Citation
Henderson, Michael Edwin (1985) Complex Bifurcation. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/JF821T64. https://resolver.caltech.edu/CaltechETD:etd03262008112516
Abstract
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): 
 
Thesis Committee: 
 
Defense Date:  28 September 1984  
Funders: 
 
Record Number:  CaltechETD:etd03262008112516  
Persistent URL:  https://resolver.caltech.edu/CaltechETD:etd03262008112516  
DOI:  10.7907/JF821T64  
Default Usage Policy:  No commercial reproduction, distribution, display or performance rights in this work are provided.  
ID Code:  1159  
Collection:  CaltechTHESIS  
Deposited By:  Imported from ETDdb  
Deposited On:  04 Apr 2008  
Last Modified:  21 Dec 2019 04:19 
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