Numerical Methods for the Continuation of Invariant Tori

dc.contributor.advisor Dieci, Luca
dc.contributor.author Rasmussen, Bryan Michael en_US
dc.contributor.committeeMember Goldsztein, Guillermo
dc.contributor.committeeMember Harrell, Evans
dc.contributor.committeeMember Mischaikow, Konstantin
dc.contributor.committeeMember Szymczak, Andrzej
dc.contributor.department Mathematics en_US
dc.date.accessioned 2005-03-04T16:39:27Z
dc.date.available 2005-03-04T16:39:27Z
dc.date.issued 2003-11-24 en_US
dc.description.abstract This thesis is concerned with numerical techniques for resolving and continuing closed, compact invariant manifolds in parameter-dependent dynamical systems with specific emphasis on invariant tori under flows. In the first part, we review several numerical methods of continuing invariant tori and concentrate on one choice called the ``orthogonality condition'. We show that the orthogonality condition is equivalent to another condition on the smooth level and show that they both descend from the same geometrical relationship. Then we show that for hyperbolic, periodic orbits in the plane, the linearization of the orthogonality condition yields a scalar system whose characteristic multiplier is the same as the non-unity multiplier of the orbit. In the second part, we demonstrate that one class of discretizations of the orthogonality condition for periodic orbits represents a natural extension of collocation. Using this viewpoint, we give sufficient conditions for convergence of a periodic orbit. The stability argument does not extend to higher-dimensional tori, however, and we prove that the method is unconditionally unstable for some common types of two-tori embedded in R^3 with even numbers of points in both angular directions. In the third part, we develop several numerical examples and demonstrate that the convergence properties of the method and discretization can be quite complicated. In the fourth and final part, we extend the method to the general case of p-tori in R^n in a different way from previous implementations and solve the continuation problem for a three-torus embedded in R^8. en_US
dc.description.degree Ph.D. en_US
dc.format.extent 3838090 bytes
dc.format.mimetype application/pdf
dc.identifier.uri http://hdl.handle.net/1853/5273
dc.language.iso en_US
dc.publisher Georgia Institute of Technology en_US
dc.subject Invariant tori en_US
dc.subject Numerical analysis
dc.subject Invariant manifolds
dc.subject Flows
dc.subject.lcsh Numerical analysis en_US
dc.subject.lcsh Invariant manifolds Mathematical models en_US
dc.title Numerical Methods for the Continuation of Invariant Tori en_US
dc.type Text
dc.type.genre Dissertation
dspace.entity.type Publication
local.contributor.advisor Dieci, Luca
local.contributor.corporatename College of Sciences
local.contributor.corporatename School of Mathematics
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relation.isOrgUnitOfPublication 84e5d930-8c17-4e24-96cc-63f5ab63da69
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