scholarly journals The Duffing Oscillator Equation and Its Applications in Physics

2021 ◽  
Vol 2021 ◽  
pp. 1-13
Author(s):  
Alvaro Humberto Salas Salas ◽  
Jairo Ernesto Castillo Hernández ◽  
Lorenzo Julio Martínez Hernández
Keyword(s):  
Initial Conditions ◽  
Duffing Oscillator ◽  
Duffing Equation ◽  
Soliton Theory ◽  
The Way

In this paper, we solve the Duffing equation for given initial conditions. We introduce the concept of the discriminant for the Duffing equation and we solve it in three cases depending on sign of the discriminant. We also show the way the Duffing equation is applied in soliton theory.

scholarly journals On the Analytical Solutions of the Forced Damping Duffing Equation in the Form of Weierstrass Elliptic Function and its Applications

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
S. A. El-Tantawy ◽  
Alvaro H. Salas ◽  
M. R. Alharthi
Keyword(s):  
Analytical Solution ◽  
Elliptic Function ◽  
Initial Conditions ◽  
Duffing Oscillator ◽  
Duffing Equation ◽  
Weierstrass Elliptic Function ◽  
Pendulum Equation ◽  
Rlc Circuit ◽  
Linear Damping ◽  
Forced Term

In this study, a novel analytical solution to the integrable undamping Duffing equation with constant forced term is obtained. Also, a new approximate analytical (semianalytical) solution for the nonintegrable linear damping Duffing oscillator with constant forced term is reported. The analytical solution is given in terms of the Weierstrass elliptic function with arbitrary initial conditions. With respect to it, the semianalytical solution is constructed depending on a new ansatz and the exact solution of the standard Duffing equation (in the absence of both damping and forced terms). A comparison between the obtained solutions and the Runge–Kutta fourth-order (RK4) is carried out. Moreover, some complicated oscillator equations such as the constant forced damping pendulum equation, forced damping cubic-quintic Duffing equation, and constant forced damping Helmholtz–Duffing equation are reduced to the forced damping Duffing oscillator, in which its solution is known. As a practical application, the proposed techniques are applied to investigate the characteristics behavior of the signal oscillations arising in the RLC circuit with externally applied voltage.

scholarly journals Effect of Nonlinear Dissipation on the Basin Boundaries of a Driven Two-Well Modified Rayleigh–Duffing Oscillator

2015 ◽  
Vol 25 (02) ◽  
pp. 1550024 ◽  
Author(s):  
C. H. Miwadinou ◽  
A. V. Monwanou ◽  
J. B. Chabi Orou
Keyword(s):  
Initial Conditions ◽  
Duffing Oscillator ◽  
Excitation Amplitude ◽  
Nonlinear Damping ◽  
Homoclinic Bifurcation ◽  
Nonlinear Dissipation ◽  
Transition To Chaos ◽  
Melnikov Criterion ◽  
Quadratic Nonlinearities ◽  
Basin Boundaries

This paper considers the effect of nonlinear dissipation on the basin boundaries of a driven two-well modified Rayleigh–Duffing oscillator where pure cubic, unpure cubic, pure quadratic and unpure quadratic nonlinearities are considered. By analyzing the potential, an analytic expression is found for the homoclinic orbit. The Melnikov criterion is used to examine a global homoclinic bifurcation and transition to chaos. Unpure quadratic parameter and parametric excitation amplitude effects are found on the critical Melnikov amplitude μ cr . Finally, the phase space of initial conditions is carefully examined in order to analyze the effect of the nonlinear damping, and particularly how the basin boundaries become fractalized.

A PICTURE BOOK OF TWO FAMILIES OF CUBIC MAPS

1993 ◽  
Vol 04 (03) ◽  
pp. 553-568 ◽  
Author(s):  
FERNANDO CABRAL ◽  
ALEXANDRE LAGO ◽  
JASON A. C. GALLAS
Keyword(s):  
Dynamical Systems ◽  
High Resolution ◽  
Parameter Space ◽  
Picture Book ◽  
Initial Conditions ◽  
One Dimensional ◽  
One Dimensional Maps ◽  
Two Parameters ◽  
The Way

This paper reports high-resolution isoperiodic diagrams for two model-families of dynamical systems characterised by one-dimensional maps depending on two parameters. We present a comparison of both diagrams, investigating the way in which initial conditions affect isoperiodic sets in the parameter space of both systems and the similarities between them. Although both models represent quite different dynamical systems, they are found to have many properties in common in their space of parameters.

scholarly journals Parameters Approach Applied on Nonlinear Oscillators

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Najeeb Alam Khan ◽  
Fatima Riaz ◽  
Nadeem Alam Khan
Keyword(s):  
Natural Frequency ◽  
Initial Conditions ◽  
Duffing Oscillator ◽  
Nonlinear Oscillators ◽  
Solution Procedure ◽  
Power Exponent ◽  
Restoring Force ◽  
Arbitrary Power ◽  
Collocation Points ◽  
Approximate Frequency

We applied an approach to obtain the natural frequency of the generalized Duffing oscillatoru¨+u+α3u3+α5u5+α7u7+⋯+αnun=0and a nonlinear oscillator with a restoring force which is the function of a noninteger power exponent of deflectionu¨+αu|u|n−1=0. This approach is based on involved parameters, initial conditions, and collocation points. For any arbitrary power ofn, the approximate frequency analysis is carried out between the natural frequency and amplitude. The solution procedure is simple, and the results obtained are valid for the whole solution domain.

Modeling of a Vibration-Based Piezomagnetoelastic Energy Harvesting System by Using the Duffing Equation

2012 ◽  
Author(s):  
Henrik Westermann ◽  
Marcus Neubauer ◽  
Jörg Wallaschek
Keyword(s):  
Energy Harvesting ◽  
Multiple Scales ◽  
Duffing Oscillator ◽  
Harmonic Balance Method ◽  
Equation Of Motion ◽  
Duffing Equation ◽  
Restoring Force ◽  
Piezoelectric Cantilever ◽  
Energy Harvesting System ◽  
Tip Mass

This article illustrates the modeling of a piezomagnetoelastic energy harvesting system. The generator consists of a piezoelectric cantilever with a magnetic tip mass. A second oppositely poled magnet is attached near the free end of the beam. Due to the nonlinear magnetic restoring force the system exhibits two symmetric stable equilibrium positions and one instable equilibrium position. The equation of motion is derived and it is shown that the system can be modeled as Duffing oscillator. An analytical approach is given to derive the Duffing parameters from the system parameters. The Duffing equation is solved for an oscillation around both equilibrium positions by using the harmonic balance method. For small orbit oscillations the equation of motion is solved by applying the fourth-order multiple scales method.

scholarly journals Identification of nonlinear dynamical systems with time delay

2021 ◽  
Author(s):  
Ghazaale Leylaz ◽  
Shuo Wang ◽  
Jian-Qiao Sun
Keyword(s):  
Time Delay ◽  
Dynamical Systems ◽  
Nonlinear Dynamical Systems ◽  
Initial Conditions ◽  
Duffing Oscillator ◽  
Algebraic Operation ◽  
Computationally Efficient ◽  
Nonlinear Dynamical ◽  
Automatic Model Selection ◽  
Robust To Noise

AbstractThis paper proposes a technique to identify nonlinear dynamical systems with time delay. The sparse optimization algorithm is extended to nonlinear systems with time delay. The proposed algorithm combines cross-validation techniques from machine learning for automatic model selection and an algebraic operation for preprocessing signals to filter the noise and for removing the dependence on initial conditions. We further integrate the bootstrapping resampling technique with the sparse regression to obtain the statistical properties of estimation. We use Taylor expansion to parameterize time delay. The proposed algorithm in this paper is computationally efficient and robust to noise. A nonlinear Duffing oscillator is simulated to demonstrate the efficiency and accuracy of the proposed technique. An experimental example of a nonlinear rotary flexible joint is presented to further validate the proposed method.

scholarly journals An Exact Solution to the Quadratic Damping Strong Nonlinearity Duffing Oscillator

2021 ◽  
Vol 2021 ◽  
pp. 1-8 ◽  
Author(s):  
Alvaro H. Salas ◽  
S. A. El-Tantawy ◽  
Noufe H. Aljahdaly
Keyword(s):  
Elliptic Function ◽  
Initial Conditions ◽  
Exact Analytical Solution ◽  
Duffing Oscillator ◽  
Higher Order ◽  
Strong Nonlinearity ◽  
Classical Dynamics ◽  
Elementary Functions ◽  
Weierstrass Elliptic Function ◽  
Quadratic Damping

The nonlinear equations of motion such as the Duffing oscillator equation and its family are seldom addressed in intermediate instruction in classical dynamics; this one is problematic because it cannot be solved in terms of elementary functions before. Thus, in this work, the stability analysis of quadratic damping higher-order nonlinearity Duffing oscillator is investigated. Hereinafter, some new analytical solutions to the undamped higher-order nonlinearity Duffing oscillator in the form of Weierstrass elliptic function are obtained. Posteriorly, a novel exact analytical solution to the quadratic damping higher-order nonlinearity Duffing equation under a certain condition (not arbitrary initial conditions) and in the form of Weierstrass elliptic function is derived in detail for the first time. Furthermore, the obtained solutions are camped to the Runge–Kutta fourth-order (RK4) numerical solution.

Initial Conditions and Emergence of Corporate Reorganization in the United States and England

2020 ◽  
pp. 21-46
Author(s):  
Sarah Paterson
Keyword(s):  
United States ◽  
Twentieth Century ◽  
Initial Conditions ◽  
Explanatory Power ◽  
The United States ◽  
Institutional Logics ◽  
Corporate Reorganization ◽  
Selective Review ◽  
The Us ◽  
The Way

This chapter is a scene-setting exercise, offering a brief and highly selective review of almost one hundred years of corporate reorganization in the US and England. It seeks to provide some explanation for the very different ways in which corporate reorganization developed in each jurisdiction. Overall, its purpose is to help to sketch out the conditions which prevailed when the account in the book really begins in the 1970s, and how they offer significant explanatory power for the way in which corporate reorganization law and practice emerges in each jurisdiction. Specifically, the chapter investigates the relatively stable corporate reorganization law and practice which prevailed in each jurisdiction for much of the twentieth century, and, in each case, the institutional logics, practices, and identities which gave rise to it.

Aftershock: Business relocation decisions in the wake of the February 2011 Christchurch earthquake

2011 ◽  
Vol 17 (6) ◽  
pp. 857-863 ◽  
Author(s):  
Stephen Bowden
Keyword(s):  
Initial Conditions ◽  
Professional Services ◽  
Limited Information ◽  
Business Location ◽  
Factors Influencing ◽  
Professional Services Firms ◽  
Business Relocation ◽  
The Impact ◽  
The Way

AbstractThis paper examines four businesses that were located in Christchurch's CBD prior to the earthquake on February 22, 2011. Immediately following the quake Christchurch's CBD was cordoned off and for many Christchurch businesses it was necessary to find new premises in an environment of extreme uncertainty and very limited information. The four businesses used in this study were all required to relocate, however the specifics of their situations differed. All four businesses examined were larger professional services firms (lawyers, accountants, architects or engineers) who had a national presence. Given the limited number of suitable and available properties speed was of the essence in relocating. Through the four cases we explore how rapidly each firm initiated and completed the relocation process and the factors influencing their speed. We examine the means by which new premises were secured and the plans in regard to immediate and longer term business location. Finally, we explore the impact of relocation and the earthquake more broadly for these firms. The results obtained highlight the differences between the firms based on both initial conditions and the way each firm managed the process.

CHAOTIC DETECTOR FOR BPSK SIGNALS IN VERY LOW SNR CONDITIONS

2012 ◽  
Vol 22 (06) ◽  
pp. 1250144 ◽  
Author(s):  
SONG ZHANG ◽  
GUO-SHENG RUI
Keyword(s):  
Phase Transition ◽  
Initial Conditions ◽  
Signal To Noise Ratio ◽  
Duffing Oscillator ◽  
Energy Operator ◽  
Chaotic Oscillator ◽  
Error Performance ◽  
Weak Signals ◽  
Low Snr ◽  
Coherent Demodulation

Chaotic detection of weak signals based on Duffing oscillator uses the property of sensitive dependence on initial conditions (SDIC). A small signal can cause a transition between the states of the system and thus be detected. Different from the early works, we concentrate on using chaotic oscillator as a detector for BPSK signals in very low signal-to-noise ratio (SNR) conditions. Phase transition identification is the key step of weak signals detection by using Duffing oscillator. In this paper, we expose a novel algorithm to use Teager energy operator (TEO) to identify the phase transition, which is more easily to be calculated than the usually used methods. According to this algorithm, a methodology is proposed for detection for BPSK signals using Duffing oscillator. A powerline carrier communication system is studied as an example to illustrate the bit error performance of the proposed chaotic detector. The simulation results show that the proposed detector works much better than the traditional coherent demodulation in strong background noise, and it can improve the error performance of uncoded BPSK signal approaching the Shannon limit curve. The proposed chaotic detector gives us another way to approach the Shannon limit without using any complex channel code technology.

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