LCDS Seminars 2001

Monday, January 29, 2001

Lefschetz Center for Dynamical Systems Seminar

Abstract :   Using two examples, the Kot-Shaffer growth-dispersal model for plants and the Kuramoto-Sivashinsky equation, I will describe a program designed to facilitate computer assisted proofs for infinite dimensional systems. The goal is to develop methods that are both computationally cheap and accurate. The techniques are based on algebraic topology and in particular the Conley index theory. We will finish with some open questions in computational geometry that are relevant to the development of algorithms for approximating the vector field.

Monday, February 5, 2001

Lefschetz Center for Dynamical Systems Seminar

Monday, February 12, 2001

Lefschetz Center for Dynamical Systems Seminar

Abstract :   Two-dimensional turbulence is characterized by the ceaseless creation of filamentary structures and intense gradients of vorticity and passive scalars. This is the manifestation of the tracer cascade down to small scales and it can be explained in terms of ``chaotic advection'' (Aref 1984): the advection by vortices is responsible of chaotic Lagrangian trajectories of particles which tends to stretch and fold the tracer field. A natural way to study this cascade is to examine the dynamics of the tracer gradient. Earlier results on this topic have shown that the velocity gradient tensor can help to characterize the cascade properties in the physical space (Okubo 1970, Weiss 1981). Recently, it has been shown (Basdevant and Philipovitch 1994, Hua and Klein 1998) that the cascade properties are also determined by the acceleration gradient tensor in two-dimensional turbulence. Here we examine the tracer gradient dynamics taking into account the velocity and acceleration gradient tensors. We show that this dynamics can be explained in terms of the dynamics of the orientation of the tracer gradient with respect to strain axes. We stress the importance of two mechanisms: the competition between strain rate and ``effective'' rotation (i.e. rotation due to both vorticity and strain axes rotation), and the time evolution of the strain rate. Because of these mechanisms the tracer gradients tend to align with directions related to the velocity and acceleration gradient tensors. These results are confirmed by numerical simulations of two-dimensional turbulence and some interpretations are discussed.

Monday, February 19, 2001

Lefschetz Center for Dynamical Systems Seminar

Abstract :   I will describe a set of equations that describe rotating fluid flows dominated by body forces. Two types of instabilities are present in such flows, the so-called ``modal stall'' and ``spike stall'', distinguished by the time-scales of their evolution. The dynamics of the system of equations resembles closely the dynamics of reaction-diffusion systems and I will discuss the parallels. The simplest model is a one-dimensional non-local reaction-diffusion equation for which the standard existence and uniqueness results can be shown. The extensions involve two and three-dimensional equations in an annular domain and I will present numerical simulations of the dynamics. The issue of control of instabilities becomes important when application to turbomachinery (axial compressors) is considered. I'll present both linear and nonlinear control strategies, compare them and discuss the issues of (finite-dimensional) implementation. In particular, I'll discuss the issue of the so-called control spillover in this nonlinear problem - a phenomenon when finite-dimensional control stabilizes some modes but destabilizes others.

Thursday, February 22, 2001

Special LCDS Seminar/Applied Mathematics Colloquium

Abstract :   Mixing is important in many areas of science involving fluids, and can be produced by either chaotic advection or weak turbulence. In this talk, the two processes are compared and contrasted. We study transient mixing in thin fluid layers of which one half is initially labeled by a fluorescent dye. In the chaotic case (time-periodic velocity field), the scalar evolves to a complex recurrent pattern that subsequently decays without change of form, as first noted in a numerical simulation by Pierrehumbert. The pattern simply decays slowly like the grin on the Cheshire Cat. The typical path length per cycle of the forcing and the Reynolds number are shown to govern the decay rate, but the dependence is strikingly non-monotonic in these variables. The time evolution of various statistical measures of the scalar field provides a quantitative description of the interplay between stretching and molecular diffusion. It is surprising to note that diffusion does not broaden the striations of the scalar field. We have explored the effects of many flow variables including periodic and nonperiodic forcing in both space and time. Particle tracking over long periods of time is also used to study the transient mixing process. Weakly turbulent flows (obtained by reducing the viscosity) are shown to mix much more efficiently than chaotic flows in the same geometry.

Monday, February 26, 2001

Lefschetz Center for Dynamical Systems Seminar

Abstract :   The study of many-body coupled systems is a problem of common interest to fluid dynamics, plasma physics, and statistical mechanics. Some examples include vortices in two-dimensional fluids, mutually interacting charged particles, and spin systems. The dynamics of these systems is self-consistent in the sense that the evolution of each member of the ensemble is determined by the collective effects of all the other members in the ensemble. In this talk we discuss a mean-field, self-consistent model that describes the weakly nonlinear dynamics of marginally stable fluids and plasmas. The model also describes globally coupled oscillators. With the model we study the role of self-consistent Hamiltonian chaos in the formation of coherent structures, and the problem of chaotic vorticity mixing in dynamically consistent fields.

Monday, March 5, 2001

Lefschetz Center for Dynamical Systems Seminar

Abstract :   We investigate the role of interacting oscillatory instabilities in the nonlinear dynamics and control of axial compressors and airfoil flutter problems. Hopf-Hopf bifurcations generically are capable of generating an exceedingly rich set of dynamic phenomena. I shall present which behaviors appear in the engineering systems and discuss examples illustrating how they might be exploited for control purposes.

Friday, March 16, 2001

* Special Joint LCDS/Scientific Computing Seminar *

Abstract :   The effects of Rossby wave -- turbulence interactions on particle dispersion are investigated in a Lagrangian analysis of the potential vorticity equation for idealized two dimensional turbulence on a $\beta $--plane. The goal is the development of realizable turbulence models for large scale geophysical flows. A Lagrangian analysis produces several exact statistical results for fluid particle dispersion which have direct consequences for simple eddy-viscosity like approximations. Indeed, it seems plausible that the vanishing of the meridional eddy viscosity plays a large role in the establishment of highly stable zonal jets so often observed. In the inviscid problem the first integral time scale of the meridional velocity is found to be zero and the meridional particle dispersion is bounded. The second integral time scale, which determines the magnitude of the bound, is shown to depend explicitly on $\beta $, the enstrophy and the energy of the meridional velocity. The applicability of these predictions is verified in a series of numerical simulations. For a range of $\beta $ values, the meridional extent of quasi-steady, alternating zonal jets appearing in the numerical solutions is seen to scale with the length scale given by the meridional particle dispersion. The relation between this inherent dispersion scale and the dynamic length given by the Rhines ' ``classical'' theory is discussed

Special LCDS Lecture Series and PDE Seminar
PLEASE NOTE CHANGE IN TIME AND PLACE

Abstract :   The magnetization of ferromagnetic materials forms complex structures of different dimensionality and on a broad range of length scales: domains, walls of different internal structure, Bloch lines and vortices. Predicting the formation of these structures, and understanding their interaction and overall effects is crucial for key technological applications. The rich source of experimental data and the simple mathematical form of the well-accepted model makes this analysis of good model problem to develop new mathematical tools for multiscale problems in material science. We report on some new analysis of cross-tie walls and on new dimensionally reduced theories for thin films, their numerical simulation and experimental validation.

Wednesday, April 4, 2001

Special LCDS Lecture Series and PDE Seminar

Abstract :   The capillarity-driven spreading of a thin droplet of a viscous liquid on a solid plane is modelled by the lubrication approximation, an evolution equation for the film height {\it h}. However, as a consequence of the no-slip boundary condition for the liquid at the solid plane, logarithmic divergences in the viscous dissipation rate occur if the support of {\it h} changes.

This well-known singularity is removed by relaxing the no-slip condition, thereby introducing a microscopic lengthscale {\it b}. Matched asymptotics suggests a relationship (Tanner's law) between the speed of the contact line (the boundary of the support of {\it h}) and the macroscopic contact angle (the slope of {\it h} near the boundary of its support), modulo a logarithm involving {\it b}. This dynamic contact angle condition, which balances viscous forces and surface tension, is quite different from the static contact angle condition (Young's law), which balances just the surface tensions.

Tanner's law predicts a specific scaling for the spreading of the droplet. In a joint work with L. Giacomelli, we rigorously derive the scaling of the spreading, which is consistent with the one predicted based on Tanner's law, including the logarithmic terms. Mathematically speaking, this amounts to estimates of

Monday, April 9, 2001

Lefschetz Center for Dynamical Systems Seminar

Abstract :   In the focusing problem, we solve the initial value problem for the porous medium equation with an initial distribution whose support lies outside a compact set $ K $. At a finite positive time $ T $ the exterior of the support of the solution shrinks to a point, and we are interested in the asymptotic of the process in the neighborhood of the focusing point for times near $ T $. If $ K $ is radially symmetric the asymptotic behavior is described by a one parameter family of self-similar solutions. These radial solutions are unstable to non-radial perturbations and there is a sequence of symmetry breaking bifurcations leading to new families of non-radial self-similar solution. We discuss this bifurcation structure as well as a class of non-radial non-self-similar patterns.

Monday, April 23, 2001

Lefschetz Center for Dynamical Systems Seminar

Abstract :   Invariant tori of dynamical systems occur both in the dissipative and in the conservative context. We focus in this talk on the latter, where the tori are intrinsically parametrised by the actions $ I_1, \ldots, I_n $ conjugate to the angles $ \theta_1, \ldots, \theta_n $ on the torus. The distribution of maximal tori in a nearly integrable Hamiltonian system is governed by the invariant tori of co-dimension one. The different Cantor families of maximal tori shrink down to normally elliptic tori and are separated by the web formed by stable and unstable manifolds of normally hyperbolic tori. The lower dimensional

Monday, April 30, 2001

Lefschetz Center for Dynamical Systems Seminar

Abstract :   We study slow modulation of patterns, or quasi-steady solutions, in forced dispersive systems as arise in models of optical parametric processes. The linearization about a pattern yields a family of non-self adjoint operators. The localization of the point spectrum of these operators requires new techniques. Moreover we obtain a rigorous decomposition of the flow near the pattern from a renormalization group method, which permits us to overcome not only the weak smoothing properties of the linear semi-groups, but also their possible secular behavior. We obtain a family of ODEs which describe the

Monday, May 7, 2001

Lefschetz Center for Dynamical Systems Seminar

Abstract :   A relatively new technology (HF Radar) is now providing us with near-surface velocity data in littoral areas. Dynamical systems tools can be used to identify Lagrangian coherent structures which have significant influence on mixing kinematics in these littorals. we compare and apply a variety of these tools to HF radar of Monterey Bay to data from both Auguest 1997 and August 2000: those which locate a hyperbolic trajectory associated with a partical stagnation point (e.g. SP-DHT) and then grow associated manifolds, and those which use a computable criteria to immediately identify manifolds (e.g. direct Lyaponov exponent). Although HF radar offers a valuable data source, there can be a significant amount of measurement error and gaps in the data. We discuss methods for defining our dynamical system as an incomplete data set: projecting the available data onto subsets of the function space spanned by select eigenfunction and how to optimize the amplitude choice in order to minimize the error in the Lagrangian coherent structures. We will discuss some practical uses of such technology in littoral areas, such as distribution of sensor arrays and optimal release of contaminants.

Monday, May 14, 2001

Lefschetz Center for Dynamical Systems Seminar

Abstract :   A dispersion-managed (DM) system, which is a system with periodical variation of dispersion along an optical fiber, is one of the key components of current development of ultrafast high-bit-rate optical communication lines. DM system is governed by a nonlinear Schroedinger equation with variable dispersion. Provided nonlinearity is small enough this equation can be averaged over period of dispersion variation resulting in path-averaged integro-differential Gabitov-Turitsyn equation. The soliton solution is obtained by iterating the path-averaged equation analytically and numerically. An efficient numerical algorithm for obtaining of DM soliton shape is developed. The envelope of soliton oscillating tails is found to decay exponentially in time while the oscillations are described by a quadratic law.

Wednesday, July 25, 2001

Lefschetz Center for Dynamical Systems Seminar REVISED...REVISED...REVISED...REVISED...

Monday, September 17, 2001

Lefschetz Center for Dynamical Systems Seminar

Abstract :   I will explain some recent work with Thierry Gallay on the construction of finite dimensional invariant manifolds in the phase space of the Navier-Stokes equation on ${\bf R}^n$. These manifolds control the long-term behavior of small solutions, give geometric insight into the host of existing results on the asymptotics of such solutions, and allow one to extend those results in a number of ways. Our results also allow us to prove the stability of certain vortex

Monday, September 24, 2001

Lefschetz Center for Dynamical Systems Seminar

Abstract :   I will present a numerical model of optical parametric oscillators that takes into account multi-mode operation of the OPO and the leading order effect of thermal deposition of optical field energy in the quadratic medium. Even though the model captures transient behaviour of the OPO on the order of the nonlinear mixing scale (nanoseconds), a novel and self-consistent advancement method is used to run simulations for durations commensurate with the thermal diffusion scale (milliseconds) and to steady-state (seconds). Using the model, I demonstrate self-induced thermal lensing of the optical fields and temperature-biased competition between two signal/idler pairs.

Monday, October 1, 2001

Lefschetz Center for Dynamical Systems Seminar

Abstract :   We study linear functional differential equations of mixed type, and obtain a fundamental decomposition of the state space into forward and backward semiflows which satisfy the estimates of an exponential dichotomy. We obtain a corresponding factorization of the characteristic function analogous to a Wiener-Hopf factorization. The theory allows us to analyze boundary value problems on infinite intervals (heteroclinic orbits), and long finite intervals ("truncated" heteroclinic orbits).

Monday, October 15, 2001

Lefschetz Center for Dynamical Systems Seminar

Abstract :   Most assimilation schemes in meteorology and oceanography use model variables computed on a fixed grid in space. Lagrangian observations in the ocean do not give the data directly in terms of model variables (e.g. velocities) and are distributed non-uniformly all over the space. We present a scheme for assimilating drifter/float positions observed at discrete times directly into the model. The assimilations is carried out using the extended Kalman filter.

The technique is tested for point vortex flows. An N point vortex system with a Gaussian noise term representing unresolved processes is modeled by the same system without the noise. Positions of L tracer particles are observed and assimilated. Numerical examples show that if the observations are reasonably frequent and accurate, the assimilating model stays close to the original system.

Monday, October 22, 2001

Lefschetz Center for Dynamical Systems Seminar

Abstract :   From the dynamical systems point of view, the behavior of the Earth's atmosphere is extremely high dimensional (e.g., a realistic atmospheric model based on a modal expansion would necessarily include many modes). In spite of the atmosphere's high dimensionality, in this talk I will demonstrate that, in a suitable sense, the local finite-time atmospheric dynamics is often low dimensional. Furthermore, I will show how this finding has important implications for weather forecasting. More generally, this behavior may be common to other physical spatio-temporally chaotic systems, and these systems may also be amenable to the type of analysis that is introduced for the atmosphere.

Monday, October 29, 2001

Lefschetz Center for Dynamical Systems Seminar

Abstract :   Gap solitons are a species of travelling wave that propagate through specialized Bragg Grating optical fibers. They have the enticing property of propagating at any speed between zero and the speed of light. Technical difficulties have prevented experimentalists from actually creating light pulses with zero velocity, which could someday be useful in optical communications devices.

I have been investigating an alternative approach, in which specialized defects are designed to "trap" the light at a designated location. I study this from a variety of angles. First, through extensive simulations of the interaction of the solitons and defects, I develop a theory that predicts under what circumstances light energy is trapped by the defect. Second, I derive finite dimensional (ODE) models for related systems which we can analyze more completely using techniques from dynamical systems theory.

Joint work with Michael Weinstein, Phil Holmes, Dick Slusher.

Thursday, November 8, 2001

Special Joint LCDS/Stochastic Systems Seminar

Abstract :   We will discuss some recent results about stochastic averaging of Hamiltonian systems. We will argue that the appropriate spaces for looking at such problems are stratified spaces. We will understand various behaviors at the junction of these strata (i.e., glueing conditions) and connect some ideas of Khasminskii to the glueing conditions. The results will combine ideas from stochastic processes with dynamical systems.

Monday, November 12, 2001

Lefschetz Center for Dynamical Systems Seminar

Abstract :   The transfer operator is important in many questions of dynamical systems and statistical mechanics. It is particularly helpful for studying the mixing and statistical properties of measures, investigations of zeta functions and Fredholm determinants, piecewise monotong transformations, etc. Estimates for the essential spectral radius of the transfer operator were obtained by V. Baladi, P. Collet, S. Isola, D. Ruelle, and many others. In this talk we give an exact formula for the essential spectral radius of the matrix coefficient transfer operator.

We study Ruell's transfer operator $\mathcal{R}$ induced by a $C^{\mathbf{r}+1}$--smooth expanding map $\varphi$ of a smooth manifold and a $C^{\mathbf{r}}$-- smooth bundle automorphism $\Phi$ of a real vector bundle. We prove the following formula for the essential spectral radius of $\mathcal{R}$ on the space $\mathbf{r}$-times continuously differentiable sections of the bundle with $\alpha$-H"{o}lder $\mathbf{r}$-th derivative: \[ \text{ress} (\mathcal{R})=\exp\left(\sup_{\nu\in\text{Erg }} \{h_\nu+\lambda_\nu-(\mathbf{r}+\alpha)\chi_\nu\} \right) , \] where $\text{Erg }$ is the set of $\varphi$-ergodic measures, $h_\nu$ the entrophy of $\varphi$ with respect to $\nu$, $\lambda_\nu$ the largest Lyapunov exponent of the cocycle induced by $\Phi$, and $\chi_\nu$ the smallest Lyapunov exponent for the differential $D\varphi$.

Monday, November 19, 2001

Lefschetz Center for Dynamical Systems Seminar

Abstract :   Intermittency is one of the important phenomena in turbulence. Simply put intermittency is the fact that
the probability distribution functions (PDF's) for quantities transported by a turbulent flow are asymptotically
broad - wider than a Gaussian distribution.

We present some work (with R.M. McLaughlin (UNC)) on a model of passive scalar intermittency originally due to Majda: \[T_t = \gamma(t) x \frac{\partial T}{\partial y} + \Delta T \] where $\gamma(t)$ is a random process, and $T$ is a passive scalar (for instance a dye) which is advected by the random (shear) flow. Majda was able to explicitly calculate moments of the distribution of the scalar $T$. McLaughlin and B. were able to calculate the large $N$ asymptotics of the moments of the distribution and, by a large deviations/Tauberian type argument calculate the distribution of the quantity $T$.

I will also talk about some recent work on a generalization of this model. A similar calculation can be done for this generalized model, which involves calculating the asymptotics of a certain compact eigenvalue problem. As a by-product of this calculation one finds the (previously unknown) optimal constants in a certain probabilistic "small ball" estimate for the probability that a fractional Brownian motion stays in a small ball in $L_2$.

Monday, November 26, 2001

Lefschetz Center for Dynamical Systems Seminar

Abstract :   Semiconductor lasers are key components of a variety of modem photonic systems such as fiber, diode pumped solid state devices and Raman amplifiers. However when they are exposed to even small amounts of feedback or external injection from a distant reflector or another laser, then radiation emitted from these lasers exhibits a plethora of instabilities and irregular chaotic transitions. Recent experiments in semiconductor lasers subject to optical feedback as well as experiments with pairs of mutually coupled diode lasers have renewed the interest to dissect these experimetal findings with simple delay rate equations. We will review two novel cases. Diodes pumped close to threshold with long cavities and biased well above threshold with ultrashort cavities. When diode lasers are biased near threshold and subject to moderate optical feedback, low frequency fluctuations appear in their radio-frequency spectrum that are evident as dropout events in the intensity time traces. Traditionally these events were obseved to occur at sporadic time intervals. However, recent experimental measurements have shown that there are regions of the pumping current where these events appear at regular time intervals. The case of ultrashort cavities presents us with similar but ultimately more dramatic phenomena, such as periodic self pulsations and high speed oscillations. In addition, work on optical injection, for applications on bandwidth enhancement and optical generation of microwaves will be reviewed.

Monday, December 3, 2001

Lefschetz Center for Dynamical Systems Seminar

Abstract :   One of the classical linear instabilities is that of two fluids of different densities, accelerated towards each other, which is called Rayleigh-Taylor instability. It will be shown in the talk that smooth steady states are indeed nonlinearly unstable in a dynamical setting.

Monday, December 10, 2001

Lefschetz Center for Dynamical Systems Seminar

Abstract :   We present a mathematical theory of time-reversal experiments. In such experiments a signal is emitted by a localized source, propagated through a medium and recorded on a small array of receivers-transducers. The signal is re-emitted into the medium reversed in time, that is, the part of the signal recorded first is re-emitted last and vice versa. The re- propagated signal approximately refocuses back on the original source. This is somewhat surprising since the recording array has a small finite size. It is also observed that refocusing is significantly better in a random medium. We will give an explanation for refocusing and explain why random media are good for refocusing, as are ergodic billiards.

This is a joint work with Guillaume Bal.