Copyright (c) 2009 All rights reserved. http://works.bepress.com/chad_topaz Recent documents in Chad Topaz en-us Wed, 15 Jul 2009 11:40:25 PDT 3600 http://works.bepress.com/chad_topaz/16 http://works.bepress.com/chad_topaz/16 Thu, 09 Jul 2009 09:33:01 PDT We study time-periodic forcing of spatially-extended patterns near a Turing-Hopf bifurcation point. A symmetry-based normal form analysis yields several predictions, including that (i) weak forcing near the intrinsic Hopf frequency enhances or suppresses the Turing amplitude by an amount that scales quadratically with the forcing strength, and (ii) the strongest effect is seen for forcing that is detuned from the Hopf frequency. To apply our results to specific models, we perform a perturbation analysis on general two-component reaction-diffusion systems, which reveals whether the forcing suppresses or enhances the spatial pattern. For the suppressing case, our results explain features of previous experiments on the CDIMA chemical reaction. However, we also find examples of the enhancing case, which has not yet been observed in experiment. Numerical simulations verify the predicted dependence on the forcing parameters. Chad M. Topaz http://works.bepress.com/chad_topaz/15 http://works.bepress.com/chad_topaz/15 Thu, 07 Aug 2008 09:12:16 PDT Chad M. Topaz http://works.bepress.com/chad_topaz/14 http://works.bepress.com/chad_topaz/14 Thu, 07 Aug 2008 09:06:56 PDT Participants: 318 international students at Macalester College screened for latent tuberculosis infection via tuberculin skin tests (TST) from 2004-2008. Objective: To estimate positive TST probability based on countries of residency and history of BCG vaccination. Methods and Results: 52% of students had a positive TST. Logistic regression modeling shows that a positive TST is insignificantly correlated with history of BCG vaccination (p=0.910) but is significantly negatively correlated with residence in the European Union (p=0.00502) and the Middle East (p=0.00451), and positively correlated with residence in North America (p=0.00151). Conclusions: There is a significant relationship between TST result and residence in three regions. The surprising positive correlation for North America is explained by examining those 52 students who resided in North America, 7 of whom had only resided in the United States in spite of being born abroad. For more useful predictive models, we recommend additional patient history data to gather. Grace Goetzke http://works.bepress.com/chad_topaz/13 http://works.bepress.com/chad_topaz/13 Sat, 26 Apr 2008 14:55:46 PDT Recent experiments [A. Kudrolli, B. Pier, J.P. Gollub, Physica D 123 (1998) 99-111] on two-frequency parametrically excited surface waves produce an intriguing "superlattice" wave pattern near a codimension-two bifurcation point where both subharmonic and harmonic waves onset simultaneously, but with different spatial wavenumbers. The superlattice pattern is synchronous with the forcing, spatially periodic on a large hexagonal lattice, and exhibits small-scale triangular structure. Similar patterns have been shown to exist as primary solution branches of a generic 12-dimensional ${\rm D}_6\dot{+}{\rm T}^2$-equivariant bifurcation problem, and may be stable if the nonlinear coefficients of the bifurcation problem satisfy certain inequalities [M. Silber, M.R.E. Proctor, Phys. Rev. Lett. 81 (1998) 2450-2453]. Here we use the spatial and temporal symmetries of the problem to argue that weakly damped harmonic waves may be critical to understanding the stabilization of this pattern in the Faraday system. We illustrate this mechanism by considering the equations developed by Zhang and Vinals [J. Fluid Mech. 336 (1997) 301-330] for small amplitude, weakly damped surface waves on a semi-infinite fluid layer. We compute the relevant nonlinear coefficients in the bifurcation equations describing the onset of patterns for excitation frequency ratios of 2/3 and 6/7. For the 2/3 case, we show that there is a fundamental difference in the pattern selection problems for subharmonic and harmonic instabilities near the codimension-two point. Also, we find that the 6/7 case is significantly different from the 2/3 case due to the presence of additional weakly damped harmonic modes. These additional harmonic modes can result in a stabilization of the superpatterns. Mary Silber http://works.bepress.com/chad_topaz/12 http://works.bepress.com/chad_topaz/12 Sat, 26 Apr 2008 14:49:28 PDT We investigate the role weakly damped modes play in the selection of Faraday wave patterns forced with rationally related frequency components $m\omega$ and $n\omega$. We use symmetry considerations to argue for the special importance of the weakly damped modes oscillating with twice the frequency of the critical mode, and those oscillating primarily with the "difference frequency" $|n−m| \omega$ and the "sum frequency" $(n+m) \omega$. We then perform a weakly nonlinear analysis using equations of Zhang and Viņals [J. Fluid Mech. 336 (1997) 301] which apply to small-amplitude waves on weakly inviscid, deep fluid layers. For weak damping and forcing and one-dimensional waves, we perform a perturbation expansion through fourth-order which yields analytical expressions for onset parameters and the cubic bifurcation coefficient that determines wave amplitude as a function of forcing. For stronger damping and forcing we numerically compute these same parameters, as well as the cubic cross-coupling coefficient for competing standing waves oriented at an angle $\theta$ relative to each other. The resonance effects predicted by symmetry are borne out in the perturbation results for one spatial dimension, and agree with the numerical results for two dimensions. The difference frequency resonance plays a key role in stabilizing superlattice patterns of the SL-I type observed by Kudrolli et al. [Physica D 123 (1-4) (1998) 99]. Chad M. Topaz http://works.bepress.com/chad_topaz/11 http://works.bepress.com/chad_topaz/11 Sat, 26 Apr 2008 14:42:49 PDT This dissertation investigates pattern selection in two-frequency forced Faraday waves. In this system, a fluid layer is subjected to a periodic vertical acceleration $g_z[\cos(\chi)\cos (m\omega t) + \sin(\chi) (n \omega t + \phi)]$, where $m$ and $n$ are co-prime integers. For sufficiently large $g_z$, standing waves form on the free surface. Experiments have produced exotic patterns, including spatially-periodic "superlattice" (SL) patterns (Kudrolli, Pier and Gollub, Physica D, 1998) which contain two length scales. This dissertation determines the selection mechanism for the length scale ratio for the SL-I superlattice pattern.A 12-dimensional $\group{D}_6 \dot{+} \group{T}^2$ equivariant bifurcation theoretic framework (Dionne, Silber and Skeldon, Nonlinearity, 1997) is used to study the competition of stripes, hexagons, rhombs, and SL-I superlattice patterns. Symmetry considerations are used to demonstrate how weakly damped harmonic modes may affect the stability of SL-I patterns through spatiotemporally-resonant triad interactions, which produce resonant contributions to coefficients in the bifurcation equations. To demonstrate this effect explicitly, the coefficients are numerically calculated via a perturbation calculation on partial differential equations of Zhang and Vinals (J. Fluid Mech., 1997) which describe Faraday waves on a deep layer of weakly viscous fluid. A bifurcation analysis reveals that a weakly damped harmonic mode may help stabilize an SL-I superlattice pattern.For weak damping and forcing, symmetry considerations also determine which particular damped harmonic modes have the most significant effect. These are: (i) modes oscillating with twice the frequency of the pattern modes, (ii) "difference frequency" modes oscillating with dominant frequency $|m-n|\omega$ and (iii) "sum frequency" modes oscillating with dominant frequency $(m+n)\omega$. For weak damping and forcing and one dimensional waves, an analytical expression is derived for the cubic self-interaction term. For stronger damping and forcing and two-dimensional waves, the remaining coefficients are computed numerically. Both calculations yield results in good agreement with those obtained from symmetry arguments.A bifurcation analysis demonstrates that the difference frequency modes help stabilize the SL-I pattern and determines the length scale ratio, which is well-predicted by the gravity-capillary wave dispersion relation. The SL-I stabilization effect may be enhanced by appropriate choice of periodic forcing functions with more than two frequency components. Chad M. Topaz http://works.bepress.com/chad_topaz/9 http://works.bepress.com/chad_topaz/9 Sat, 26 Apr 2008 14:34:33 PDT Autonomous underwater vehicles are gradually being recognized as key assets in future combat systems. Central to this attitude is the realization that teams of vehicles acting in concerted fashion can accomplish tasks that are either too costly or simply outside the range of capabilities of single vehicles. The VSW-MCM target reacquisition problem is the primary driver of underwater multi-agent research. Because of the VSW's inherent high vehicle attrition rate and unreliable communication, it is felt that vehicle coordination must be done off-site. In this paper, we suggest an alternative to this which permits on-site coordination despite loss of vehicles and communication. Because of its generality, this approach might also be valuable in land-based and aerial applications. Ben Cook http://works.bepress.com/chad_topaz/8 http://works.bepress.com/chad_topaz/8 Sat, 26 Apr 2008 14:29:54 PDT We use symmetry considerations to investigate control of a class of resonant three-wave interactions relevant to pattern formation in weakly damped, parametrically forced systems near onset. We classify and tabulate the most important damped, resonant modes and determine how the corresponding resonant triad interactions depend on the forcing parameters. The relative phase of the forcing terms may be used to enhance or suppress the nonlinear interactions. We compare our symmetry-based predictions with numerical and experimental results for Faraday waves. Our results suggest how to design multifrequency forcing functions that favor chosen patterns in the lab. Jeff Porter http://works.bepress.com/chad_topaz/7 http://works.bepress.com/chad_topaz/7 Sat, 26 Apr 2008 14:19:12 PDT We construct a continuum model for the motion of biological organisms experiencing social interactions and study its pattern-forming behavior. The model takes the form of a conservation law in two spatial dimensions. The social interactions are modeled in the velocity term, which is nonlocal in the population density and includes a parameter that controls the interaction length scale. The dynamics of the resulting partial integrodifferential equation may be uniquely decomposed into incompressible motion and potential motion. For the purely incompressible case, the model resembles one for fluid dynamical vortex patches. There exist solutions which have constant population density and compact support for all time. Numerical simulations produce rotating structures which have circular cores and spiral arms and are reminiscent of naturally observed phenomena such as ant mills. The sign of the social interaction term determines the direction of the rotation, and the interaction length scale affects the degree of spiral formation. For the purely potential case, the model resembles a nonlocal (forwards or backwards) porous media equation. The sign of the social interaction term controls whether the population aggregates or disperses, and the interaction length scale controls the balance between transport and smoothing of the density profile. For the aggregative case, the population clumps into regions of high and low density. The characteristic length scale of the density pattern is predicted and confirmed by numerical simulations. Chad M. Topaz http://works.bepress.com/chad_topaz/6 http://works.bepress.com/chad_topaz/6 Wed, 23 Apr 2008 04:42:17 PDT We show how pattern formation in Faraday waves may be manipulated by varying the harmonic content of the periodic forcing function. Our approach relies on the crucial influence of resonant triad interactions coupling pairs of critical standing wave modes with damped, spatiotemporally resonant modes. Under the assumption of weak damping and forcing, we perform a symmetry-based analysis that reveals the damped modes most relevant for pattern selection, and how the strength of the corresponding triad interactions depends on the forcing frequencies, amplitudes, and phases. In many cases, the further assumption of Hamiltonian structure in the inviscid limit determines whether the given triad interaction has an enhancing or suppressing effect on related patterns. Surprisingly, even for forcing functions with arbitrarily many frequency components, there are at most five frequencies that affect each of the important triad interactions at leading order. The relative phases of those forcing components play a key role, sometimes making the difference between an enhancing and suppressing effect. In numerical examples, we examine the validity of our results for larger values of the damping and forcing. Finally, we apply our findings to one-dimensional periodic patterns obtained with impulsive forcing and to two-dimensional superlattice patterns and quasipatterns obtained with multifrequency forcing. Chad M. Topaz