Slow divergence integral and its application to classical Liénard equations of degree 5 Chengzhi Li1 and Kening Lu2 1School of Mathematical Sciences, Peking University, Beijing 100871, China 2Department of Mathematics, Brigham Young University, Provo, UT 84602, USA
The slow divergence integral is a crucial tool to study the cyclicity of a slow-fast cycle for singularly perturbed planar vector fields. In this talk, we first introduce a useful form of this integral, then use it to prove that the slow divergence integral along any non-degenerate slow-fast cycle for singular perturbations of classical Li?enard equations of degree 5 has at most one zero, and the zero is simple if it exists; hence its cyclicity in this class of equations is at most two. Up to now there are many interesting results about Li?enard equations of degree 3, 4 and  6, but almost nothing is known about degree 5. This result can be seen as a first step to study the uniform property for classical Li?enard equations of degree 5.
Singular perturbations and ionchannel problem Weishi Liu University of Kansas, USA I channel problems concern macroscopic properties of ionic flow through nano-scale ion channels .It is no coincidence that singularly perturbedsystemsserve as suitablemodels for ananlyzing these multi-reveals special structures (idealized physcial situations) of multi-scale phenomena and allows one to extract concrete informations for specific problems .this is the case for the poisson-Nernst-Plank(PNP)systems as primitive models for ionic flows. In this talk ,wewill describethe geometric singular perturbation framework for ananalysis of PNP systems and report a number of concrete results that are directly relevant to central topics of ion channel problems. the talk is based on works with several collaborations.
A Step-type Solution for the Affine Singularly Perturbed Optimal Control Problem Mingkang Ni Department of Mathematics, East China Normal University, Shanghai 200241 Limeng Wu School of Mathematics and Information Technology, Hebei Normal University of Science and Technology, Qinhuangdao Hebei 066004 In this paper, we consider the boundary layer and internal layer for a class of affine singularly perturbed optimal control problem. Based on the geometrical theory, we study the dynamical behavior of the solution for the optimal control system. Through the geometrical analysis, we obtain that the internal layer solution for the singularly pertu |