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Series: CDSNS Colloquium

In the talk I will discuss the periodicity of solutions to the classical forced pendulum equation y" + A sin y = f(t) where A= g/l is the ratio of the gravity constant and the pendulum length, and f(t) is an external periodic force with a minimal period T. The major concern is to characterize conditions on A and f under which the equation admits periodic solutions with a prescribed minimal period pT, where p>1 is an integer. I will show how the new approach, based on the critical point theory and an original decomposition technique, leads to the existence of such solutions without requiring p to be a prime as imposed in most previous approaches. In addition, I will present the first non-existence result of such solutions which indicates that long pendulum has a natural resistance to oscillate periodically.

Series: CDSNS Colloquium

I will discuss new computational tools based on topological methods that extracts coarse, but rigorous, combinatorial descriptions of global dynamics of multiparameter nonlinear systems. These techniques are motivated by the fact that these systems can produce an wide variety of complicated dynamics that vary dramatically as a function of changes in the nonlinearities and the following associated challenges which we claim can, at least in part, be addressed. 1. In many applications there are models for the dynamics, but specific parameters are unknown or not directly computable. To identify the parameters one needs to be able to match dynamics produced by the model against that which is observed experimentally. 2. Experimental measurements are often too crude to identify classical dynamical structures such as fixed points or periodic orbits, let alone more the complicated structures associated with chaotic dynamics. 3. Often the models themselves are based on nonlinearities that a chosen because of heuristic arguments or because they are easy to fit to data, as opposed to being derived from first principles. Thus, one wants to be able to separate the scientific conclusions from the particular nonlinearities of the equations. To make the above mentioned comments concrete I will describe the techniques in the context of a simple model arising in population biology.

Series: CDSNS Colloquium

I will speak on the dispersive character of waves on the interface between vacuum and water under the influence of gravity and surface tension. I will begin by giving a precise account of the formulation of the surface water-wave problem and discussion of its distinct features. They include the dispersion relation, its severe nonlinearity, traveling waves and the Hamiltonian structure. I will describe the recent work of Hans Christianson, Gigliola Staffilani and myself on the local smoothing effect of 1/4 derivative for the fully nonlinear problem under surface tension with some detail of the proof. If time permits, I will explore some open questions regarding long-time behavior and stability.

Series: CDSNS Colloquium

Allocation of service capacity ('staffing') at stations in queueing networks is both of fundamental and practical interest. Unfortunately, the problem is mathematically intractable in general and one therefore typically resorts to approximations or computer simulation. This talk describes work in progress with M. Squillante and S. Ghosh (IBM Research) on an algorithm that serves as an approximation for the 'best' capacity allocation rule. The algorithm can be interpreted as a discrete-time dynamical system, and we are interested in uniqueness of a fixed point and in convergence properties. No prior knowledge on queueing networks will be assumed.

Series: CDSNS Colloquium

In this talk we will review results on local entropy theory for the past 15 years, introduce the current development and post some open questions for the further study.

On the theory and applications of the longtime dynamics of 3-dimensional fluid flows on thin domains

Series: CDSNS Colloquium

The current theory of global attractors for the Navier-Stokes equations on thin 3D domains is motivated by the desire to better understand the theory of heat transfer in the oceans of the Earth. (In this context, the thinness refers to the aspect ratio - depth divided by expanse - of the oceans.) The issue of heat transfer is, of course, closely connected with many of the major questions concerning the climate. In order to exploit the tools of modern dynamical systems in this study, one needs to know that the global attractors are "good" in the sense that the nonlinearities are Frechet differentiable on these attractors. About 20 years ago, it was discovered that on certain thin 3D domains, the Navier-Stokes equations did possess good global attractors. This discovery, which was itself a major milestone in the study of the 3D Navier-Stokes equations, left open the matter of extending the theory to cover oceanic-like regions with the appropriate physical boundary behavior. In this lecture, we will review this theory, and the connections with climate modeling, while placing special emphasis on the recent developments for fluid flows with the Navier (or slip) boundary conditions

Series: CDSNS Colloquium

Motivated by Smale's work on smooth dynamical systems, David Ruelle introduced the notion of Smale spaces. These are topological dynamical systems which are hyperbolic in the sense of having local coordinates of contracting and expending directions. These include hyperbolic automorphisms of tori, but typically, the spaces involved have a fractal nature. An important subclass are the shifts of finite type which are symbolic systems described by combinatorial data. These are also precisely the Smale spaces which are totally disconnected. Rufus Bowen showed that every Smale space is the image of shift of finite type under a finite-to-one factor map. In the 1970's, Wolfgang Krieger introduced a beautiful invariant for shifts of finite type. The aim of this talk is to show how a refined version of Bowen's theorem may be used to extend Krieger's invariant to all (irreducible) Smale spaces. The talk will assume no prior knowledge of these topics - we begin with a discussion of Smale spaces and shifts of finite type and then move on to Krieger's invariant and its extension.

Series: CDSNS Colloquium

The Cohen-Gallavotti Fluctuation theorem is a result describing the behaviour of simple hyperbolic dynamical systems. It was introduced to illustrate, in a somewhat simpler context, anomalies in the second law of thermodynamics. I will describe the mathematical formulation of this Fluctuation Theorem, and some variations on it.

Series: CDSNS Colloquium

A plasma is a completed ionized gas. In many applications such as in nuclear fusion or astrophysical phenomena, the plasma has very high temperature and low density, thus collisions can be ignored. The standard kinetic models for a collisionless plasma are the Vlasov- Maxwell and Vlasov-Poisson systems. The Vlasov-Poisson system is also used to model galaxy dynamics, where a star plays the role of a particle. There exists infinitely many equilibria for Vlasov models and their stability is a very important issue in physics. I will describe some of my works on stability and instability of various Vlasov equilibria.

Series: CDSNS Colloquium

Permutation entropy was introduced as a complexity measure of time series. Formally, it replaces the symbol blocks in the definition of Shannon entropy by the so-called ordinal patterns –a digest of the ups-and-downs along a finite orbit in a totally ordered state space. Later, this concept was extended to self maps of n-dimensional intervals, in metric and topological versions. It can be proven that, under some assumptions, the metric and topological permutation entropy coincide with their corresponding conventional counterparts. Besides its use as an entropy estimator, permutation entropy has found some interesting applications. We will talk about the detection of determinism in noisy time series, and the recovery of the control parameter from the symbolic sequences of a unimodal map (which allows to cryptanalize some chaotic ciphers).