Upcoming Events

Monday, February 15, 2010 at 3:00 PM in SAS 4201
Boris Gershgorin, Courant Institute, NYU
Climate response and fluctuation-dissipation theorem
The fluctuation-dissipation theorem (FDT) provides an attractive
perspective to address climate change in an atmosphere-ocean system
(AOS). The theorem states that in order to predict the linear response of a
dynamical system in equilibrium to a small perturbation in external
forcing, it is sufficient to find the appropriate correlation function
in equilibrium without the need for perturbing the system. An attractive
feature of applying the FDT to low-frequency climate variables of
an AOS is that the linear response operator computed by the
theorem can then be utilized for multiple climate change scenarios
without the need of running the complex climate model in each
individual case. In this talk, we focus on the performance of FDT
and its approximations in predicting climate change in idealized
models. On the one hand these models are simple enough to be
solved either analytically or numerically and on the other hand they
mimic some of the key properties of a complex AOS. In particular,
we consider three different nonlinear test models that allow us to
investigate such important practical issues of assessing climate
change via FDT as the performance of linear regressions, the
subtle departures from Gaussianity and the time-dependent
statistics in the model. The results of our study should provide
useful guidelines for applying the FDT to more complex realistic
systems.

(Joint with Probability Seminar)

Thursday, February 18, 2010 at 3:00 PM in SAS 4201
Alexandros Sopasakis, UNC Charlotte
Capturing stochastic contributions in noise systems
We introduce and study a class of model prototype hybrid systems, with
multiscale coupling which essentially describe a wide range of physical
applications. The challenge in these hybrid problems is two fold:
a) The direct numerical simulation of realistic size systems due to
scale disparities between the discrete stochastic microscopic model and
the continuum macroscopic equations.
b) The stochasticity inherited from the microscopic model can play a
subtle but important role in the dynamic behavior of the overall system
due to the nonlinear interaction of the coupling.
In this work we address directly or indirectly both issues in the context
of proposed prototype mathematical models of a deterministic ODE coupled
with a stochastic spin flip / spin exchange Ising model that capture
essential features of complex hybrid systems.

Wednesday, February 24, 2010 at 3:00 PM in SAS 4201
Thomas Ward, NC State Mechanical and Aerospace Engineering
Mixing by Periodic Shear
TBA

Tuesday, March 2, 2010 at 3:00 PM in SAS 4201
Yi Sun, NC State Mathematics
Network dynamics of Hodgkin-Huxley neurons
The reliability and predictability of neuronal network dynamics is a central question in neuroscience. We present a numerical analysis of the dynamics of all-to-all pulsed-coupled Hodgkin-Huxley (HH) neuronal networks. Since this is a non-smooth dynamical system, we propose a pseudo-Lyapunov exponent (PLE) that captures the long-time predictability of HH neuronal networks. The PLE can capture very well the dynamical regimes of the network. Furthermore, we present an efficient library-based numerical method for simulating HH neuronal networks. Our pre-computed high resolution data library can allow us to avoid resolving the spikes in detail and to use large numerical time steps for evolving the HH neuron equations. By using the library-based method, we can evolve the HH networks using time steps one order of magnitude larger than the typical time steps used for resolving the trajectories without the library, while achieving comparable resolution in statistical quantifications of the network activity. Moreover, our large time steps using the library method can overcome the stability requirement of standard ODE methods for the original dynamics.

Wednesday, March 10, 2010 at 3:00 PM in SAS 4201
Haitao Fan, Georgetown University
TBA
TBA

Wednesday, March 24, 2010 at 3:00 PM in SAS 4201
Boaz Ilan, University of California, Merced
Dynamics of Nonlinear Bound States in Inhomogeneous Media
Nonlinear Schrodinger (NLS) equations are used to model physical phenomena such as intense laser beams and ultra-cold matter waves. These waves can be better confined and controlled when using spatially inhomogeneous media. Examples include light propagation in Photonic Crystal Fibers and Bose-Einstein condensate waves inside an external potential. Nonlinear bound state solutions (often called solitary waves or solitons) offer new insight into the salient features of these complex physical systems. This talk will present recent rigorous, asymptotic and computational results on existence and dynamics of such solutions. The bifurcation of bound states from the edge of spectral bands is analyzed in detail. We prove that in the L2-critical case, perturbed bound states with frequency near the band edge do not undergo wave collapse, yet they are nonlinearly unstable. The ensuing dynamics is elucidated using computations of NLS equations.

Wednesday, March 31, 2010 at 3:00 PM in SAS 4201
Stéphane Lafortune, College of Charleston
TBA
TBA

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Seminar Organizer: Mark Hoefer