Finding and exploiting structure in complex systems via geometric and statistical methods

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dc.contributor.authorGrover, Piyushen
dc.contributor.committeechairRoss, Shane D.en
dc.contributor.committeememberKriz, Ronald D.en
dc.contributor.committeememberStremler, Mark A.en
dc.contributor.committeememberDankowicz, Harry J.en
dc.contributor.committeememberWoolsey, Craig A.en
dc.contributor.departmentEngineering Science and Mechanicsen
dc.date.accessioned2014-03-14T20:12:57Zen
dc.date.adate2010-07-06en
dc.date.available2014-03-14T20:12:57Zen
dc.date.issued2010-06-01en
dc.date.rdate2010-07-06en
dc.date.sdate2010-06-11en
dc.description.abstractThe dynamics of a complex system can be understood by analyzing the phase space structure of that system. We apply geometric and statistical techniques to two Hamiltonian systems to find and exploit structure in the phase space that helps us get qualitative and quantitative results about the phase space transport. While the structure can be revealed by the study of invariant manifolds of fixed points and periodic orbits in the first system, there do not exist any fixed points (and hence invariant manifolds) in the second system. The use of statistical (or measure theoretic) and topological methods reveals the phase space structure even in the absence of fixed points or stable and unstable invariant manifolds. The first problem we study is the four-body problem in the context of a spacecraft in the presence of a planet and two of its moons, where we exploit the phase space structure of the problem to devise an intelligent control strategy to achieve mission objectives. We use a family of analytically derived controlled Keplerian Maps in the Patched-Three-Body framework to design fuel efficient trajectories with realistic flight times. These maps approximate the dynamics of the Planar Circular Restricted Three Body Problem (PCR3BP) and we patch solutions in two different PCR3BPs to form the desired trajectories in the four body system. The second problem we study concerns phase space mixing in a two-dimensional time dependent Stokes flow system. Topological analysis of the braiding of periodic points has been recently used to find lower bounds on the complexity of the flow via the Thurston-Nielsen classification theorem (TNCT). We extend this framework by demonstrating that in a perturbed system with no apparent periodic points, the almost-invariant sets computed using a transfer operator approach are the natural objects on which to pin the TNCT.en
dc.description.degreePh. D.en
dc.identifier.otheretd-06112010-162808en
dc.identifier.sourceurlhttp://scholar.lib.vt.edu/theses/available/etd-06112010-162808/en
dc.identifier.urihttp://hdl.handle.net/10919/28019en
dc.publisherVirginia Techen
dc.relation.haspartGrover_P_D_2010.pdfen
dc.rightsIn Copyrighten
dc.rights.urihttp://rightsstatements.org/vocab/InC/1.0/en
dc.subjectset-oriented methodsen
dc.subjectbraid bifurcationen
dc.subjectPerron-Frobenius operatoren
dc.subjectghost rodsen
dc.subjectbraidsen
dc.subjectfluid mixingen
dc.subjectmulti-moon orbiteren
dc.subjectlow energy mission designen
dc.subjectbraiding of almost-invariant setsen
dc.titleFinding and exploiting structure in complex systems via geometric and statistical methodsen
dc.typeDissertationen
thesis.degree.disciplineEngineering Science and Mechanicsen
thesis.degree.grantorVirginia Polytechnic Institute and State Universityen
thesis.degree.leveldoctoralen
thesis.degree.namePh. D.en

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