A new method for homoclinic solutions of ordinary differential equations

Consideration is given to the homoclinic solutions of ordinary differential equations. We first review the Melnikov analysis to obtain Melnikov function, when the perturbation parameter is zero and when the differential equation has a hyperbolic equilibrium. Since Melnikov analysis fails, using Homotopy Analysis Method (HAM, see [Liao SJ. Beyond perturbation: introduction to the homotopy analysis method. Boca Raton: Chapman & Hall/CRC Press; 2003; Liao SJ. An explicit, totally analytic approximation of Blasius’ viscous flow problems. Int J Non-Linear Mech 1999;34(4):759–78; Liao SJ. On the homotopy analysis method for nonlinear problems. Appl Math Comput 2004;147(2):499–513] and others [Abbasbandy S. The application of the homotopy analysis method to nonlinear equations arising in heat transfer. Phys Lett A 2006;360:109–13; Hayat T, Sajid M. On analytic solution for thin film flow of a forth grade fluid down a vertical cylinder. Phys Lett A, in press; Sajid M, Hayat T, Asghar S. Comparison between the HAM and HPM solutions of thin film flows of non-Newtonian fluids on a moving belt. Nonlinear Dyn, in press]), we obtain homoclinic solution for a differential equation with zero perturbation parameter and with hyperbolic equilibrium. Then we show that the Melnikov type function can be obtained as a special case of this homotopy analysis method. Finally, homoclinic solutions are obtained (for nontrivial examples) analytically by HAM, and are presented through graphs.     

一类Lotka—Volterra型捕食与被捕食系统的Flip分岔

本文利用中心流形定理和映射的分岔理论,研究了一类离散的捕食与被捕食系统的Flip分岔的存在性与稳定性,并给出了数值模拟的结果。     

Corrected explicit-implicit domain decomposition algorithms for two-dimensional semilinear parabolic equations

Corrected explicit-implicit domain decomposition (CEIDD) algorithms are studied for parallel approximation of semilinear parabolic problems on distributed memory processors. It is natural to divide the spatial domain into some smaller parallel strips and cells using the simplest straight-line interface (SI). By using the Leray-Schauder fixed-point theorem and the discrete energy method, it is shown that the resulting CEIDD-SI algorithm is uniquely solvable, unconditionally stable and convergent. The CEIDD-SI method always suffers from the globalization of data communication when interior boundaries cross into each other inside the domain. To overcome this disadvantage, a composite interface (CI) that consists of straight segments and zigzag fractions is suggested. The corresponding CEIDD-CI algorithm is proven to be solvable, stable and convergent. Numerical experiments are presented to support the theoretical results.     

A constructive proof on the existence of globally exponentially attractive set and positive invariant set of general Lorenz family

In this paper, we give a constructive proof on the existence of globally exponentially attractive set and positive invariant set of general Lorenz family, which contains four independent parameters and is more general than any Lorenz systems studied so far in the literature. The system considered in this paper not only contains the classical Lorenz system and the generalized Lorenz family as special cases, but also provides three new Lorenz systems, which do not belong to the generalized Lorenz system, but the general Lorenz system. The results presented in this paper contain all the existing relative results as special cases.     

On impulsive control for synchronization and its application to the nuclear spin generator system

The issue of impulsive control theory for the synchronization of the nuclear spin generator (NSG) chaotic systems is developed. We propose an impulsive control scheme for the complete synchronization of the NSG system including chaotic systems. A sufficient condition for the impulsive control is derived, with an easily calculated maximum impulsive interval. The proposed impulsive control scheme is applied to the NSG system and the simulation result demonstrates the effectiveness of the method.     

Existence of positive solutions of nonlinear fractional delay differential equations

This paper investigates the existence and uniqueness of positive solutions for a class of nonlinear fractional delay differential equations. Using a nonlinear alternative of Leray-Schauder type, we show the existence of positive solutions for the equations in question.     

High performance PtRuIr catalysts supported on carbon nanotubes for the anodic oxidation of methanol

We report the formation of a new PtRuIr catalyst using an organic colloid synthesis method, involving acetone as the solvent, ethylene glycol as the reducing agent, citrate as a complexing agent and stabilizer, and multiwall carbon nanotubes (CNT, diameter 8-10 nm) as the support. This catalyst has a very high real surface area and is highly active toward the oxidation of methanol, relevant to fuel cell applications. The Ir component appears to act as a promoter, and the splitting of the Pt(111) XRD feature into four peaks and the shift to larger d spacing reflect the high dispersion of the metallic components.     

Ln2(C2O4)2(pyzc)2(H2O)2]n[Ln = Pr (1), Er (2)]: novel two-dimensional lanthanide coordination polymers with 2-pyrazinecarboxylate and oxalate

We report for the first time hydrothermal synthesis of lanthanide-pyzcH polymers, [Ln2(C2O4)2(pyzc)2 (H2O)2]n [Ln = Pr (1), Er (2)], in which pyzcH was decomposed into C2O42- and thus caused novel two-dimensional hexagonal lattice networks. The magnetic property of polymer 1 has been studied by an approximate treatment being enlightened by McPherson et al. (Inorg. Chim. Acta 1988, 148, 265), leading to Delta = -4.3 cm-1, zJ' = -11.73 cm-1, and g = 0.79. Complex 2 displays an intense room-temperature, liquid-state luminescent emission.     

Hybrids of colloidal silica and waterborne polyurethane

Waterborne polyurethane (WPU) was synthesized and followed by adding colloidal silica to prepare WPU-silica hybrids. The silica content in the hybrid thin films was varied from 0 to 50 wt%. The experimental results revealed that the viscosity of these hybrid solutions increased with increasing silica content and resulted in the aggregation of silica particle in the hybrid films. The latter result was evidenced by SEM examination. The effect of interaction between silica particle and urethane polymer chains is more significant with increasing silica content. The prepared hybrid films show much better thermal stability and mechanical properties than pure WPU. The optical transparence did not linearly decrease with increasing the silica fraction in the hybrid thin film. At below 20% silica content, the film transparence decreased with increasing silica content; the converse is true at above 20% silica content. These results showed that the prepared hybrid films demonstrated tunable transparence with the silica fraction in the films.     

CH...pi interaction for rhenium-based rectangles: an interaction that is rarely designed into a host-guest pair

Alkoxy- and thiolato-bridged Re(I) molecular rectangles [{(CO)3Re(mu-ER)2Re(CO)3}2(mu-bpy)2] (ER = SC4H9, 1a; SC8H17, 1b; OC4H9, 2a; OC12H25, 2b; bpy = 4,4'-bipyridine) exhibit strong interactions with several planar aromatic molecules. The nature of their binding was studied by spectral techniques and verified by X-ray diffraction analysis. Standard absorption and fluorescence titrations showed that a relatively strong 1:1 interaction occurs between aromatic guests such as pyrene and these rectangles. The results of a single-crystal X-ray diffraction analysis show that the recognition of 1 with a pyrene molecule is mainly due to CH...pi interactions and the face of the guest pyrene is located over the edges of the bpy linkers of 1. This is a fairly novel example of an interaction that is rarely designed into a host-guest pair. Furthermore, the interaction of 1 with Ag+ results in the self-organization of supramolecular arrays, as revealed by solid-state data.