The Homoclinic Tangle of A 1:2 Resonance in a 2-D Hamiltonian System

We study in great detail the geometry of the homoclinic tangle, with respect to the energy, corresponding to an unstable periodic orbit of type 1:2, on a surface of section representing a 2-D Hamiltonian system. The tangle consists of two resonance areas, in contrast with the tangles of type-l or -{l, m, k, x = 0} considered in previous studies, that consist of only one resonance area. We study the intersections of the inner and outer lobes of the same resonance area and of the two resonance areas. The intersections of the lobes follow certain rules. The detailed study of these rules allows us to derive quantitative relations about the number of intersections and to understand the complex behavior of the higher order lobes by studying the lower order lobes. We find 1st, 2nd, 3rd, etc. order intersections formed by lobes making 1, 2, 3, etc. turns around an island. After a sufficiently high order of iterations a lobe may intersect its image and thus produce a Poincaré recurrence. Numerical results for a wide interval of energies are presented. The number of intersections changes through tangencies. In any finite interval of the energy between two tangencies of 1st order, an infinite number of higher order tangencies occur and thus, according to the Newhouse theorem, there exist nearby islands of stability.     

THE EFFECTS OF EXTERNAL FORCING,DISSIPATION AND NONLINEARITY ON THE SOLUTIONS OF ATMOSPHERIC EQUATIONS*

Based on the primitive equations of the atmosphere,we study the effects of external forcing.dissipation and nonlinearity on the solutions of stationary motion and non-stationary motion.The results show that the asymptotic behavior of solutions of the forced dissipative nonlinear system is essentially different from that of the adiabatic non-dissipative system,the adiabatic dissipative system,the diabatic non-dissipative system and the diabatic dissipative linear system,and that the joint action of external forcing,dissipation and nonlinearity is the source of multiple equilibria.From this we can conclude that the important actions of diabatic heating and dissipation must be considered in the models of the long-term weather and the climate.     

急倾斜煤层开采地表沉陷的渐近灰色预测

煤层开采所引起的地表沉陷是一种严重的矿区地质灾害。煤层的倾角、厚度等物理条件是地表沉陷的主要影响因素。在水平煤层或缓倾斜煤的开采过程中，由于地层倾角小，地表沉陷具有较完整的规律性，其预测效果也比较理想。但是，在急倾斜煤层的开采中，由于地层倾角较大，赋存条件和地质体物理力学性质的差异，增强了地表下沉的非线性特征，使地表沉陷具有不确定的表现规律。文章对重庆市南桐矿区东林矿的地表沉陷非线性特征进行了探讨，得到了东林煤矿地表下沉曲线的分形维数是1．07。在岩层移动这个系统当中，既含有已知的又含有未知的或非确定的信息，可以作为一个灰色系统来研究。岩层控制系统的状态、结构和边界条件难以精确描述，属本征性灰色系统。文章针对东林煤矿地表下沉曲线非线性较弱的性质，提出用一种基于GM(1，1)的渐近预测模型对东林煤矿42个月的地表下沉量时间序列进行探讨。结果表明，这种模型对急倾斜层开采地表沉陷的预测是一种行之有效的方法。通过对其他工程实例的应用分析，进一步证明这种渐近的灰色预测方法具有相对较高的精度，是一种比较实用的地表沉陷预测方法，具有广泛的工程实用空间。     

Why asymptotic limit of the acid dissolution capacity can lead to a sharp dissolution front in chemical dissolution of porous rocks?

The use of the asymptotic limit can greatly simplify the theoretical analysis of chemical dissolution front instabilities in fluid‐saturated rocks and therefore make it possible to obtain mathematical solutions, which often play a crucial role in understanding the propagation behavior of chemical dissolution fronts in chemical dissolution systems. However, there has been a debate in recent years that the asymptotic limit of the acid dissolution capacity (i.e., the acid dissolution capacity number approaching zero) alone cannot lead to a sharp dissolution front of the Stefan type in the acidization dissolution system, in which the dissolvable minerals of carbonate rocks are chemically dissolved by the injected acid flow. The acid dissolution capacity number is commonly defined as the ratio of the volume of the carbonate rock dissolved by an acid to that of the acid. In this paper, we use four different proof methods, including (i) direct use of the fundamental concepts; (ii) use of the mathematical governing equations of an acidization dissolution system; (iii) use of the different time scaling approach; and (iv) use of a moving coordinate system approach, to demonstrate that the asymptotic limit of the acid dissolution capacity can indeed lead to sharp dissolution fronts of the Stefan type in acidization dissolution systems on a much larger time scale (than the dissolution time scale). Our new finding is that on the reaction time scale, the condition of the conventional time derivative of porosity approaching zero alone can ensure that the acidization dissolution front has a sharp shape of the Stefan type. Copyright © 2017 John Wiley & Sons, Ltd.     

Lienard方程零解渐近稳定性的一个弱条件

研究Liemrd方程零解的渐近稳定性.在Liapunov函数Ⅴ满足dV/dt≤0的情况下,给出了进一步判断其渐近稳定的一个条件.     

第一类算子方程的吉洪诺夫正则解的收敛性和渐近阶估计

本文对于第一类算子方程当右端近似给定和当算子与右端皆近似分别给定时,讨论如何确定正则参数。并给出吉洪诺夫正则解的收敛性和渐近阶估计。