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Series: CDSNS Colloquium

Many dynamical systems may be subject to stochastic excitations, so to find an efficient method to analyze the stochastic system is very important. As for the complexity of the stochastic systems, there are not any omnipotent methods. What I would like to present here is a brief introduction to quasi-non-integrable Hamiltonian systems and stochastic averaging method for analyzing certain stochastic dynamical systems. At the end, I will give some examples of the method.

Series: CDSNS Colloquium

Stable sets and unstable sets of a dynamical system with positive entropy
are investigated. It is shown that in any invertible system with positive entropy,
there is a measure-theoretically ?rather big? set such that for any point from the
set, the intersection of the closure of the stable set and the closure of the
unstable set of the point has positive entropy.
Moreover, for several kinds of specific systems, the lower bound of Hausdorff
dimension of these sets is estimated. Particularly the lower bound of the Hausdorff
dimension of such sets appearing in a positive entropy diffeomorphism on a smooth
Riemannian manifold is given in terms of the metric entropy and of Lyapunov exponent.

Series: CDSNS Colloquium

Despite advances in treatment of chronic hepatitis B virus (HBV) infection,
liver transplantation remains the only hope for many patients with end-stage
liver disease due to HBV. A complication with liver transplantation,
however, is that the new liver is eventually reinfected in chronic HBV
patients by infection in other compartments of the body. We have formulated
a model to describe the dynamics of HBV after liver transplant, considering
the liver and the blood of areas of infection. Analyzing the model, we
observe that the system shows either a transcritical or a backward
bifurcation. Explicit conditions on the model parameters are given for the
backward bifurcation to be present, to be reduced, or disappear.
Consequently, we investigate possible factors that are responsible for
HBV/HCV infection and assess control strategies to reduce HBV/HCV
reinfection and improve graft survival after liver transplantation.

Series: CDSNS Colloquium

This talk continues from last week's colloquium.

Fourier's Law assert that the heat flow through a point in a solid is proportional to the temperature gradient at that point. Fourier himself thought that this law could not be derived from the mechanical properties of the elementary constituents (atoms and electrons, in modern language) of the solid. On the contrary, we now believe that such a derivation is possible and necessary. At the core of this change of opinion is the introduction of probability in the description. We now see the microscopic state of a system as a probability measure on phase space so that evolution becomes a stochastic process. Macroscopic properties are then obtained through averages. I will introduce some of the models used in this research and discuss their relevance for the physical problem and the mathematical results one can obtain.

Series: CDSNS Colloquium

Series: CDSNS Colloquium

We study the behavior of the asymptotic dynamics of a dissipative reaction-diffusion equation in a dumbbell domain, which, roughly speaking, consists of two fixed domains joined by a thin channel. We analyze the behavior of the stationary solutions (solutions of the elliptic problem), their local unstable manifold and the attractor of the equation as the width of the connecting channel goes to zero.

Series: CDSNS Colloquium

The Bendixson conditions for general nonlinear differential equations in Banach spaces are developed in terms of stability of associated compound differential equations. The generalized Bendixson criterion states that, if some measure of 2-dimensional surface area tends to zero with time, then there are no closed curves that are left invariant by the dynamics. In particular, there are no nontrivial periodic orbits, homoclinic loops or heteroclinic loops. Concrete conditions that preclude the existence of periodic solutions for a parabolic PDE will be given. This is joint work with Professor James S. Muldowney at University of Alberta.

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.