Frequency formula for a class of fractal vibration system

Four fractal nonlinear oscillators (The fractal Duffing oscillator, fractal attachment oscillator, fractal Toda oscillator, and a fractal nonlinear oscillator) are successfully established by He’s fractal derivative in a fractal space, and their variational principles are obtained by semi-inverse transform method. The approximate frequency of the four fractal oscillators are found by a simple frequency formula. The results show the frequency formula is a powerful and simple tool to a class of fractal oscillators.

An Effective Solution of the Cube-Root Truly Nonlinear Oscillator: Extended Iteration Procedure

The cube-root truly nonlinear oscillator and the inverse cube-root truly nonlinear oscillator are the most meaningful and classical nonlinear ordinary differential equations on behalf of its various applications in science and engineering. Especially, the oscillators are used widely in the study of elastic force, structural dynamics, and elliptic curve cryptography. In this paper, we have applied modified Mickens extended iteration method to solve the cube-root truly nonlinear oscillator, the inverse cube-root truly nonlinear oscillator, and the equation of pendulum. Comparison is made among iteration method, harmonic balance method, He’s amplitude-frequency formulation, He’s homotopy perturbation method, improved harmonic balance method, and homotopy perturbation method. After comparison, we analyze that modified Mickens extended iteration method is more accurate, effective, easy, and straightforward. Also, the comparison of the obtained analytical solutions with the numerical results represented an extraordinary accuracy. The percentage error for the fourth approximate frequency of cube-root truly nonlinear oscillator is 0.006 and the percentage error for the fourth approximate frequency of inverse cube-root truly nonlinear oscillator is 0.12.

Cavitation of Multiscale Vortices in Circular Cylinder Wake at Re = 9500

Cavitation characteristics in the wake of a circular cylinder, which contains multiscale vortices, are numerically investigated via Large Eddy Simulation (LES) in this paper. The Reynolds number is 9500 based on the inlet velocity, the cylinder diameter and the kinematic viscosity of the noncavitation liquid. The Schneer–Sauer (SS) model is applied to cavitation simulation because it is more sensitive to vapor–liquid two-phase volume fraction than the Zwart–Gerber–Belamri (ZGB) model, according to theoretical analyses. The wake is quasiperiodic, with an approximate frequency of 0.2. It is found that the cavitation of vortices could inhibit the vortex shedding. Besides, the mutual aggregation of small-scale vortices in the vortex system or the continuous stripping of small-scale vortices at the edge of large-scale vortices could induce the merging or splitting of cavities in the wake.

He’s frequency formula to fractal undamped Duffing equation

Nonlinear oscillation is an increasingly important and extremely interesting topic in engineering. This article completely reviews a simple method proposed by Ji-Huan He and successfully establishes a fractal undamped Duffing equation through the two-scale fractal derivative in a fractal space. Its variational principle is established, and the two-scale transform method and the fractal frequency formula are adopted to find the approximate frequency of the fractal oscillator. The numerical result shows that He’s frequency formula is a unique tool for the fractal equations.

Archaisms in D. Lyndsay’s play of the Early Modern English period “A Satire of the Three Estates”: problem of identification and stylistic assessment

The subject of this research is the method of identification and problem of statistical interpretation of archaic lexis in literary texts of the Early Modern English period, namely in D. Lyndsay’s play “A Satire of the Three Estates”. The article discusses the capacity of attracting the data from diachronic corpus and corpus-based dictionaries for determination of archaic elements in literary works of the XVI century. Leaning on the commentaries of reputable philologists of the Early Modern English period and modern research, the article explores the trends of relevant perception pertaining to introduction of archaisms into the literary text by the authors of that time. An algorithm is proposed for identification of outdated units in literary material, created on Scottish (considering the obsolescent character of this dialect), based on juxtaposition of the approximate frequency indexes of reproduction of the element in the Early Modern English and Scottish sections of Helsinki corpus, as well as data from the corpus-based dictionaries of Middle English and Scottish. For stylistic assessment of archaisms in the play of D. Lyndsay, the author applies the method of linguo-stylistic analysis, in which linguistic element is viewed from the contextual perspective. The scientific novelty is substantiated by the fact that for solution of the problem of analysis of archaisms in the texts of Early Modern English period, the author elaborates an algorithm that allows clarifying if a unit is obsolescent in a specific moment of development of the English language. It is established that the prevalent proportion of words in Lyndsay’s play was not archaic within the framework of Scottish dialect; and only few of the involved units were outdated at the moment of creation of text in the standard Early Modern English. The archaisms determined in the play allows stating the Lyndsay considered archaisms a part of elevated lexis. The practical values of this work is defined by the possibility to apply the proposed algorithm for detecting archaisms in any literary text of the Early Modern English period, as well as conduct stylistic assessment of archaisms from the perspective of the trends of their perception by the authors of the XVI century.

A powerful and simple frequency formula to nonlinear fractal oscillators

In this work, a fractal nonlinear oscillator is successfully established by fractal derivative in a fractal space, and its variational principle is obtained by semi-inverse transform method. The variational principle can provide conservation laws in an energy form. The approximate frequency of the fractal oscillator is found by a simple fractal frequency formula. An example shows the fractal frequency formula is a powerful and simple tool to fractal oscillators.

On the analytical approximation of the quadratic non-linear oscillator by modified extended iteration method

A modified analytical solution of the quadratic non-linear oscillator has been obtained based on an extended iteration method. In this study, truncated Fourier terms have been used in each step of iterations. The frequencies obtained by this technique show good agreement with the exact frequency. The percentage of error between the exact frequency and our third approximate frequency is as low as 0.001%. There is no algebraic complexity in our calculation, which is why this technique is very easy. The results have been compared with the exact and other existing results, which are both convergent and consistent.

A Modified Solution of the Nonlinear Singular Oscillator by Extended Iteration Procedure

A modified solution of the nonlinear singular oscillator has been obtained based on the extended iteration procedure. We have used an appropriate truncation of the obtained Fourier series in each step of iterations to determine the approximate analytic solution of the oscillator. The third approximate frequency of the nonlinear singular oscillator shows a good agreement with its exact values. Earlier different authors presented the analytic solution of the oscillator by using various types of methods. We have compared the results obtained by the modified technique with some of the existing results. We see that some of their techniques deviate from higher-order approximations and the present technique performs comparatively better. The rate of change of percentage of error of the presented modified solution shows the validity of convergence.

Electromagnetic radiation modulates synchronization in cortical neurons of Macaque brain

We study dynamical synchronization in a model of a neural system representing 47 cortical regions of Macaque brain under the effect of an electromagnetic field. This system is constituted by local networks of densely interconnected excitatory and inhibitory neurons. Coupling between the local networks is introduced through sparsely distributed excitatory connectivity. Voltage- and ligand-gated ion channels determine the neural dynamics of the networks. The effect of electromagnetic field on the neural system is studied by modulating magnetic flux on the membrane potential using memristor coupling. With the application of electromagnetic field and the modulation of long-range synaptic coupling, the system easily makes transition to synchronization. It is found that the threshold for synchrony between coupled local networks is lowered by the applied electromagnetic field. Also, electromagnetic field causes the neural subsystems to make low amplitude oscillations with an approximate frequency of 130 Hz. This indicates that electromagnetic field gives rise to high-gamma activity in the cortical regions of the brain which increases selective attention. This may facilitate adaptive brain function by giving rise to a rich collection of dynamics and contribute to the origin of complex patterns observed in the EEG.

On the Analytical Approximation of the Nonlinear Cubic Oscillator by an Iteration Method

A modified approximate analytic solution of the cubic nonlinear oscillator “ ” has been obtained based on an iteration procedure. Here we have used the truncated Fourier series in each iterative step. The approximate frequencies obtained by this technique show a good agreement with the exact frequency. The percentage of error between exact frequency and our fifth approximate frequency is as low as 0.009%. The calculation with this technique is very easy. This easily-calculated modified technique accelerates the rapid convergence, reduces the error and increases the validity range of the solution.