A spectral element approach for the stability analysis of time-periodic delay equations with multiple delays

This paper describes a general spectral element approach to study the stability of multiple time delay systems (MTDS). We show, for the first time, how this approach can be applied to periodic MTDS where the delays and the period are incommensurate. In contrast to prior works on MTDS, the spectral element approach is applicable to both autonomous as well as non-autonomous MTDS. Both MTDS of first order or higher can be obtained and systems with or without damping can be investigated. Since the spectral element approach uses efficient interpolation and a set of well-distributed interpolation points, the size of the matrices necessary for convergence is kept small. Further, since the spectral element approach is a semi-analytical procedure, it avoids the need to use tedious time marching algorithms to explore the stability behavior of the system.     

Response behavior of bi-stable point wave energy absorbers under harmonic wave excitations

Nonlinear Dynamics - To expand the narrow response bandwidth of linear point wave energy absorbers (PWAs), a few research studies have recently proposed incorporating a bi-stable restoring force in...     

Assessing Logo programming among Jordanian seventh grade students through turtle geometry

The present study is concerned with assessing Logo programming experiences among seventh grade students. A formal multiple-choice test and five performance tasks were used to collect data. The results provided that students’ performance was better than the expected score by the probabilistic laws, and a very low correlation between their Logo programming performance and school mathematics achievement was revealed. Most of the made misconceptions were due to geometrical aspects rather than Logo primitives, and were concentrated on the angle of rotation, the angle of complete rotation and the angle of regular polygon. In addition, students’ problem-solving ability was limited while conducting some Logo programming tasks, and acceptable in others. In regard to the results, it is recommended that teaching Logo programming should be used in different contexts that enhance students’ learning, and develop problem-solving processes.     

Affinity screening by packed capillary high performance liquid chromatography using molecular imprinted sorbents. II. Covalent imprinted polymers

This study concentrates on the production of covalent molecular imprint polymers (MIPs) as highly selective sorbents for nortriptyline (NOR), a representative tricyclic antidepressant (TCA). The functionalized template contains a polymerizable 4-vinylphenyl carbamate moiety used to bind the template molecule to the polymer matrix. Polymerization with a cross-linker followed by hydrolytic cleavage of the labile carbamate functionality leaves an MIP with selective binding sites capable of binding template through hydrogen bonding interactions. Demonstrated chromatographically through a "selection index", these MIPs showed high selectivity for the template molecule (NOR) among a library of structurally similar compounds. The recognition was found to correlate with structural similarity to the template compound. A direct comparison between covalent and non-covalent molecular imprinting strategies reveals a great deal of improvement in the peak shape of the retained compound resulting from covalent imprinting (evidenced by peak asymmetry factors A.).     

Room-temperature fluorescence spectrometry of some pharmaceutical compounds

Room-temperature fluorescence of 66 pharmaceutical drugs are presented using a mixture of ethanol/water as solvent. The analytical figures of merit for the fluorescent species are reported with limits of detection ranging between 1 and 30 ng/ml for most of them and with average blank standard deviation of 7.6%. The spectral band half-widths are also reported.     

Room-temperature phosphorimetry of polyaromatic hydrocarbons with organized media and paper substrate: A comparative study

Room-temperature phosphorescence of some polyaromatic hydrocarbons as model compounds with stabilization by use of a microemulsion of a non-polar solvent, has been examined. Some advantage in the preparation of the solutions is achieved. A comparative study of room-temperature phosphorimetry in microemulsion and micellar media, on paper substrates and at low temperature has also been made. The spectral characteristics and analytical figures of merit are given.     

Parameter identification in periodic delay differential equations with distributed delay

In this study, a parameter identification approach for identifying the parameters of a periodic delayed system with distributed delay is introduced based on time series analysis and spectral element analysis. Using this approach the parameters of the distributed delayed system can be identified from the time series of the response of the system. The experimental or numerical data of the response is examined with Floquet theory and time series analysis techniques to estimate a reduced order dynamics, or truncated state space to identify the Floquet multipliers. Parameter identification is then completed using a dynamic map developed for the assumed model of the system which can relate the Floquet multipliers to the unknown parameters in the model. The parameter identification technique is validated numerically for first and second order delay differential equations with distributed delay.     

Periodic solutions of nonlinear delay differential equations using spectral element method

We extend the temporal spectral element method further to study the periodic orbits of general autonomous nonlinear delay differential equations (DDEs) with one constant delay. Although we describe the approach for one delay to keep the presentation clear, the extension to multiple delays is straightforward. We also show the underlying similarities between this method and the method of collocation. The spectral element method that we present here can be used to find both the periodic orbit and its stability. This is demonstrated with a variety of different examples, namely, the delayed versions of Mackey–Glass equation, Van der Pol equation, and Duffing equation. For each example, we show the method’s convergence behavior using both p and h refinement and we provide comparisons between equal size meshes that have different distributions.     

On the comparison of semi-analytical methods for the stability analysis of delay differential equations

This paper provides the first comparison of the semi-discretization, spectral element, and Legendre collocation methods. Each method is a technique for solving delay differential equations (DDEs) as well as determining regions of stability in the DDE parameter space. We present the necessary concepts, assumptions, and equations required to implement each method. To compare the relative performance between the methods, the convergence rate and computational time for each method is compared in three numerical studies consisting of a ship stability example, the delayed damped Mathieu equation, and a helicopter rotor control problem. For each study, we present one or more stability diagrams in the parameter space and one or more convergence plots. The spectral element method is demonstrated to have the quickest convergence rate while the Legendre collocation method requires the least computational time. The semi-discretization method on the other hand has both the slowest convergence rate and requires the most computational time.