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76 DS05 Abstracts CP1 and the CAP-dependent activation complex. Short-Strand Dna Renaturation by An Active Micro-Mixer David Swigon Department of Mathematics Understanding renaturation of short (40-50 mer long) ss- University of Pittsburgh DNA strands in flow is of interest from both the theoretical [email protected] perspective and from that of experimentalists studying me- chanical properties of DNA. Since the size of these strands is less that the persistence length of ssDNA, we model the CP2 strands as rods. Assuming that two rods combine to form Patched Heteroclinic Orbits in a Family of Scalar dsDNA if they are close together and are closely aligned, Wave Equations and considering the dilute limit, we model the process as a reaction-diffusion-advection system in position-orientation We analyze traveling wave solutions of the following family of scalar wave equations: space where the reaction term has a particularly simple 2 2 form. The model contains avenues for control in the form ut + uux = −γα G ∗ ux , (1) of the advective velocity fields and applied external poten- x tials. We employ the model to study the enhancement of for γ>0, where G is the Green’s function of the Helmholtz 2 reaction rate by an active micro-mixer consisting of chan- operator Hu = u−α uxx. We find a novel class of patched nel that is 200 x 100 microns in cross section and is a few heteroclinic orbits which appear to cross a line of singu- hundred microns in length. Two streams containing each larities in the (u,u’) phase plane. As γ increases past the type of ssDNA are introduced at the inlet and are agitated critical value γ = 1, we observe a qualitative change in the by fluid in side-channels (50microns wide) before reaching behavior of trajectories near the singularity line. These the outlet. We perform 2-D as well as 3-D analysis using orbits correspond to shock-like traveling wave solutions of analytical expressions for the underlying velocity field. We (1), which in the zero-α limit, are global weak solutions of also consider another process, where streams of dye and Burgers’ equation. dsDNA are introduced into the micro-mixer. The reaction consists of the dye attaching itself between the strands of Razvan C. Fetecau the DNA, subsequent to which the dye fluoresces and the Stanford University intensity of fluorescence can thus indicate amount of bound Department of Mathematics dye. This process is treated as a special case of the same [email protected] model. Harish S. Bhat Igor Mezic, Thomas John California Institute of Technology University of California, Santa Barbara Control and Dynamical Systems [email protected], [email protected] [email protected] CP2 CP1 Time Simulation-Based Bifurcation Analysis of A Dynamical Systems Approach to Reconstructing Waves Repetitive Dna In Nonlinearity 16:1257-1275, 2003, we presented a sym- Mathematical methods in biology, especially regarding metry reduction of a PDE which can be used to ”freeze” analysis of DNA sequence information has led to major waves and self-similar solutions. In this presentation, we results in recent years. Still, the task of determining the will review strategies based on time simulation to com- DNA sequence of an organism, genome assembly, remains pute wave solutions, some based on the above symmetry- an expensive and lengthy process. Repetitive regions of reduced PDE. We will also argue that a generalized eigen- DNA are especially difficult to assemble. We develop a value problem is a better way to determine the stability method inspired by ideas from Symbolic Dynamical Sys- since it can filter out the neutral eigenvalues. tems to reconstruct repetitive DNA. We the validity of our method by analyzing two different assemblies of Drosophila Clarence Rowley melanogaster (fruit fly). We conclude by describing ways Princeton University our method can be used to improve genome assembly. Department of Mechanical and Aerospace Engineering [email protected] Suzanne Sindi University of Maryland, College Park Ioannis Kevrekidis [email protected] Dept. of Chemical Engineering Princeton University CP1 [email protected] Mesoscale Modeling of Protein-Dna Complexes Kurt Lust DNA transcription begins with the formation of a complex University of Groningen of DNA and proteins. Recently developed base-pair level Institute of Mathematics and Computing Science theory of DNA elasticity enables construction of structural [email protected] and dynamical models of such complexes that yield new information about the role of the promoter sequence in the mechanism of gene regulation. Presented will be applica- CP2 tions to two complexes important for the regulation of the Traveling Waves for Differential-Difference Equa- Lac operon in E. coli: the LacR-DNA promoter complex tions with Inhomogeneous Diffusion Traveling wave solutions for lattice differential equations DS05 Abstracts 77 are defined by boundary value problems with advances and [email protected] delays on an unbounded domain. Fourier transform tech- niques and Jacobi operator theory allow one to obtain an- Elbert E. Macau alytic results for a problem of this type with a diffusion LAC - Laboratory for Computing and Applied coefficient that is varied in some interval on the lattice. Mathematics Of particular interest is the issue of propagation failure INPE - Brazilian Institute for Space Research of these traveling waves depending on the inhomogeneous [email protected] diffusion. Tony R. Humphries Celso Grebogi McGill University Instituto de Fisica - IF Mathematics & Statistics Universidade de Sao Paulo/ USP [email protected] [email protected] Brian E. Moore CP3 Department of Mathematics and Statistics Dynamics of Adaptive Delayed-Feedback Control McGill University Systems [email protected] First, we derive an adaptive control method for discrete Erik Van Vleck time maps, by extending delayed-feedback controls already Department of Mathematics proposed. Then, we study dynamics of adaptive control University of Kansas systems. In particular, we apply our method to the H´enon [email protected] map, and numerically show the following two character- istics: Power law decay of distribution of control times. Almost zero finite-time Lyapunov exponent. With an an- CP3 alytical treatment, we show the simplest control system Feedback Linearization of Chemostats becomes neutrally stable, which well explains these char- acteristics. The chemostat is a rare example of a class of biological sys- tems that has real experimental and industrial applications Asaki Saito and also admits a modeling paradigm that yields rigor- Future University - Hakodate ously analyzable mathematical models. We apply the dif- [email protected] ferential geometry-based methodology of nonlinear control to mathematical models of chemostats. We show that by properly choosing control parameters these models can be CP4 made equivalent to linear dynamical systems and suitably The Use of Iterated Function Systems in Cryptog- chosen control objectives can be met via linear methods. raphy and Steganography We stress that this is not an approximation, but actually amounts to an analytic equivalence of the full nonlinear Iterated Function Systems (IFS)has found its applications system with controls to a controllable linear system. in many areas of mathematics and our lives. In this talk we show that one can hide data such as Cryptography Mary Ballyk Keys and messages in the attarctors of IFS such that the New Mexico State U. intended observers or recipients can reterive the message Dept of Mathematical Sciences from the atractor. The algorithm along with several exam- [email protected] ples will be disscussed. Ernest Barany Mohammad Khadivi New Mexico State University Jackson State University Dept of Mathematical Sciences Department of Mathematics [email protected] [email protected] CP3 CP4 Geometric Control and Chaotic Synchronization Bounded Nonwandering Sets for Polynomial Maps In this paper, we consider a class of polynomial maps We present a new parameter estimation procedure for non- m m linear systems. Such technique is based on the synchroniza- on R or C which is defined by the assumption that tion between the model and the system whose unknown the delay equations induced by the maps have leading parameter is wanted. Synchronization is accomplished by monomials of single variable. We show that for any map controlling the model to make it follow the system. We use from this class, the nonwandering set is bounded while geometric nonlinear control techniques to design the con- for all unbounded orbits, some kind of monotonicity takes trol system. These techniques allow us to derive necessary place. The class under consideration is proved to contain in and sufficient conditions for synchronization and hence for particular the generalized H´enon maps and the Arneodo- proper parameter estimation. As an example, this proce- Coullet-Tresser maps. dure is used to estimate a parameter of an example serving Mikhail Malkin as a model. Department of Mathematics, Nizhny Novgorod State Ubiratan Freitas University LAC - Laboratory for Computing and Applied Nizhny Novgorod, RUSSIA Mathematics [email protected] INPE - Brazilian Institute for Space Research 78 DS05 Abstracts Ming-Chia Li is illustrated in several examples. Dept. of Math., National Changhua University of Education Yilian Zhang [email protected] University of South Carolina Aiken [email protected] CP5 Adrian Nachman Turbulent Spin Dynamics under the Joint Action of University of Toronto the Distant Dipolar Field and Radiation Damping [email protected] Nonlinear evolution of the microscopic magnetization and experimentally measured net magnetization in high-field CP7 solution magnetic resonance is analyzed. These dynam- Basin Hopping in a Gearbox Model ics arise from the joint action of the distant dipolar field and radiation damping and have been recently used to en- We analyse the consequences of noise in a model for rat- hance sensitivity and contrast in magnetic resonance spec- tling in a single-stage gearbox with a bachlash.
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