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Complex Bifurcation


Henderson, Michael Edwin (1985) Complex Bifurcation. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/JF82-1T64.


Real equations of the form g(x,λ) = 0 are shown to have a complex extension G(u,λ) = 0, defined on the complex Banach space 𝔹 ⊕ i𝔹. At a singular point of the real equation this extension has solution branches corresponding to both the real and imaginary roots of the Algebraic Bifurcation Equations (ABE's).

We solve the ABE's at simple quadratic folds, quadratic bifurcation points, and cubic bifurcation points, and show that these are complex bifurcation points. We also show that at a Hopf bifurcation point of the real equation there are two families of complex periodic orbits, parametrized by three real parameters.

By taking sections of solutions of complex equations with two real parameters, we show that complex branches may connect disjoint solution branches of the real equation. These complex branches provide a simple and practical means of locating disjoint branches of real solutions.

Finally, we show how algorithms for computing real solutions may be modified to compute complex solutions. We use such an algorithm to find solutions of several example problems, and locate two sets of disjoint real branches.

Item Type:Thesis (Dissertation (Ph.D.))
Subject Keywords:Applied Mathematics
Degree Grantor:California Institute of Technology
Division:Physics, Mathematics and Astronomy
Major Option:Applied Mathematics
Awards:The W. P. Carey & Co., Inc. Prize in Applied Mathematics
Thesis Availability:Public (worldwide access)
Research Advisor(s):
  • Keller, Herbert Bishop
Thesis Committee:
  • Kreiss, Heinz-Otto (chair)
  • Saffman, Philip G.
  • Cohen, Donald S.
  • Keller, Herbert Bishop
Defense Date:28 September 1984
Funding AgencyGrant Number
Department of Energy (DOE)DE-AM03-76SF00767
Record Number:CaltechETD:etd-03262008-112516
Persistent URL:
Default Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:1159
Deposited By: Imported from ETD-db
Deposited On:04 Apr 2008
Last Modified:21 Dec 2019 04:19

Thesis Files

PDF (Henderson_me_1985.pdf) - Final Version
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