Point Spectrum: Reduction to Finite-Rank Eigenvalue Problems

  • Todd Kapitula
  • Keith Promislow
Part of the Applied Mathematical Sciences book series (AMS, volume 185)


The word bifurcation refers to changes in the number and stability of equilibria supported by a governing system as its parameters are varied. The classical bifurcation problem begins with an analysis of the point spectrum of the linearized operator associated with the equilibria under investigation. In this chapter we investigate finite-rank bifurcations for which a finite number of point eigenvalues cross the imaginary axis, either transversely or more degenerately, as the system parameters are varied. In particular, we derive the perturbative motion of such point spectra. This analysis is most informative in those cases for which the associated linearized operator initially has purely imaginary eigenvalues, and a small change in parameters moves the eigenvalues decisively off the imaginary axis.


Imaginary Axis Point Spectrum Translational Symmetry Orbital Stability Simple Eigenvalue 
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Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Todd Kapitula
    • 1
  • Keith Promislow
    • 2
  1. 1.Department of Mathematics and StatisticsCalvin CollegeGrand RapidsUSA
  2. 2.Department of MathematicsMichigan State UniversityEast LansingUSA

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