Using the homotopy method to find periodic solutions of forced nonlinear differential equations

This paper discusses a result of Li and Shen which proves the existence of a unique periodic solution for the differential equation x + k [xdot] + g(x,t) = ε(t) where k is a constant; g is continuous, continuously differentiable with respect to x, and is periodic of period P in the variable t; ε(t) is continuous and periodic of period P, and when ?g/?x satisfies some additional boundedness conditions. This means that there exist initial values x(0) = α* and [xdot] (0) = β* so that the solution to the corresponding initial value problem is periodic of period P and is unique (up to a translation of the time variable) with this property. The proof of this result is constructive, so that starting with any initial conditions x(0) = α and [xdot](0) = β, a path in the phase plane can be produced, starting at (α, β) and terminating at (α*, β*). Both the theoretical proof and a constructive proof are discussed and a Mathematica implementation developed which yields an algorithm in the form of a Mathematica notebook (which is posted on the webpage http://pax.st.usm.edu/downloads). The algorithm is robust and can be used on differential equations whose terms do not satisfy Li and Shen's hypotheses. The ideas used reinforce concepts from beginning courses in ordinary differential equations, linear algebra, and numerical analysis.     

Coupled N-dimensional harmonic oscillator systems

A general decoupling procedure is described for systems of N interacting harmonic oscillators coupled by some bilinear perturbing potentials, and explicit conditions are derived which must be fulfilled in order to obtain physical bound states. Illustrative examples include a general treatment of the 2-oscillator system, some particular extensions to systems containing 3 oscillators, and a class of N-oscillator models.     

Trivalent Copper Complexes with 1,4,8,11-Tetramethyl-1,4,8,11-tetraazacyclotetradecane in Aqueous Solutions. A Pulse Radiolysis and Electrochemical Study

The mechanisms and kinetics of the formation and decomposition of the complexesCu^IIIL_aq and [LCu^IIIN₃]²⁺ are reportedand discussed.     

Synthesis of nitroxide-functionalized phthalocyanines

A nitroxide-functionalized phthalonitrile and a diimino-isoindoline were prepared via Pd(0)-catalyzed cyanation from the corresponding dibromide and subsequent addition of ammonia, without interference from the radical moieties. The structures of these radicals were confirmed by single-crystal X-ray structure determinations. Metal-templated macrocyclization of these species under standard Linstead conditions or in 2-(dimethylamino)ethanol at reflux gave the corresponding metallated and free-base phthalocyanines, which were characterized using UV-vis, FTIR and EPR spectroscopy as well as mass spectrometry.     

Synthesis of the dysiherbaine tetrahydropyran core employing a tethered aminohydroxylation reaction

A stereocontrolled and scalable synthesis of an advanced intermediate of the dysiherbaine tetrahydropyran core has been achieved in 11 steps and 27% overall yield. The key feature of this synthetic approach is the application of the Donohoe tethered aminohydroxylation reaction to install the amino diol and establish the four contiguous syn stereocenters on the tetrahydropyran ring.     

PbTiO3在压力下的异常相变和极强压电效应

运用第一性原理计算预言了在一纯化合物中可由压力诱导出顺序为叫方晶-单斜体-菱形体-立方体的新的相变，而且存在有变形相界面。在相变区有可与在复杂的单晶固溶体压电材料，如人们期待在机电应用方面引起革命性变化的Pb（Mg1/3Nb2/3）O3-PbTiO3可比的，极大的介电和压电耦合常数。我们的结果表明变形相界面和巨压电效应并不需要内禀的无序，并打开了在简单系统中研究这一效应的可能性，     

Linearization of a class of non-linear systems modelled by multibond graphs

Based on non-linear systems described by multibond graphs, a procedure designed to present symbolic linearization of these multibond graphs, is presented in this paper. Firstly, a junction structure of a multibond graph with multiport gyrators that represent Eulerian junction structures is proposed. In addition, non-linear multiport resistors are considered. By knowing the non-linear causal paths and loops of the non-linear multibond graph, the linearization is obtained by two steps: (1) The original multibond graph on the nominal operating point is evaluated; (2) New and additional paths based on the non-linear causal paths and loops are included. The state space representation of the linearized multibond graph using the corresponding junction structure is presented. An advantage of this methodology is its ability to allow the user to define a nominal operating point in which the linearization will be carried out.

In order to apply the proposed methodology, two physical systems are modelled and linearized by multibond graphs: a synchronous generator and a two degrees of freedom PUMA. Simulation results of these non-linear and linearized systems are shown.