmetric operators for which $ L_\lambda $ is invertible for $ \lambda \ne \lambda _0,$ $ L_{\lambda _0}$ is Fredholm, and the spectral flow of $ L\colon I\to {\mathcal L}(H)$ across $ \lambda _0$ is zero, then there is an open interval $ J$ that contains the point $ \lambda _0$, an open ball $ B$ about the origin, and a family $ \psi \colon J\times B\to \ R$ of $ C^2$ functions such that, for each $ \lambda \in J, $ $ \nabla _x\psi (\lambda ,0)=0$ and $ \mathop {\rm Hessian}\nolimits _x\psi (\lambda ,\,0)= L_\lambda ,$ but $ \nabla _x\psi (\lambda ,x)\ne 0$ if $ x\ne 0.$ Therefore, at an isolated singular point of the path of linearizations of the gradient, under the sole further assumption that the linearization at the singular point is Fredholm, spectral flow is a complete invariant for the detection of bifurcation of nontrivial critical points.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 46T99, 47J15, 58E05, 58E07

Retrieve articles in all journals with MSC (2010): 46T99, 47J15, 58E05, 58E07


Additional Information

James C. Alexander
Affiliation: Department of Mathematics, University of Maryland, College Park, Maryland 20742

Patrick M. Fitzpatrick
Affiliation: Department of Mathematics, University of Maryland, College Park, Maryland 20742

DOI: https://doi.org/10.1090/tran/6474
Received by editor(s): May 15, 2013
Received by editor(s) in revised form: May 1, 2014
Published electronically: January 13, 2016
Article copyright: © Copyright 2016 American Mathematical Society