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.

Typescript (photocopy) / 198 leaves ; 30 cm. / Title page, contents and abstract only. The complete thesis in print form is available from the University Library. / Thesis (Ph.D.)--University of Adelaide, Dept. of Mathematical Physics, 1983

Thesis (Ph.D.) -- University of Adelaide, Dept of Mathematical Physics, 1983. / Typescript (photocopy).

Thesis (Ph. D.)--Physics, Georgia Institute of Technology, 2006. / Mustafa Aral, Committee Member ; John Wood, Committee Member ; Kurt Wiesenfeld, Committee Member ; M. Raymond Flannery, Committee Member ; Turgay Uzer, Committee Chair.

In this dissertation we study the Hamiltonicity and the uniform-Hamiltonicity of subset graphs, subspace graphs, and their associated bipartite graphs. In 1995 paper "The Subset-Subspace Analogy," Kung states the subspace version of a conjecture. The study of this problem led to a more general class of graphs. Inspired by Clark and Ismail's work in the 1996 paper "Binomial and Q-Binomial Coefficient Inequalities Related to the Hamiltonicity of the Kneser Graphs and their Q-Analogues," we defined subset graphs, subspace graphs, and their associated bipartite graphs. The main emphasis of this dissertation is to describe those graphs and study their Hamiltonicity. The results on subset graphs are presented in Chapter 3, on subset bipartite graphs in Chapter 4, and on subspace graphs and subspace bipartite graphs in Chapter 5. We conclude the dissertation by suggesting some generalizations of our results concerning the panciclicity of the graphs.

Numerical methods are usually necessary in solving Hamiltonian systems since there is often no closed-form solution. By utilizing a general property of Hamiltonians, namely the symplectic property, all of the qualities of the system may be preserved for indefinitely long integration times because all of the integral (Poincare) invariants are conserved. This allows for more reliable results and frequently leads to significantly shorter execution times as compared to conventional methods. The resonant triad Hamiltonian with one degree of freedom will be focused upon for most of the numerical tests because of its difficult nature and, moreover, analytical results exist whereby useful comparisons can be made.

The hamiltonian number problem is a generalization of hamiltonian cycle problem in graph theory. It is well known that the hamiltonian cycle problem in graph theory is NP-complete [16]. So the hamiltonian number problem is also NP-complete. On the other hand, the hamiltonian number problem is the traveling salesman problem with each edge having weight 1. A hamiltonian walk of a graph G is a closed spanning walk of minimum length. The length of a hamiltonian walk in G is called the hamiltonian number. For graphs, we give some bounds for hamiltonian numbers of graphs. First, we improve some results in [14] and give a necessary and sufficient condition for h(G) < e(G) where e(G) is the minimum length of a closed walk passing through all edges of G. Next, we prove that if two nonadjacent vertices u and v satisfying that deg(u)+deg(v) ≥ |G|, then h(G) = h(G + uv). This result generalizes a theorem of Bondy and Chv¡¬atal for the hamiltonian cycle. Finally, we show that if 0 ≤ k ≤ n − 2 and G is a 2-connected graph of order n satisfying deg(u) + deg(v) + deg(w) ≥ 3n−k−2 for every independent set {u, v,w} of three vertices in G, then h(G) ≤ n+k. It is a generalization of a Bondy¡¦s result. For digraphs, we give some bounds for hamiltonian numbers of digraphs first. We prove that if a digraph D of order n is strongly connected, thenn ≤ h(D) ≤ ⌊(n+1)^2/4 ⌋. Next, we also present some digraphs of order n ≥ 5 which have hamiltonian number k for n ≤ k ≤ ⌊(n+1)^2/4 ⌋. Finally, we study hamiltonian numbers of M¡Lobius double loop networks. We introduce M¡Lobius double loop network and every strongly connected double loop network is isomorphic to some M¡Lobius double loop network. Next, we give an upper bound for the hamiltonian numbers of M¡Lobius double loop networks. Then, we find some necessary and sufficient conditions for M¡Lobius double loop networks MDL(d, m, ℓ) to have hamiltonian numbers dm, dm + 1 or dm + 2.

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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