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

@article{Alexander2016SpectralFI,
  title={Spectral flow is a complete invariant for detecting bifurcation of critical points},
  author={James C. Alexander and Patrick M. Fitzpatrick},
  journal={Transactions of the American Mathematical Society},
  year={2016},
  volume={368},
  pages={4439-4459}
}
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… 
1 Citations
Bifurcation of critical points along gap-continuous families of subspaces
We consider the restriction of twice differentiable functionals on a Hilbert space to families of subspaces that vary continuously with respect to the gap metric. We study bifurcation of branches of

References

SHOWING 1-10 OF 28 REFERENCES
Spectral Flow and Bifurcation of Critical Points of Strongly-Indefinite Functionals Part I. General Theory☆
Abstract Spectral flow is a well-known homotopy invariant of paths of selfadjoint Fredholm operators. We describe here a new construction of this invariant and prove the following
Spectral Flow and Bifurcation of Critical Points of Strongly Indefinite Functionals
Abstract Our main results here are as follows: Let X λ be a family of 2 π -periodic Hamiltonian vectorfields that depend smoothly on a real parameter λ in [ a ,  b ] and has a known, trivial, branch
Bifurcation of critical points for continuous families of C2 functionals of Fredholm type
Given a continuous family of C2 functionals of Fredholm type, we show that the nonvanishing of the spectral flow for the family of Hessians along a known (trivial) branch of critical points not only
A bifurcation theorem for potential operators
Generalized Jordan chains and two bifurcation theorems of Krasnoselskii
Spectral flow and bifurcation of critical points
Bifurcation theory and the type numbers of marston morse.
  • M. Berger
  • Mathematics
    Proceedings of the National Academy of Sciences of the United States of America
  • 1972
TLDR
It is shown that the type numbers of Marston Morse for an isolated critical point can be used to prove the existence of a point of bifurcation at (x(0),lambda(0)).
Spectral asymmetry and Riemannian geometry I
1. Introduction . The main purpose of this paper is to present a generalization of Hirzebruch's signature theorem for the case of manifolds with boundary. Our result is in the framework of Riemannian
On conjugate loci and cut loci of compact symmetric spaces II
In the present paper, we shall first study the topological structures of the cut locus and the conjugate locus of a compact symmetric space. We shall show that the cut locus C is the disjoint union
Uniqueness of spectral flow
...
1
2
3
...