On the persistence of elliptic lower-dimensional tori in Hamiltonian systems under the first Melnikov condition and Rüssmann’s non-degeneracy condition

In this paper we formulate a theorem on the persistence of elliptic lower-dimensional invariant tori for nearly integrable analytic Hamiltonian systems under the first Melnikov condition and Rüssmann’s non-degeneracy condition, and give the measure estimates of parameters for the non-resonance conditions under Rüssmann’s non-degeneracy condition, which is essential for the proof of our result.     

混合溶液中CARS过程的非共振信号实验研究

相干反斯托克斯拉曼散射是一种非线性四波混频效应,但是通过探测共振信号不易进行物质成分的定量光谱分析。本文利用单频相干反斯托克斯拉曼散射光谱分析法,对不同体积比混合的乙醇溶液在远离双光子共振跃迁及其远离溶质与溶剂的特征拉曼共振态位置进行了光谱探测。通过对实验结果进行分析发现,在共振位置2 876 cm-1的信号强度随混合溶液中的乙醇的体积比增加而增加,呈二次方关系。而在远离共振态位置的非共振信号强度随混合溶液中乙醇的体积比增加而增加,且呈线性关系,进而说明了非共振信号强度随着分子数浓度N呈线性变化关系。因此,通过探测非共振信号强度与分子数浓度关系可以为混合物中特定成分的定量光谱分析提供一种研究途径。     

粗糙银电极上高铁血红素的非共振表面增强拉曼光谱研究

以632.8nm为激发线,研究了高铁血红素的非共振表面增强拉曼光谱。结果表明,高铁血红素的氧化和还原状态的拉曼谱图以B_(1g)振动模式为主;血红素以丙酸侧基直接吸附于电极表面,叶啉环则以垂直于电极表面的形式存在。B_(1g)振动模式v_(15)对血红素中心铁的氧化还原状态十分敏感,被指认为非共振条件下血红素氧化还原状态的特征谱峰。     

Computation of the normal forms for general M-DOF systems using multiple time scales. Part I: autonomous systems

This paper is concerned with the symbolic computation of the normal forms of general multiple-degree-of-freedom oscillating systems. A perturbation technique based on the method of multiple time scales, without the application of center manifold theory, is generalized to develop efficient algorithms for systematically computing normal forms up to any high order. The equivalence between the perturbation technique and Poincaré normal form theory is proved, and general solution forms are established for solving ordered perturbation equations. A number of cases are considered, including the non-resonance, general resonance, resonant case containing 1:1 primary resonance, and combination of resonance with non-resonance. “Automatic” Maple programs have been developed which can be executed by a user without knowing computer algebra and Maple. Examples are presented to show the efficiency of the perturbation technique and the convenience of symbolic computation. This paper is focused on autonomous systems, and non-autonomous systems are considered in a companion paper.     

Codimension 3 Non-resonant Bifurcations of Rough Heteroclinic Loops with One Orbit Flip

Heteroclinic bifurcations in four dimensional vector fields are investigated by setting up a local coordinates near a rough heteroclinic loop. This heteroclinic loop has a principal heteroclinic orbit and a non-principal heteroclinic orbit that takes orbit flip. The existence, nonexistence, coexistence and uniqueness of the 1-heteroclinic loop, 1-homoclinic orbit and 1-periodic orbit are studied. The existence of the two-fold or three-fold 1-periodic orbit is also obtained.