【作者】 陈益辉；

【导师】 秦文新；

【作者基本信息】 苏州大学 ， 应用数学， 2003， 硕士

【摘要】 在许多科学模型中，耦合格点系统扮演着非常重要的角色。例如：某些化学反应[1-2]；影像处理和花纹的确认[3-7]；分子科学[8]；以及生物科学[9-17]都有类似的问题出现。由于生物和电学（Josephson Junctions）的需要，人们对一维耦合振子链做了许多研究，其中包括许多理论的结果和数值上的分析。而在花纹的确认以及分子科学的许多模型中，很多是高维耦合格点系统，就比如[17]中的cardiac模型。 我们在这篇文章里主要比较两类性质不同的局部函数，从反可积的极限出发，随着耦合系数的增大，讨论平衡解的延拓和分支。 第一种情况：局部函数为周期的情况。对于反可积极限下的平衡解σ，我们定义了b（σ）。当b（σ）＜∞。时，∈（σ）＞0，σ可以延拓至|∈|＜∈（σ）。当|∈|大到一定程度，σ必然发生分支。 第二种情况：局部函数仅有有限个零点。此时存在一致的∈₀＞0，当耦合系数|∈|＜∈₀时，反可积极限下的平衡解都可以延拓；若将局部函数的条件再加强一点，我们可以得到∈₁，满足0＜∈₁＜∈₀，当|∈|＜∈₁时，不会有新的平衡解产生。 我们还就一些特殊的局部函数讨论平衡解的分支。更多还原

【Abstract】 In many scientific models, lattices play an important and in some cases essential role, typically modelling an underlying spatial structure in the problem. We mention in particular models arising in chemical reactions [1-2], image processing and pattern recognition [3-7], material science [8], and biology[9-17]. Much theoretical work in lattice differential equations concerns one-dimensional lattices, often with weak coupling between lattice sites. By contrast, we are more concerned with lattice systems in two-dimensions. This is certainly the case in the pattern recognition and material science models above, and as well in the cardiac model[17].In this paper, we compare two kinds of local maps. From the anti-integrability limit, we study the question whether the equilibria at the anti-integrability limit can persist or not with growing coupling coefficient e.Case one: the local map is periodic and has infinitely many zeros. For each equilibrium a at the anti-integrability limit, we define a number b( }. When b( ) < , we find ( ) > 0 and a persists for . But if e is big enough, a undergoes bifurcation.Case two: the local map has finite zeros. In this case, we find the uniform critical value 0 > 0. All the equilibria without coupling persist for . If we impose an additional condition on the local map, we can find < , such there are no new-born equilibria for < 1.We also study in detail the bifurcations of the equilibria for some special local maps.更多还原