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Global paths of time-periodic solutions of the Benjamin-Ono equation connecting arbitrary traveling waves

Description: We classify all bifurcations from traveling waves to non-trivial time-periodic solutions of the Benjamin-Ono equation that are predicted by linearization. We use a spectrally accurate numerical continuation method to study several paths of non-trivial solutions beyond the realm of linear theory. These paths are found to either re-connect with a different traveling wave or to blow up. In the latter case, as the bifurcation parameter approaches a critical value, the amplitude of the initial condition grows without bound and the period approaches zero. We propose a conjecture that gives the mapping from one bifurcation to its counterpart on the other side of the path of non-trivial solutions. By experimentation with data fitting, we identify the form of the exact solutions on the path connecting two traveling waves, which represents the Fourier coefficients of the solution as power sums of a finite number of particle positions whose elementary symmetric functions execute simple orbits in the complex plane (circles or epicycles). We then solve a system of algebraic equations to express the unknown constants in the new representation in terms of the mean, a spatial phase, a temporal phase, four integers (enumerating the bifurcation at each end of the path) and one additional bifurcation parameter. We also find examples of interior bifurcations from these paths of already non-trivial solutions, but we do not attempt to analyze their algebraic structure.
Date: December 11, 2008
Creator: Ambrose, David M. & Wilkening, Jon
Partner: UNT Libraries Government Documents Department

An infinite branching hierarchy of time-periodic solutions of the Benjamin-Ono equation

Description: We present a new representation of solutions of the Benjamin-Ono equation that are periodic in space and time. Up to an additive constant and a Galilean transformation, each of these solutions is a previously known, multi-periodic solution; however, the new representation unifies the subset of such solutions with a fixed spatial period and a continuously varying temporal period into a single network of smooth manifolds connected together by an infinite hierarchy of bifurcations. Our representation explicitly describes the evolution of the Fourier modes of the solution as well as the particle trajectories in a meromorphic representation of these solutions; therefore, we have also solved the problem of finding periodic solutions of the ordinary differential equation governing these particles, including a description of a bifurcation mechanism for adding or removing particles without destroying periodicity. We illustrate the types of bifurcation that occur with several examples, including degenerate bifurcations not predicted by linearization about traveling waves.
Date: July 1, 2008
Creator: Wilkening, Jon
Partner: UNT Libraries Government Documents Department

Crystallographic texture effects on mixed-mode strain localization for lower-symmetry metals

Description: A bifurcation analysis is used to understand the damage realized in flatplate gas-gun specimens that were machined from a highly-textured plate stock of Zr. These low-symmetry material specimens were tested to insipient failure and subsequently soft-recovered. Post-mortem data sets consisting of EBSD imaging of metallographic samples cut from the recovered targets show very different texture-dependent damage morphologies depending on the initial texturekarget orientation at impact.
Date: January 1, 2003
Creator: Maudlin, P. J. (Paul J.); Mason, T. A. (Thomas A.); Gray, G. T. (George T.), III; Bourne, N. K. & Bingert, J. F. (John F.)
Partner: UNT Libraries Government Documents Department

Nonlinear Dynamics of Parametrically Excited Gyroscopic Systems

Description: The primary objective of this project is to determine how some of the powerful geometric methods of dynamical systems can be applied to study nonlinear gyroscopic systems. We proposed to develop techniques to predict local and global behavior and instability mechanisms and to analyze the interactions between noise, stability, and nonlinearities inherent in gyroscopic systems. In order to obtain these results we use the method of normal forms, global bifurcation techniques, and various other dynamical systems tools.
Date: June 1, 2001
Creator: Namachchivaya, N. S.
Partner: UNT Libraries Government Documents Department

CSR Interaction for a 2D Energy-Chirped Bunch on a General Orbit

Description: When an electron bunch with initial linear energy chirp traverses a bunch compression chicane, the bunch interacts with itself via coherent synchrotron radiation (CSR) and space charge force. The effective longitudinal CSR force for such kind of 2D bunch on a circular orbit has been analyzed earlier [1]. In this paper, we present the analytical results of the effective longitudinal CSR force for a 2D energy-chirped bunch going through a general orbit, which includes the entrance and exit of a circular orbit. In particular, we will show the behavior of the force in the last bend of a chicane when the bunch is under extreme compression. This is the condition when bifurcation of bunch phase space occurs in many CSR measurements. [1] R. Li, Phys. Rev. ST Accel. Beams 11, 024401 (2008)
Date: May 1, 2009
Creator: Li, Rui
Partner: UNT Libraries Government Documents Department

Turbulent fluctuations in the main core of TFTR plasmas with negative magnetic shear

Description: Turbulent fluctuations in plasmas with reversed magnetic shear have been investigated in TFTR. Under intense auxiliary heating, these plasmas are observed to bifurcate into two states with different transport properties. In the state with better confinement, it has been found that the level of fluctuations is very small throughout most of the region with negative shear. By contrast, the state with lower confinement is characterized by large bursts of fluctuations which suggest a competition between the driving and the suppression of turbulence. These results are consistent with the suppression of turbulence by the E x B velocity shear.
Date: September 1997
Creator: Mazzucato, E.; Beer, M. A.; Bell, M. G. & Batha, S. H.
Partner: UNT Libraries Government Documents Department

Bifurcation Mode of Relativistic and Charge-Displacement Self-Channeling

Description: Stable self-channeling of ultra-powerful (P{sub 0} - 1 TW -1 PW) laser pulses in dense plasmas is a key process for many applications requiring the controlled compression of power at high levels. Theoretical computations predict that the transition zone between the stable and highly unstable regimes of relativistic/charge-displacement self-channeling is well characterized by a form of weakly unstable behavior that involves bifurcation of the propagating energy into two powerful channels. Recent observations of channel instability with femtosecond 248 nm pulses reveal a mode of bifurcation that corresponds well to these theoretical predictions. It is further experimentally shown that the use of a suitable longitudinal gradient in the plasma density can eliminate this unstable behavior and restore the efficient formation of stable channels.
Date: July 20, 2000
Creator: BORISOV,A.B.; CAMERON,STEWART M.; LUK,TING S.; NELSON,THOMAS R.; VAN TASSLE,A.J.; SANTORO,J. et al.
Partner: UNT Libraries Government Documents Department

Reexamination of fault angles predicted by shear localization theory

Description: This paper reexamines orientations of shear bands (fault angles) predicted by a theory of shear localization as a bifurcation from homogeneous deformation. In contrast to the Coulomb prediction, which does not depend on deviatoric stress state, the angle between the band normal and the least (most compressive) principal stress increases as the deviatoric stress state varies from axisymmetric compression to axisymmetric extension. This variation is consistent with the data of Mogi (1967) on Dunham dolomite for axisymmetric compression, extension and biaxial compression, but the predicted angles are generally less than observed. This discrepancy may be due to anisotropy that develops due to crack growth in preferred orientations. Results from specialized constitutive relations for axisymmetric compression and plane strain that include this anisotropy indicate that it tends to increase the predicted angles. Measurements for a weak, porous sandstone (Castlegate) indicate that the band angle decreases with increasing inelastic compaction that accompanies increasing mean stress. This trend is consistent with the predictions of the theory but, for this rock, the observed angles are less than predicted.
Date: February 1, 1998
Creator: Rudnicki, J.W. & Olsson, W.A.
Partner: UNT Libraries Government Documents Department

{Delta}I = 4 bifurcation: Origins and criteria

Description: The new {gamma}-ray detector arrays have demonstrated that rotational sequences in certain superdeformed bands with angular momentum differing by two can split into two branches. This is commonly called {Delta}I = 4 bifurcation, and has attracted considerable interest in the nuclear structure community. An alternative approach for the {Delta}I = 4 bifurcation phenomenon has been presented without introducing either a Y{sub 44} deformation or an I{sup 4} term in the Hamiltonian explicitly. The optimal criteria for observing the phenomenon have been discussed as well.
Date: December 31, 1995
Creator: Zhang, J.Y.; Sun, Y. & Guidry, M.
Partner: UNT Libraries Government Documents Department

A Theoretical Investigation of Mode-Locking Phenomena in Reversed Field Pinches

Description: OAK-B135 This paper investigates the formation and breakup of the ''slinky mode'' in an RFP using analytic techniques previously employed to examine mode locking phenomena in tokamaks. The slinky mode is a toroidally localized, coherent interference pattern in the magnetic field which co-rotates with the plasma at the reversal surface. This mode forms, as a result of the nonlinear coupling of multiple m = 1 core tearing modes, via a bifurcation which is similar to that by which toroidally coupled tearing modes lock together in a tokamak. The slinky mode breaks up via a second bifurcation which is similar to that by which toroidally coupled tearing modes in a tokamak unlock. However, the typical m = 1 mode amplitude below which slinky breakup is triggered is much smaller than that above which slinky formation occurs. Analytic expressions for the slinky formation and breakup thresholds are obtained in all regimes of physical interest. The locking of the slinky mode to a static error-field is also investigated analytically. Either the error-field arrests the rotation of the plasma at the reversal surface before the formation of the slinky mode, so that the mode subsequently forms as a non-rotating mode, or the slinky mode forms as a rotating mode and subsequently locks to the error-field. Analytic expressions for the locking and unlocking thresholds are obtained in all regimes of physical interest. The problems associated with a locked slinky mode can be alleviated by canceling out the accidentally produced error-field responsible for locking the slinky mode, using a deliberately created ''control'' error-field. Alternatively, the locking angle of the slinky mode can be swept toroidally by rotating the control field.
Date: April 7, 2004
Creator: Fitzpatrick, Richard
Partner: UNT Libraries Government Documents Department

A theoretical investigation of mode-locking phenomena in reversed field pinches

Description: OAK-B135 This paper investigates the formation and breakup of the ''slinky mode'' in an RFP using analytic techniques previously employed to examine mode locking phenomena in tokamaks. The slinky mode is a toroidally localized, coherent interference pattern in the magnetic field which co-rotates with the plasma at the reversal surface. This mode forms, as a result of the nonlinear coupling of multiple m = 1 core tearing modes, via a bifurcation which is similar to that by which toroidally coupled tearing modes lock together in a tokamak. The slinky mode breaks up via a second bifurcation which is similar to that by which toroidally coupled tearing modes in a tokamak unlock. However, the typical m = 1 mode amplitude below which slinky breakup is triggered is much smaller than that above which slinky formation occurs. Analytic expressions for the slinky formation and breakup thresholds are obtained in all regimes of physical interest. The locking of the slinky mode to a static error-field is also investigated analytically. Either the error-field arrests the rotation of the plasma at the reversal surface before the formation of the slinky mode, so that the mode subsequently forms as a non-rotating mode, or the slinky mode forms as a rotating mode and subsequently locks to the error-field. Analytic expressions for the locking and unlocking thresholds are obtained in all regimes of physical interest. The problems associated with a locked slinky mode can be alleviated by canceling out the accidentally produced error-field responsible for locking the slinky mode, using a deliberately created ''control'' error-field. Alternatively, the locking angle of the slinky mode can be swept toroidally by rotating the control field.
Date: March 17, 2004
Creator: Fitzpatrick, Richard
Partner: UNT Libraries Government Documents Department

Bifurcation Theory of the Transition to Collisionless Ion-temperature-gradient-driven Plasma Turbulence

Description: The collisionless limit of the transition to ion-temperature-gradient-driven plasma turbulence is considered with a dynamical-systems approach. The importance of systematic analysis for understanding the differences in the bifurcations and dynamics of linearly damped and undamped systems is emphasized. A model with ten degrees of freedom is studied as a concrete example. A four-dimensional center manifold (CM) is analyzed, and fixed points of its dynamics are identified and used to predict a ''Dimits shift'' of the threshold for turbulence due to the excitation of zonal flows. The exact value of that shift in terms of physical parameters is established for the model; the effects of higher-order truncations on the dynamics are noted. Multiple-scale analysis of the CM equations is used to discuss possible effects of modulational instability on scenarios for the transition to turbulence in both collisional and collisionless cases.
Date: September 22, 2005
Creator: Kolesnikov, R.A. & Krommes, J.A.
Partner: UNT Libraries Government Documents Department

Post-Treatment Hemodynamics of a Basilar Aneurysm and Bifurcation

Description: Aneurysm re-growth and rupture can sometimes unexpectedly occur following treatment procedures that were initially considered to be successful at the time of treatment and post-operative angiography. In some cases, this can be attributed to surgical clip slippage or endovascular coil compaction. However, there are other cases in which the treatment devices function properly. In these instances, the subsequent complications are due to other factors, perhaps one of which is the post-treatment hemodynamic stress. To investigate whether or not a treatment procedure can subject the parent artery to harmful hemodynamic stresses, computational fluid dynamics simulations are performed on a patient-specific basilar aneurysm and bifurcation before and after a virtual endovascular treatment. The simulations demonstrate that the treatment procedure produces a substantial increase in the wall shear stress. Analysis of the post-treatment flow field indicates that the increase in wall shear stress is due to the impingement of the basilar artery flow upon the aneurysm filling material and to the close proximity of a vortex tube to the artery wall. Calculation of the time-averaged wall shear stress shows that there is a region of the artery exposed to a level of wall shear stress that can cause severe damage to endothelial cells. The results of this study demonstrate that it is possible for a treatment procedure, which successfully excludes the aneurysm from the vascular system and leaves no aneurysm neck remnant, to elevate the hemodynamic stresses to levels that are injurious to the immediately adjacent vessel wall.
Date: January 16, 2008
Creator: Ortega, J; Hartman, J; Rodriguez, J & Maitland, D
Partner: UNT Libraries Government Documents Department

The dynamics of unsteady detonation in ozone

Description: An ultra-fine, sub-micron discrete grid is used to capture the unsteady dynamics of a one-dimensional detonation in an inviscid O - O{sub 2} - O{sub 3} mixture. The ultra-fine grid is necessary to capture the length scales revealed by a complementary analysis of the steady detonation wave structure. For the unsteady calculations, shock-fitting coupled with a high order spatio-temporal discretization scheme combine to render numerical corruption negligible. As a result, mathematically verified solutions for a mixture initially of all O{sub 3} at one atmosphere and 298.15 K have been obtained; the solutions are converging at a rate much faster than the sub-first order convergence rate of all shock-capturing schemes. Additionally, the model has been validated against limited experimental data. Transient calculations show that strongly overdriven waves are stable and moderately overdriven waves unstable. New limit cycle behavior is revealed, and the first high resolution bifurcation diagram for etonation with detailed kinetics is found.
Date: January 1, 2008
Creator: Aslam, Tariq D & Powers, Joseph M
Partner: UNT Libraries Government Documents Department

Experimental Characterization of the Transverse Phase Space of a 60-MeV Electron Beam Through a Compressor Chicane

Description: Space charge and coherent synchrotron radiation may deteriorate electron beam quality when the beam passes through a magnetic bunch compressor. This paper presents the transverse phase-space tomographic measurements for a compressed beam at 60 MeV, around which energy the first stage of magnetic bunch compression takes place in most advanced linacs. Transverse phase-space bifurcation of a compressed beam is observed at that energy, but the degree of the space charge-induced bifurcation is appreciably lower than the one observed at 12 MeV.
Date: February 12, 2007
Creator: Zhou, F.; Kabel, A.; Rosenzweig, J.; Agustsson, R.; Andonian, G.; Cline, D. et al.
Partner: UNT Libraries Government Documents Department

Large-Scale Eigenvalue Calculations for Stability Analysis of Steady Flows on Massively Parallel Computers

Description: We present an approach for determining the linear stability of steady states of PDEs on massively parallel computers. Linearizing the transient behavior around a steady state leads to a generalized eigenvalue problem. The eigenvalues with largest real part are calculated using Arnoldi's iteration driven by a novel implementation of the Cayley transformation to recast the problem as an ordinary eigenvalue problem. The Cayley transformation requires the solution of a linear system at each Arnoldi iteration, which must be done iteratively for the algorithm to scale with problem size. A representative model problem of 3D incompressible flow and heat transfer in a rotating disk reactor is used to analyze the effect of algorithmic parameters on the performance of the eigenvalue algorithm. Successful calculations of leading eigenvalues for matrix systems of order up to 4 million were performed, identifying the critical Grashof number for a Hopf bifurcation.
Date: August 1, 1999
Creator: Lehoucq, Richard B. & Salinger, Andrew G.
Partner: UNT Libraries Government Documents Department

Bifurcations and Patterns in Nonlinear Dissipative Systems

Description: This project consists of experimental investigations of heat transport, pattern formation, and bifurcation phenomena in non-linear non-equilibrium fluid-mechanical systems. These issues are studies in Rayleigh-B\'enard convection, using both pure and multicomponent fluids. They are of fundamental scientific interest, but also play an important role in engineering, materials science, ecology, meteorology, geophysics, and astrophysics. For instance, various forms of convection are important in such diverse phenomena as crystal growth from a melt with or without impurities, energy production in solar ponds, flow in the earth's mantle and outer core, geo-thermal stratifications, and various oceanographic and atmospheric phenomena. Our work utilizes computer-enhanced shadowgraph imaging of flow patterns, sophisticated digital image analysis, and high-resolution heat transport measurements.
Date: May 27, 2005
Creator: Ahlers, Guenter
Partner: UNT Libraries Government Documents Department

LOCA 1.0 Library of Continuation Algorithms: Theory and Implementation Manual

Description: LOCA, the Library of Continuation Algorithms, is a software library for performing stability analysis of large-scale applications. LOCA enables the tracking of solution branches as a function of a system parameter, the direct tracking of bifurcation points, and, when linked with the ARPACK library, a linear stability analysis capability. It is designed to be easy to implement around codes that already use Newton's method to converge to steady-state solutions. The algorithms are chosen to work for large problems, such as those that arise from discretizations of partial differential equations, and to run on distributed memory parallel machines. This manual presents LOCA's continuation and bifurcation analysis algorithms, and instructions on how to implement LOCA with an application code. The LOCA code is being made publicly available at www.cs.sandia.gov/loca.
Date: March 1, 2002
Creator: SALINGER, ANDREW G.; BOU-RABEE, NAWAF M.; BURROUGHS,ELIZABETH A.; PAWLOWSKI, ROGER P.; LEHOUCQ, RICHARD B.; ROMERO, LOUIS et al.
Partner: UNT Libraries Government Documents Department

Theoretical studies in nuclear structure. Final progress report, June 1, 1991--July 31, 1996

Description: The general purview of the project is the theory of collective motion in atomic nuclei. The chief aim is to elucidate the phenomena of (1) anharmonic multiphonon excitations, and (2) collective tilted rotation, both of which are topics of considerable current interest. In the primary stage of an investigation it is often necessary to develop appropriate mathematical tools, as was the case here. In the next stage, the formalism must be tested on simple soluble models. The work described here is mainly concerned with these two stages. The final stage of realistic applications will require more time, manpower and, of course, the necessary funding. Some planning for this last stage has been carried out and anticipated problems axe briefly discussed. As it turns out, both of the above topics can be approached within the unified framework of a theorem that I developed, called the Cranking Bifurcation Theorem (CBT) to be described below. The CBT can be regarded as an outgrowth of the boson expansion method, which provides a general, and, in principal, exact formalism for treating collective excitations. We begin with a brief discussion of the CBT and then continue on to the applications.
Date: June 1, 1997
Creator: Marshalek, E. R.
Partner: UNT Libraries Government Documents Department

Discontinuous Bifurcation Analysis of the Gurson Model for Ductile Void Growth and a Local Material Failure Model.

Description: The Gurson constitutive equation and variations of it have been used extensively to model ductile damage as represented by the growth in porosity. The model has been widely used in large numerical simulations of dynamic impact and penetration. Because the growth of porosity causes softening there is always a possibility that a discontinuous bifurcation will exist at a critical value of porosity. The criterion for a discontinuous bifurcation is identical to that of loss of ellipticity and, consequently, a well-posed problem becomes ill posed once a discontinuous bifurcation exists. Since the Gurson model has been, and is continuing to be, used extensively in numerical solutions for a wide range of technically complex problems, it is often considered to be not feasible to confirm well-posedness by demonstrating convergence with mesh refinement. Therefore, an analytical criterion for loss of ellipticity is of considerable value for computational purposes. The analysis demonstrates that for a particular form of the Gurson model loss of ellipticity may occur at small strains and values of porosity. The inherent implication is that numerical solutions that do not include a check for loss of ellipticity, or that do not include a convergence study, may not be valid solutions.
Date: January 1, 2002
Creator: Lewis, M. W. (Matthew W.)
Partner: UNT Libraries Government Documents Department

Experimental and Numerical Investigation of Flows in Expanding Channels

Description: We present an experimental realization of the classical Jeffery-Hamel flows inside a wedge-shaped channel. We compare the measured velocity fields with the predictions of Jeffery-Hamel theory. A detailed experimental study of bifurcation diagrams for the solutions reveals the absolute stability of the pure outflow solution and an interesting hysteretic structure for bifurcations. We also observe a multiple vortex flow regime predicted earlier numerically and analytically. Experimental studies of the stability of the flow to perturbations at the channel exit are also conducted.
Date: October 24, 2008
Creator: Vorobieff, Peter & Putkaradze, Vakhtang
Partner: UNT Libraries Government Documents Department