The diffusion-driven instability and complexity for a single-handed discrete Fisher equation

For a reaction diffusion system, it is well known that the diffusion coefficient of the inhibitor must be bigger than that of the activator when the Turing instability is considered. However, the diffusion-driven instability/Turing instability for a single-handed discrete Fisher equation with the Neumann boundary conditions may occur and a series of 2-periodic patterns have been observed. Motivated by these pattern formations, the existence of 2-periodic solutions is established. Naturally, the periodic double and the chaos phenomenon should be considered. To this end, a simplest two elements system will be further discussed, the flip bifurcation theorem will be obtained by computing the center manifold, and the bifurcation diagrams will be simulated by using the shooting method. It proves that the Turing instability and the complexity of dynamical behaviors can be completely driven by the diffusion term. Additionally, those effective methods of numerical simulations are valid for experiments of other patterns, thus, are also beneficial for some application scientists

Existence of time periodic solutions for a class of non-resonant discrete wave equations

The final publication is available at Springer via http://dx.doi.org/10.1186/s13662-015-0457-zIn this paper, a class of discrete wave equations with Dirichlet boundary conditions are obtained by using the center-difference method. For any positive integers m and T, when the existence of time mT-periodic solutions is considered, a strongly indefinite discrete system needs to be established. By using a variant generalized weak linking theorem, a non-resonant superlinear (or superquadratic) result is obtained and the Ambrosetti-Rabinowitz condition is improved. Such a method cannot be used for the corresponding continuous wave equations or the continuous Hamiltonian systems; however, it is valid for some general discrete Hamiltonian systems

Research of Proxy Cache Algorithm in Multi-media Education System

Multi-media education system is more and more widely used in all levels of education. In order to decrease cost of multi-media system and keep efficiency with increasing multi-media materials, proxy cache algorithm has been widely studied. Based on analysis of existing research of proxy cache results, an improved proxy coaching strategy of prefix cache and postfix merging is proposed. The strategy can dynamically adjust prefix cache size with the object access change. A more effective method of steaming merging has been proposed with multicast used in postfix portion. The results show that the improved strategy can effectively utilize proxy cache resource, shorten time delay and save band width