In this talk we will first describe how the clocks synchronization problem can be modeled as a consensus algorithm for heterogeneous double integrators. We show two possible implementations of this consensus based clocks synchronization. The dynamics of one of these can be described as a discrete time nonlinear system, which is a perturbation of the linear system. Hence, using the center manifo...

A class of discrete-time Cohen-Grossberg neural networkswith delays is investigated in this paper. By analyzing the corresponding characteristic equations, the asymptotical stability of the null solution and the existence of Neimark-Sacker bifurcations are discussed. By applying the normal form theory and the center manifold theorem, the direction of the Neimark-Sacker bifurcation and the stabi...

Let M be a closed oriented 3-manifold which can be triangulated with N simplices. We prove that any map from M to a genus 2 surface has Hopf invariant at most C . Let X be a closed oriented hyperbolic 3-manifold with injectivity radius less than ǫ at one point. If there is a degree non-zero map from M to X, then we prove that ǫ is at least C . In the 1970’s, Thurston invented simplex straighten...

We give a brief introduction to re-publication of Chapter 2 of the author’s 1964 Dissertation. The result has come to be known as the Neimark-Sacker bifurcation theorem. Some of the applications of the theorem are cited and a brief overview of the reduction of the mapping to normal form is given in 3-dimensions. By introducing weighted monomials the reduction method is carried out without the u...

Following Almgren’s construction of the center manifold in his Big regularity paper, we show the C regularity of area-minimizing currents in the neighborhood of points of density one, without using the nonparametric theory. This study is intended as a first step towards the understanding of Almgren’s construction in its full generality. 0. Introduction In this note we consider area-minimizing i...

The aim of this chapter is to introduce tools from bifurcation theory which will be necessary in the following sections for the study of neural field equations (NFE) set in the primary visual cortex. In a first step, we deal with elementary bifurcations in low dimensions such as saddle-node, transcritical, pitchfork and Hopf bifurcations. NFEs are dynamical systems defined on Banach spaces and ...

In this study, a discrete-time prey-predator model based on the Allee effect is presented. We examine parametric conditions for local asymptotic stability of fixed points model. Furthermore, with use center manifold theorem and bifurcation theory, we analyze existence directions period-doubling Neimark-Sacker bifurcations. The plots maximum Lyapunov exponents provide indications complexity chao...

An analytic pair of dimension n and center V is a pair (V,M) where M is a complex manifold of (complex) dimension n and V ⊂ M is a closed totally real analytic submanifold of dimension n. To an analytic pair (V,M) we associate the class U (V,M) of the functions u : M → [0,π/4[ which are plurisubharmonic in M and such that u(p)= 0 for each p ∈ V . If U (V,M) admits a maximal function u, the trip...