I. Asymptotic boundary conditions for ordinary differential equations. II. Numerical Hopf bifurcation

Citation

Jepson, Allan Douglas (1981) I. Asymptotic boundary conditions for ordinary differential equations. II. Numerical Hopf bifurcation. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/C2RJ-6F27. https://resolver.caltech.edu/CaltechETD:etd-08182005-153922

Abstract

Part I. "Asymptotic Boundary Conditions for Ordinary Differential Equations" The numerical solution of two point boundary value problems on semi-infinite intervals is often obtained by truncating the interval at some finite point. In this thesis we determine a hierarchy of increasingly accurate boundary conditions for the truncated interval problem. Both linear and nonlinear problems are considered. Numerical techniques for error estimation and the determination of an appropriate truncation point are discussed. A Fredholm theory for boundary value problems on semi-infinite intervals is developed, and used to prove the stability of our numerical methods. Part II. "Numerical Hopf Bifurcation" Several numerical methods for locating a Hopf bifurcation point of a system of o.d.e.'s or p.d.e.'s are discussed. A new technique for computing the Hopf bifurcation parameters is also presented. Finally, well-known numerical techniques for simple bifurcation problems are adapted for Hopf bifurcation problems. This provides numerical techniques for computing the bifurcating branch of periodic solutions, possibly including turning points and simple bifurcation points. The stability of the periodic solutions is also discussed.

Thesis Files