|
Dr. Shangjiang Guo's Homepage
|
|
กก |
Welcome to my Web site! I completed my B.Sc. in Mathematics at National University of Defense Technology in 1996, and M.Sc. in Applied Mathematics at Hunan University (China) in 2001, respectively. I got my Ph.D. in Applied Mathematics in 2004 under the supervision of Dr. Lihong Huang at the College of Mathematics and Econometrics,Hunan University.My brief CV is now availabel in PDF file. My Master's Report partly focuses on a delayed network of two neurons with both self-feedback and interaction described by an all-or-none threshold function. The model describes a combination of analog and digital signal processing in the network and takes the form of a system of delay differential equations with discontinuous nonlinearity. We show that the dynamics of the network can be understood in terms of the iteration of a one-dimensional map, and we obtain simple criteria for the convergence of solutions, the existence, multiplicity and attractivity of periodic solutions. In addition, I derived some sufficient conditions for the local and global exponential stability of the discrete-time Hopfield neural etworks with general activation functions, which generalize those existing results. By means of M-matrix theory and some inequality analysis techniques, the exponential convergence rate of the neural networks to the equilibrium is estimated, and,for the local exponential stability, the basin of attraction of the stable equilibrium is also characterized.I also studied a class of neural networks which includes BAM (bidirectional associative memory) networks and CNNs (cellular neural networks) as its special cases. By Brouwer's fixed point theorem, matrix theory and inequality analysis, we not only obtain some new sufficient conditions ensuring the existence, uniqueness, global attractivity and global exponential stability of the equilibrium but also estimate the exponentially convergent rate. Our results are less restrictive than previously known criteria and can be applied to neural networks with a broad range of activation functions assuming neither differentiability nor strict monotonicity. (See the abstact-master.tex for more details) The focus of my Ph.D. thesis work is to study issues related to stability and bifurcation of a ring of neurons with self-feedback and delays, which has an on-center off-surround characteristic and can be identified with a Lie group. Such a network has been found in a variety of neural structures, such as neocortex, cerebellum, hippocampus, and even in chemistry and electrical engineering, and can be studied to gain insight into the mechanisms underlying the behavior of recurrent network.Needless to say, this is a very difficult task due to the infinite-dimensional nature of the problem caused by the synaptic delay and the possible spatial structure of the system (equivariant with respect to a $\Bbb{D}_n$-action). Some general theorems are available about the existence and global continuation of periodic solutions in symmetric delay differential equations.However, applications of these general results to concrete systems such as on-center off-surround networks involve several highly nontrivial tasks: (i) distribution of zeros in characteristic equations which are usually transcendental and depend on parameters; (ii) symmetry analysis on certain generalized eigenspaces of the generator of a linearized system and identifcation of these spaces with a direct sum of two identical absolute irreducible representations of $\Bbb{D}_n$; (iii) calculation of the so-called crossing numbers which are related to the usual transversality condition in a standard Hopf bifurcation theory; (iv) a priori estimation of the period and of the norm of a periodic solution. This dissertation is organized as follows: Firstly, linear stability of the model is investigated by analyzing the associated characteristic transcendental equation. By means of space decomposition, we subtly discuss the distribution of zeros of the characteristic equation, and then we have derived some sufficient conditions to ensure that all the characteristic roots have negative real parts. Hence, the zero solution of the model is asymptotically stable. Secondly, by means of the standard Hopf bifurcation theory, we obtain a branch of periodic solutions and its continuation. Based on the normal form approach and the center manifold theory,we derive the formula for determining the properties of Hopf bifurcating periodic orbit for a ring of neurons with delays, such as the direction of Hopf bifurcation,stability of the Hopf bifurcating periodic orbits and so on. Thirdly, under some suitable conditions, such a network has a slowly oscilatory synchronous solution which is completely characterized by a scalar delay differential equation. Despite the fact that the slowly oscillatory periodic solution of the scalar equation is stable, by making use of Floquet theory and Krein-Rutman theorem, we show that the associated synchronous periodic solution is unstable if the size of the network is large.Forthly, by using of the symmetric bifurcation theory of delay differential equations coupled with representation theory of standard dihedral groups, we not only investigate the effect of synaptic delay of signal transmission on the pattern formation,but also obtain some important results about the spontaneous bifurcation of multiple branches of periodic solutions and their spatio-temporal patterns (i.e., mirror-reflecting waves, standing waves and it discrete waves). Moreover, we analyze the stability of the bifurcated periodic solutions. In addition, we show that spontaneous bifurcations of multiple branches of periodic solutions exist for all large delay (global continuation),and consider the coincidence of these periodic solutions. (See the abstact-phd.tex for more details) |