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The stability of steady-state solutions of the equations governing two-dimensional, homogeneous, incompressible fluid flow are analyzed in the context of shear-flow in a channel. Both the linear and nonlinear theories are reviewed and compared. In proving nonlinear stability of an equilibrium, emphasis is placed on using the stability algorithm developed in Holm et al. (1985). It is shown that for certain types of equilibria the linear theory is inconclusive, although nonlinear stability can be proven. Establishing nonlinear stability is dependent on the definition of a norm on the space of perturbations. McIntyre and Shepherd (1987) specifically define five norms, two for corresponding to one flow state and three to a different flow state, and suggest that still others are possible. Here, the norms given by McIntyre and Shepherd (1987) are shown to induce the same topology (for the corresponding flow states), establishing their equivalence as norms, and hence their equivalence as measures of stability. Summaries of the different types of stability and their mathematical definitions are presented. Additionally, a summary of conditions on shear-flow equilibria under which the various types of stability have been proven is presented. The Hamiltonian structure of the two-dimensional Euler equations is outlined following Olver (1986). A coordinate-free approach is adopted emphasizing the role of the Poisson bracket structure. Direct calculations are given to show that the Casimir invariants, or distinguished functionals, are time-independent and therefore are conserved quantities in the usual sense. / Graduation date: 1990

A new method for evaluating the Lyapunov exponent for a Hamiltonian system involves a spatial evaluation, rather than a numerical time integration. The introduction of a novel vector field to the phase space allows the Lyapunov exponent to be expressed in a form that does not involve time. The Lyapunov exponent is seen to be a property of the geometry and topology of ergodic regions of phase space. As a result it has a more regular behavior than previously thought. The Lyapunov exponent is found to be a differentiable function of the perturbation coupling in regions where it was previously thought to be discontinuous. Properties of the Lyapunov function once taken for granted are shown to be artifacts of the traditional computation methods. The technique is discussed with examples from a system of coupled quartic oscillators. / Graduation date: 1996

Diverse complex phenomena that are found in many fundamental problems of atomic physics and chemistry can be understood in the framework of nonlinear theory. Most of simple atomic and chemical systems are classically described by the Hamiltonian models of dimension three and higher. The multidimensional nature of these problems makes widely used diagnostics of dynamics to be impractical. We demonstrate the application of rigorous and effective computational methods to treat multidimensional systems in strongly perturbative regimes. The results of a qualitative analysis of the phase space stability structures are presented for two multidimensional non--integrable Hamiltonian systems: highly excited planar carbonyl sulfide molecule and hydrogen atom in elliptically polarized microwave fields. The molecular system of the planar carbonyl sulfide and atomic system of hydrogen in elliptically polarized microwave fields are treated for different regimes of energies including regimes of classical ionization of hydrogen and dissociation of carbonyl sulfide molecule.

Thesis (Ph. D.)--University of Texas at Austin, 2002. / Vita. Includes bibliographical references. Available also from UMI Company.

Thesis (Ph. D.)--University of Oregon, 2003. / Typescript. Includes vita and abstract. Includes bibliographical references (leaves 184-187). Also available for download via the World Wide Web; free to University of Oregon users.

Thesis (Ph.D)--Mathematics, Georgia Institute of Technology, 2009. / Committee Chair: Gangbo, Wilfrid; Committee Member: Loss, Michael; Committee Member: Pan, Ronghua; Committee Member: Swiech, Andrzej; Committee Member: Tannenbaum, Allen. Part of the SMARTech Electronic Thesis and Dissertation Collection.

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Thesis (Ph.D.)--University of Delaware, 2006. / Principal faculty advisor: Sunil K. Agrawal, Dept. of Mechanical Engineering. Includes bibliographical references.