The framework for the Hall magnetohydrodynamic (MHD) model for plasma physics is built up from kinetic theory and used to analytically solve problems of interest in the field. The Hall MHD model describes fast magnetic reconnection processes in space and laboratory plasmas. Specifically, the magnetic reconnection process at an X-type neutral point, where current sheets form and store enormous amounts of magnetic energy which is later released as magnetic storms when the sheets break up, is investigated. The phenomena of magnetic flux pile-up driving the merging of antiparallel magnetic fields at an ion stagnation-point flow in a thin current sheet, called the Parker problem, also receives rigorous mathematical analysis.

Throughout the last several decades many techniques have been developed in establishing solutions to nonlinear partial differential equations (NPDE). These techniques are characterized by their limited reach in solving large classes of NPDE. This body of work will study the analysis of NPDE using two of the most ubiquitous techniques developed in the last century. In this body of work, the analysis and techniques herein are applied to unsolved physical problems in both the fields of variational approximations and inverse scattering transform. Additionally, a new technique for estimating the error of a variational approximation is established. Note that the material in chapter 2, "Quantitative Measurements of Variational Approximations" has recently been published. Variational problems have long been used to mathematically model physical systems. Their advantage has been the simplicity of the model as well as the ability to deduce information concerning the functional dependence of the system on various parameters embedded in the variational trial functions. However, the only method in use for estimating the error in a variational approximation has been to compare the variational result to the exact solution. In this work, it is demonstrated that one can computationally obtain estimates of the errors in a one-dimensional variational approximation, without any a priori knowledge of the exact solution. Additionally, this analysis can be done by using only linear techniques. The extension of this method to multidimensional problems is clearly possible, although one could expect that additional difficulties would arise. One condition for the existence of a localized soliton is that the propagation constant does not fall into the continuous spectrum of radiation modes. For a higher order dispersive systems, the linear dispersion relation exhibits a multiple branch structure. It could be the case that in a certain parameter region for which one of the components of the solution has oscillations (i.e., is in the continuous spectrum), there exists a discrete value of the propagation constant, k(ES), for which the oscillations have zero amplitude. The associated solution is referred to as an embedded soliton (ES). This work examines the ES solutions in a CHI(2):CHI(3), type II system. The method employed in searching for the ES solutions is a variational method recently developed by Kaup and Malomed [Phys. D 184, 153-61 (2003)] to locate ES solutions in a SHG system. The variational results are validated by numerical integration of the governing system. A model used for the 1-D longitudinal wave propagation in microstructured solids is a KdV-type equation with third and fifth order dispersions as well as first and third order nonlinearities. Recent work by Ilison and Salupere (2004) has identified certain types of soliton solutions in the aforementioned model. The present work expands the known family of soliton solutions in the model to include embedded solitons. The existence of embedded solitons with respect to the dispersion parameters is determined by a variational approximation. The variational results are validated with selected numerical solutions.

The nonlinear nonplanar steady-state responses of cantilever beams to direct and parametric harmonic excitations are investigated using perturbation techniques. Modal interactions between the bending-bending and bending-bending-twisting motions are studied. Using a variational formulation, we obtained the governing equations of motion and associated boundary conditions for monoclinic composite and isotropic metallic inextensional beams. The method of multiple scales is applied either to the governing system of equations and associated boundary conditions or to the Lagrangian and virtual-work term to determine the modulation equations that govern the slow dynamics of the responses. These equations are shown to exhibit symmetry properties, reflecting the conservative nature of the beams in the absence of damping. It is popular to first discretize the partial-differential equations of motion and then apply a perturbation technique to the resulting ordinary-differential equations to determine the modulation equations. Due to the presence of quadratic as well as cubic nonlinearities in the governing system for the bending-bending-twisting oscillations of beams, it is shown that this approach leads to erroneous results. Furthermore, the symmetries are lost in the resulting equations. Nontrivial fixed points of the modulation equations correspond, generally, to periodic responses of the beams, whereas limit-cycle solutions of the modulation equations correspond to aperiodic responses of the beams. A pseudo-arclength scheme is used to determine the fixed points and their stability. In some cases, they are found to undergo Hopf bifurcations, which result in limit cycles. A combination of a long-time integration, a two-point boundary-value continuation scheme, and Floquet theory is used to determine in detail branches of periodic and chaotic solutions and assess their stability. The limit cycles undergo symmetry-breaking, cyclic-fold, and period-doubling bifurcations. The chaotic attractors undergo attractor-merging and boundary crises as well as explosive bifurcations. For certain cases, it is determined that the response of a beam to a high-frequency excitation is not necessarily a high-frequency low-amplitude oscillation. In fact, low-frequency high-amplitude components that dominate the responses may be activated by resonant and nonresonant mechanisms. In such cases, the overall oscillations of the beam may be significantly large and cannot be neglected. / Ph. D.

We created an analytical reduced-order model (macromodel) for an electrically actuated circular plate with an in-plane residual stress for applications in capacitive micromachined ultrasonic transducers (CMUTs). After establishing the equations governing the plate, we discretized the system by using a Galerkin approach. The distributed-parameter equations were then reduced to a finite system of ordinary-differential equations in time. We solved these equations for the equilibrium states due to a general electric potential and determined the natural frequencies of the axisymmetric modes for the stable deflected position. As expected, the fundamental natural frequency generally decreases as the electric forcing increases, reaching a value of zero at pull-in. However, strain-hardening effects can cause the frequencies to increase with voltage. The macromodel was validated by using data from experiments and simulations performed on silicon-based microelectromechanical systems (MEMS). For example, the pull-in voltages differed by about 1% from values produced by full 3-D MEMS simulations. The macromodel was then used to investigate the response of an electrostatically actuated clamped circular plate to a primary resonance excitation of its first axisymmetric mode. The method of multiple scales was used to derive a semi-analytical expression for the equilibrium amplitude of vibration. The plate was found to always transition from a hardening-type to a softening-type behavior as the DC voltage increases towards pull-in. Because the response of CMUTs is highly influenced by the boundary conditions, an updated reduced-order model was created to account for more realistic boundary conditions. The electrode was still considered to be infinitesimally thin, but the electrode was allowed to have general inner and outer radii. The updated reduced-order model was used to show how sensitive the pull-in voltage is with respect to the boundary conditions. The boundary parameters were extracted by matching the pull-in voltages from the macromodel to those from finite element method (FEM) simulations for CMUTs with varying outer and inner radii. The static behavior of the updated macromodel was validated because the pull-in voltages for the macromodel and FEM simulations were very close to each other and the extracted boundary parameters were physically realistic. A macromodel for CMUTs was then created that includes both the boundary effects and an electrode of finite thickness. Matching conditions ensured the continuity of displacements, slopes, forces, and moments from the composite to the non-composite regime of the CMUT. We attempted to validate this model with results from FEM simulations. In general, the center deflections from the macromodel fell below those from the FEM simulation, especially for relatively high residual stresses, but the first natural frequencies that accompany the deflections were very close to those from the FEM simulations. Furthermore, the forced vibration characteristics also compared well with the macromodel predictions for an experimental case in which the primary resonance curve bends to the right because the CMUT is a hardening-type system. The reduced-order model accounts for geometric nonlinear hardening, residual stresses, and boundary conditions related to the CMUT post, allows for general design variables, and is robust up to the pull-in instability. However, even more general boundary conditions need to be incorporated into the model for it to be a more effective design tool for capacitive micromachined ultrasonic transducers. / Ph. D.

Three types of problems in nonlinear dynamics are studied. First, we use a complex-variable invariant-manifold approach to determine the nonlinear normal modes of weakly nonlinear discrete systems with one-to-one and three-to-one internal resonances. Cubic geometric nonlinearities are considered. The system under investigation possesses similar nonlinear normal modes for the case of One-to-one internal resonance and nonsimilar nonlinear normal modes for the case of three-to-one internal resonance. In contrast with the case of no internal resonance, the number of nonlinear normal modes may be more than the number of linear normal modes. Bifurcations of the calculated nonlinear normal modes are investigated. For continuous systems without internal resonances, we consider a cantilever beam and compare two approaches for the determination of its nonlinear planar modes. In the first approach, the governing partial-differential system is discretized using the linear mode shapes and then the nonlinear mode shapes are determined from the discretized system. In the second approach, the boundary-value problem is treated directly by using the method of multiple scales. The results show that both approaches yield the same nonlinear modes because the discretization is performed using a complete set of basis functions, namely, the linear mode shapes. Second, we study the nonlinear response of multi-degree-of-freedom systems with repeated natural frequencies to various parametric resonances. The linear part of the system has a nonsemisimple one-to-one resonance. The character of the stability and various types of bifurcation are analyzed. The results are applied to the flutter of a simply supported panel in a supersonic airstream. In which case, the nonlinear ordinary differential equations governing the modulation of the amplitudes and phases of the excited modes are derived and used to calculate the equilibrium solutions and their stability and hence to identify the excitation parameters that suppress flutter and those that lead to complex motions. A combination of a shooting technique and Floquet theory is used to calculate limit cycles and their stability. The numerical results indicate the existence of a sequence of period-doubling bifurcations that culminates in chaos. The complex motions are characterized by using phase planes, power spectra, Lyapunov exponents, and dimensions. Moreover, bifurcation analyses in the case of fundamental parametric resonance reveal that the addition of quadratic nonlinearities change qualitatively as well as quantitatively the response of systems with cubic nonlinearities. The quadratic nonlinearities change the pitchfork bifurcation to a transcritical bifurcation. Cyclic-fold bifurcations, Hopf bifurcations of the nontrivial constant solutions, and period-doubling sequences leading to chaos are induced by these quadratic terms. The effects of quadratic nonlinearities for the case of principal parametric resonance are discussed. Third, we investigate the transfer of energy from high- to low-frequency modes. The method of averaging is used to analyze the response of a two-degree-of-freedom system with widely spaced modal frequencies and cubic nonlinearities to a principal parametric resonance of the high-frequency mode. The conditions under which energy can be transferred from high- to low-frequency modes, as observed in the experiments, are determined. The interactions between the widely separated modes result in various bifurcations, the coexistence of multiple attractors, and chaotic attractors. The results show that damping may be destabilizing. The analytical results are validated by numerically solving the original system. / Ph. D.

<p> A repulsive meson-meson interaction was suggested many years ago (1951 by Schiff) as a possible mechanism for nuclear saturation, but very little has been done since then. This is mainly because the meson field equation becomes nonlinear due to the meson-meson interaction. We realized that the nonlinear field equation is analytically solvable, within classical and adiabatic approximations, if the space is reduced to a one-dimensional one. Within the above context we investigate the effect of the meson-meson interaction on nuclear forces. The approximations which Schiff used are critically examined. A variational method for determining the meson field, which Schiff suggested but did not fully investigate, is found to be a very efficient approximation. Finally, quantum corrections are briefly examined.</p> / Thesis / Master of Science (MSc)

<p> Differential Dynamic. Programming is a new method, based on Bellman's principle of optimality, for determining optimal control strategies for nonlinear systems. It has originally been developed by D.H.Jacobson. </p> <p> In this thesis a result is presented for a problem with saturation characteristics in nonlinearity solved by the Jacobson's approach. In the differential dynamic programming the principle of optimality is applied to the differential change in non-optimal cost due to small changes in state; variables instead of the cost itself. This results in modest memory requirements for its defining parameters and rapid convergence. </p> / Thesis / Master of Engineering (MEngr)

<p> Transverse vibrations of a beam with one end fixed and the other supported on nonlinear spring have been studied. Theoretical analysis has been carried out for two different cases of springs, viz.; cubic nonlinear and bilinear types. </p> <p> Theoretical results for bilinear case have been compared with those obtained experimentally. The effect of end mass has also been considered in theoretical analysis. </p> / Thesis / Master of Engineering (MEngr)

The realm of aircraft flight dynamics analysis reaches from local static stability to global dynamic behavior. It includes aircraft performance issues as well as structural concerns. In the particular aspect of dynamic motions of an aircraft and how we understand them, an alternate coordinate system will be introduced that will lend insight and simplification into the understanding of these dynamic motions. The main contribution of this coordinate system is that one can easily visualize how the instantaneous velocity vector relates to the instantaneous rotation vector, the angular rate vector of the aircraft. The out-of-control motion known as the Falling Leaf will be considered under the light of this new coordinate system. This motion is not well understood and can lead to loss of the aircraft and crew. Design guidelines will be presented to predict amplitude and frequency of the Falling Leaf. NOTE: (12/2009) An updated copy of this ETD was added after there were patron reports of problems with the file. / Master of Science

Plasma waves are key drivers of the highly variable electron dynamics in Earth’s outer radiation belts. In particular, whistler mode chorus waves, which are commonly observed with intense wave amplitudes, are known to be a key driver of rapid electron acceleration and precipitation observed by many recent satellite (e.g., Arase, ELFIN, THEMIS, and Van Allen Probes) and balloon missions (BARREL). However, quantitative understanding of how electron acceleration and precipitation is modified due to the nonlinear interactions with chorus waves is limited. This dissertation systematically evaluates the nonlinear effects of chorus waves in the full electron pitch angle-energy space using test particle simulations, quasilinear models, and satellite observations. More specifically, the dependences of these nonlinear effects on the chorus wave amplitude modulation (waveform structures), as well as wave amplitude and frequency bandwidth (spectrum structures), are quantified over a wide range of wave parameters. The results show that realistic chorus wave structures tend to limit the nonlinear effects on energetic electrons. The system can be described by a diffusion model similar to quasilinear theory, but nonlinear effects alter the diffusion coefficients from quasilinear ones. Using an intriguing event observed by the Van Allen Probes, I further demonstrate that nonlinear phase trapping by the upper-band chorus waves can efficiently accelerate electrons to form the distinct butterfly pitch angle distribution within 30 seconds. The effects of nonlinear interaction (Landau trapping) on electron precipitation are also evaluated during a bursty electron precipitation event observed by the ELFIN CubeSats, in association with very oblique chorus waves observed by THEMIS near the equatorial plane. The test particle simulation results provide the first direct evidence of rapid (~5 s) electron precipitation driven by high-order resonances due to chorus waves. Overall, this dissertation provides a full quantification of nonlinear effects and their dependences on various electron and chorus wave parameters. The findings in this dissertation are crucial to our fundamental understanding of wave-particle interactions, particularly on short timescales in the Earth’s radiation belts and in other space plasma environments, such as solar wind and other planets, as well as astrophysical and laboratory plasmas.