Cheng-Hsiung Hsu

Traveling waves in cellular neural networks

In this paper, we study the structure of traveling wave solutions of Cellular Neural Networks of the advanced type. We show the existence of monotone traveling wave, oscillating wave and eventually periodic wave solutions by using shooting method and comparison principle. In addition, we obtain the existence of periodic wave train solutions.

A least-squares finite element method for incompressible flow in stress-velocity-pressure version☆

In this paper we are concerned with the incompressible flow in 2-D. Introducing additional variables of derivatives of velocity, which are called stresses here, the second-order dynamic equations are reduced into a first-order system with variables of stress, velocity and pressure. Combining the compatability conditions: and the divergence ice condition, we have a system with six first-order equations and six unknowns. Least-squares method performed over this extended system. The analysis shows that this method achieves optimal rates of convergence in the H1-norm as the h approaches to zero. Numerical experiences are also available.

CELLULAR NEURAL NETWORKS: LOCAL PATTERNS FOR GENERAL TEMPLATES

This work investigates the mosaic local patterns for cellular neural networks with general templates. Our results demonstrate that the set of templates can be divided into many nite regions. In each region, the same family of local patterns can be generated. Conversely, our results further demonstrate that some templates can realize a family of local patterns which can be linearly separated by a hyperplane in the conguration space. This study also proposes algorithms for verifying the linear separability for a given family of local patterns and, when it is separable, for obtaining the associated template.

Existence, uniqueness, monotonicity and asymptotic behaviour of travelling waves for epidemic models

The purpose of this work is to investigate the existence, uniqueness, monotonicity and asymptotic behaviour of travelling wave solutions for a general epidemic model arising from the spread of an epidemic by oral?faecal transmission. First, we apply Schauders fixed point theorem combining with a supersolution and subsolution pair to derive the existence of positive monotone monostable travelling wave solutions. Then, applying the Ikeharas theorem, we determine the exponential rates of travelling wave solutions which converge to two different equilibria as the moving coordinate tends to positive infinity and negative infinity, respectively. Finally, using the sliding method, we prove the uniqueness result provided the travelling wave solutions satisfy some boundedness conditions.

PERIODIC OSCILLATIONS ARISING AND DEATH IN DELAY-COUPLED NEURAL LOOPS

We consider the system consisting of two identical n-neural loops which are coupled via a single neuron of each loop with discrete time delay. A framework describing the linearized stability region in the parameter space for the trivial steady state solution is constructed. In the boundary of the stability region, a periodic oscillation may arise from the Hopf bifurcation while these periodic oscillations could be dead owing to the change of coupling strength, delay or meeting a pitchfork bifurcation. The examples of n = 4 and n = 5 show the phenomena of oscillation arising or death, and the effects of the coupling strengths and delay on the stability and oscillation. The criticalities of pitchfork and Hopf bifurcations are investigated by using the normal form method for retarded functional differential equations.

On camel-like traveling wave solutions in cellular neural networks

Abstract This paper is concerned with the existence of camel-like traveling wave solutions of cellular neural networks distributed in the one-dimensional integer lattice Z 1 . The dynamics of each given cell depends on itself and its nearest m left neighbor cells with instantaneous feedback. The profile equation of the infinite system of ordinary differential equations can be written as a functional differential equation in delayed type. Under appropriate assumptions, we can directly figure out the solution formula with many parameters. When the wave speed is negative and close to zero, we prove the existence of camel-like traveling waves for certain parameters. In addition, we also provide some numerical results for more general output functions and find out oscillating traveling waves numerically.

SMALE HORSESHOE OF CELLULAR NEURAL NETWORKS

The paper shows the spatial disorder of one-dimensional Cellular Neural Networks (CNN) using the iteration map method. Under certain parameters, the map is two-dimensional and the Smale horseshoe is constructed. Moreover, we also illustrate the variant of CNN, closely related to Henon-type and Belykh maps, and discrete Allen–Cahn equations.

SPATIAL DISORDER OF CNN — WITH ASYMMETRIC OUTPUT FUNCTION

This investigation will describe the spatial disorder of one-dimensional Cellular Neural Networks (CNN). The steady state solutions of the one-dimensional CNN can be replaced as an iteration map which is one dimensional under certain parameters. Then, the maps are chaotic and the spatial entropy of the steady state solutions is a three-dimensional devil-staircase like function.

Spatial disorder of cellular neural networks with biased term

This study describes the spatial disorder of one-dimensional Cellular Neural Networks (CNN) with a biased term by applying the iteration map method. Under certain parameters, the map is one-dimensional and the spatial entropy of stable stationary solutions can be obtained explicitly as a staircase function.

DIVERSITY OF TRAVELING WAVE SOLUTIONS IN DELAYED CELLULAR NEURAL NETWORKS

This work investigates the diversity of traveling wave solutions for a class of delayed cellular neural networks on the one-dimensional integer lattice ℤ1. The dynamics of a given cell is characterized by instantaneous self-feedback and neighborhood interaction with distributed delay due to, for example, finite switching speed and finite velocity of signal transmission. Applying the monotone iteration scheme, we can deduce the existence of monotonic traveling wave solutions provided the templates satisfy the so-called quasi-monotonicity condition. We then consider two special cases of the delayed cellular neural network in which each cell interacts only with either the nearest m left neighbors or the nearest m right neighbors. For the former case, we can directly figure out the analytic solution in an explicit form by the method of step with the help of the characteristic function and then prove that, in addition to the existence of monotonic traveling wave solutions, for certain templates there exist nonmonotonic traveling wave solutions such as camel-like waves with many critical points. For the latter case, employing the comparison arguments repeatedly, we can clarify the deformation of traveling wave solutions with respect to the wave speed. More specifically, we can describe the transition of profiles from monotonicity, damped oscillation, periodicity, unboundedness and back to monotonicity as the wave speed is varied. Some numerical results are also given to demonstrate the theoretical analysis.