Abstract
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.
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Kapitula, T., Promislow, K. (2013). Point Spectrum: Reduction to Finite-Rank Eigenvalue Problems. In: Spectral and Dynamical Stability of Nonlinear Waves. Applied Mathematical Sciences, vol 185. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-6995-7_6
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DOI: https://doi.org/10.1007/978-1-4614-6995-7_6
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