Journal articles: 'Duffing Oscillator'

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Fu, Yongqing, Yanan Li, Lin Zhang, and Xingyuan Li. "The DPSK Signal Noncoherent Demodulation Receiver Based on the Duffing Oscillators Array." International Journal of Bifurcation and Chaos 26, no. 13 (December 15, 2016): 1650216. http://dx.doi.org/10.1142/s0218127416502163.

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Chaotic communication requires the knowledge of corresponding phase relationship between the primary phase of Duffing oscillator’s internal driving force and the primary phase of the undetected signal. Currently, there is no method of noncoherent demodulation for DPSK (Differential Phase Shift Keying) signal and mobile communication signal by Duffing oscillator. To solve this problem, this study presents a noncoherent demodulation method based on the Duffing oscillators array and Duffing oscillator optimization. We first present the model of Duffing oscillator and its sensitivity to undetected signal primary phase. Then the zone partition is proposed to identify the Duffing oscillator’s phase trajectory, and subsequently, the mathematical model and implementation method of the Duffing oscillators array are outlined. Thirdly, the Duffing oscillator optimization and its adaptive strobe technique are proposed, also their application to DPSK signal noncoherent demodulation are discussed. Finally, the design of new concept DPSK chaotic digital receiver based on the Duffing oscillators array is presented, together with its simulation results obtained by using SystemView simulation platform. The simulation results suggest that the new concept receiver based on the Duffing oscillator optimization of Duffing oscillators array owns better SNR (signal-to-noise ratio) threshold property than typical existing receivers (chaotic or nonchaotic) in the AWGN (additive white Gaussian noise) channel and multipath Rayleigh fading channel. In addition, the new concept receiver may detect mobile communication signal.

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LUO, ALBERT C. J., and JIANZHE HUANG. "ASYMMETRIC PERIODIC MOTIONS WITH CHAOS IN A SOFTENING DUFFING OSCILLATOR." International Journal of Bifurcation and Chaos 23, no. 05 (May 2013): 1350086. http://dx.doi.org/10.1142/s0218127413500867.

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In this paper, asymmetric periodic motions in a periodically forced, softening Duffing oscillator are presented analytically through the generalized harmonic balance method. For the softening Duffing oscillator, the symmetric periodic motions with jumping phenomena were understood very well. However, asymmetric periodic motions in the softening Duffing oscillators are not investigated analytically yet, and such asymmetric periodic motions possess much richer dynamics than the symmetric motions in the softening Duffing oscillator. For asymmetric motions, the bifurcation tree from asymmetric period-1 motions to chaos is discussed comprehensively. The corresponding, unstable and stable, asymmetric and symmetric, periodic motions in the softening Duffing oscillator are presented, and numerical illustrations of stable and unstable periodic motions are completed. This investigation provides a better picture of complex motion in the softening Duffing oscillator.

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Cintra, Daniel, and Pierre Argoul. "Nonlinear argumental oscillators: A few examples of modulation via spatial position." Journal of Vibration and Control 23, no. 18 (January 22, 2016): 2888–911. http://dx.doi.org/10.1177/1077546315623888.

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Under certain conditions, an oscillator can enter a stable regime when submitted to an external harmonic force whose frequency is far from the natural frequency of the oscillator. This may happen when the external force acts on the oscillator in a way which depends on the oscillator's spatial position. This phenomenon is called “argumental oscillation”. In this paper, six argumental oscillators are described and modeled, and experimental results are given and compared to numerical simulations based on the models. A polar Van der Pol representation, with embedded time indications, is used to allow a precise comparison. The pendulums are modeled as Duffing oscillators. The six models are based on various pendulums excited by spatially localized magnetic-field sources consisting of wire coils. Each pendulum receives the excitation via a steel element, or a permanent magnet, fitted at the tip of the pendulum's rod. The spatial localization induces another nonlinearity besides the Duffing nonlinearity. A control system allowing a real-time Van der Pol representation of the motion is presented. Attractors are brought out from experimental results.

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Chen, Ting, Xiangyu Cao, and Dezhi Niu. "Model modification and feature study of Duffing oscillator." Journal of Low Frequency Noise, Vibration and Active Control 41, no. 1 (October 6, 2021): 230–43. http://dx.doi.org/10.1177/14613484211032760.

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With the development of chaos theory, Duffing oscillator has been extensively studied in many fields, especially in electronic signal processing. As a nonlinear oscillator, Duffing oscillator is more complicated in terms of equations or circuit analysis. In order to facilitate the analysis of its characteristics, the study analyzes the circuit from the perspective of vibrational science and energetics. The classic Holmes-Duffing model is first modified to make it more popular and concise, and then the model feasibility is confirmed by a series of rigorous derivations. According to experiments, the influence of driving force amplitude, frequency, and initial value on the system is finally explained by the basic theories of physics. Through this work, people can understand the mechanisms and characteristics of Duffing oscillator more intuitively and comprehensively. It provides a new idea for the study of Duffing oscillators and more.

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Kanamaru, Takashi. "Duffing oscillator." Scholarpedia 3, no. 3 (2008): 6327. http://dx.doi.org/10.4249/scholarpedia.6327.

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Alhejaili, Weaam, Alvaro H. Salas, and Samir A. El-Tantawy. "Ansatz and Averaging Methods for Modeling the (Un)Conserved Complex Duffing Oscillators." Mathematics 11, no. 9 (April 24, 2023): 2007. http://dx.doi.org/10.3390/math11092007.

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In this study, both the ansatz and averaging methods are carried out for analyzing the complex Duffing oscillators including the undamped/conserved complex Duffing oscillator (CDO) and the damped/unconserved CDO to obtain some approximate analytical solutions. To analyze the conserved CDO, it is reduced to two decoupled conserved Duffing oscillators. After that, the exact solution of the conserved Duffing oscillator is employed to derive an approximation of the conserved CDO in terms of the Jacobi elliptic function. To analyze the damped CDO, two methodologies are considered. For the first methodology, the damped CDO is reduced to two decoupled damped Duffing oscillators, and the ansatz method is devoted to analyzing the damped Duffing oscillator. Accordingly, an approximation of the damped CDO in terms of trigonometric functions is obtained. In the second methodology, the averaging method is applied directly to the damped CDO to derive an approximation in terms of trigonometric functions. All the obtained solutions are compared with the fourth-order Runge–Kutta (RK4) numerical approximations. This study may help many researchers interested in the field of plasma physics to interpret their laboratory and observations results.

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Karimov, Timur, Olga Druzhina, Valerii Vatnik, Ekaterina Ivanova, Maksim Kulagin, Veronika Ponomareva, Anzhelika Voroshilova, and Vyacheslav Rybin. "Sensitivity Optimization and Experimental Study of the Long-Range Metal Detector Based on Chaotic Duffing Oscillator." Sensors 22, no. 14 (July 12, 2022): 5212. http://dx.doi.org/10.3390/s22145212.

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Sensors based on chaotic oscillators have a simple design, combined with high sensitivity and energy efficiency. Among many developed schemes of such sensors, the promising one is based on the Duffing oscillator, which possesses a remarkable property of demonstrating chaotic oscillations only in the presence of a weak sine wave at the input. The main goal of this research was to evaluate the maximal sensitivity of a practically implemented metal detector based on the Duffing oscillator and compare its sensitivity with conventional sensors. To achieve high efficiency of the Duffing-based design, we proposed an algorithm which performs a bifurcation analysis of any chaotic system, classifies the oscillation modes and determines the system sensitivity to a change in different parameters. We apply the developed algorithm to improve the sensitivity of the electronic circuit implementing the Duffing oscillator, serving as a key part of a three-coil metal detector. We show that the developed design allows detecting the presence of metal objects near the coils more reliably than the conventional signal analysis techniques, and the developed detector is capable of sensing a large metal plate at distances up to 2.8 of the coil diameter, which can be considered a state-of-the-art result.

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Kim, Valentine, and Roman Parovik. "Mathematical Model of Fractional Duffing Oscillator with Variable Memory." Mathematics 8, no. 11 (November 19, 2020): 2063. http://dx.doi.org/10.3390/math8112063.

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The article investigates a mathematical model of the Duffing oscillator with a variable fractional order derivative of the Riemann–Liouville type. The study of the model is carried out using a numerical scheme based on the approximation of the fractional derivative of the Riemann–Liouville type by a discrete analog—the fractional derivative of Grunwald–Letnikov. The adequacy of the numerical scheme is verified using specific examples. Using a numerical algorithm, oscillograms and phase trajectories are constructed depending on the values of the model parameters. Chaotic regimes of the Duffing fractional oscillator are investigated using the Wolf–Bennetin algorithm. The forced oscillations of the Duffing fractional oscillator are investigated using the harmonic balance method. Analytical formulas for the amplitude-frequency, phase-frequency characteristics, and also the quality factor are obtained. It is shown that the fractional Duffing oscillator possesses different modes: regular, chaotic, multi-periodic. The relationship between the order of the fractional derivative and the quality factor of the oscillatory system is established.

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Wang, Ke, Xiaopeng Yan, Zhiqiang Zhu, Xinhong Hao, Ping Li, and Qian Yang. "Blind Estimation Methods for BPSK Signal Based on Duffing Oscillator." Sensors 20, no. 22 (November 10, 2020): 6412. http://dx.doi.org/10.3390/s20226412.

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To realize the blind estimation of binary phase shift keying (BPSK) signal, this paper describe a new relational expression among the state of Duffing oscillator excited by BPSK signal, the pseudo-random code of BPSK signal, and the difference frequency between the to-be-detect signal and internal drive force signal of Duffing oscillator. Two output characteristics of Duffing oscillators excited by BPSK signals named implied periodicity and pilot frequency array synchronization are presented according to the different chaotic states of Duffing oscillator. Then two blind estimation methods for the carrier frequency and pseudo-random sequence of the BPSK signal are proposed based on these two characteristics, respectively. These methods are shown to have a significant effect on the parameter estimation of BPSK signals with no prior knowledge, even at very low signal-to-noise ratios (SNRs).

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Krok, Kamila A., Artur P. Durajski, and Radosław Szczȩśniak. "The Abraham–Lorentz force and the time evolution of a chaotic system: The case of charged classical and quantum Duffing oscillators." Chaos: An Interdisciplinary Journal of Nonlinear Science 32, no. 7 (July 2022): 073130. http://dx.doi.org/10.1063/5.0090477.

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This paper proves that the Abraham–Lorentz (AL) force can noticeably modify the trajectories of the charged Duffing oscillators over time. The influence of the reaction force on the oscillator evolution is strongly enhanced if the system is considered at the level of quantum mechanics. For example, the AL force examined within the scope of Newtonian description can change the trajectory of the Duffing oscillator only if it has the mass of an electron. However, we showed that when quantum corrections along with the nondeterministic contributions are taken into account, the reaction force of the electromagnetic field affects noticeably even the oscillator with a mass equal to the mass of the [Formula: see text] ion. The charged Duffing oscillators belong to the class of systems characterized by the chaotic nondeterministic dynamics. In classical terms, the nondeterministic behavior of the discussed systems results from the breaking of the causality principle by the AL force.

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Elías-Zúñiga, Alex, and Oscar Martínez-Romero. "Transient and Steady-State Responses of an Asymmetric Nonlinear Oscillator." Mathematical Problems in Engineering 2013 (2013): 1–9. http://dx.doi.org/10.1155/2013/574696.

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We study the dynamical response of an asymmetric forced, damped Helmholtz-Duffing oscillator by using Jacobi elliptic functions, the method of elliptic balance, and Fourier series. By assuming that the modulus of the elliptic functions is slowly varying as a function of time and by considering the primary resonance response of the Helmholtz-Duffing oscillator, we derived an approximate solution that provides the time-dependent amplitude-frequency response curves. The accuracy of the derived approximate solution is evaluated by studying the evolution of the response curves of an asymmetric Duffing oscillator that describes the motion of a damped, forced system supported symmetrically by simple shear springs on a smooth inclined bearing surface. We also use the percentage overshoot value to study the influence of damping and nonlinearity on the transient and steady-state oscillatory amplitudes.

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Tian, Yi. "Frequency formula for a class of fractal vibration system." Reports in Mechanical Engineering 3, no. 1 (December 15, 2022): 55–61. http://dx.doi.org/10.31181/rme200103055y.

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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.

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Salas, Alvaro H., Mamon Abu Hammad, Badriah M. Alotaibi, Lamiaa S. El-Sherif, and Samir A. El-Tantawy. "Analytical and Numerical Approximations to Some Coupled Forced Damped Duffing Oscillators." Symmetry 14, no. 11 (November 1, 2022): 2286. http://dx.doi.org/10.3390/sym14112286.

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In this investigation, two different models for two coupled asymmetrical oscillators, known as, coupled forced damped Duffing oscillators (FDDOs) are reported. The first model of coupled FDDOs consists of a nonlinear forced damped Duffing oscillator (FDDO) with a linear oscillator, while the second model is composed of two nonlinear FDDOs. The Krylov–Bogoliubov–Mitropolsky (KBM) method, is carried out for analyzing the coupled FDDOs for any model. To do that, the coupled FDDOs are reduced to a decoupled system of two individual FDDOs using a suitable linear transformation. After that, the KBM method is implemented to find some approximations for both unforced and forced damped Duffing oscillators (DDOs). Furthermore, the KBM analytical approximations are compared with the fourth-order Runge–Kutta (RK4) numerical approximations to check the accuracy of all obtained approximations. Moreover, the RK4 numerical approximations to both coupling and decoupling systems of FDDOs are compared with each other.

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Cveticanin, Livija. "On the Van Der Pol Oscillator: an Overview." Applied Mechanics and Materials 430 (September 2013): 3–13. http://dx.doi.org/10.4028/www.scientific.net/amm.430.3.

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In this paper an overview of the self-sustained oscillators is given. The standard van der Pol and the Rayleigh oscillators are considered as basic ones. The cubic nonlinear term of Duffing type is included. The special attention is given to the various complex systems based on the Rayleighs and van der Pols oscillator which are extended with the nonlinear oscillators of Duffing type and also excited with a periodical force. The connection is with the linear elastic force or with linear damping force. The objectives for future investigation are given in this matter.

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Li, Chenjing, Xuemei Xu, Yipeng Ding, Linzi Yin, and Beibei Dou. "Weak photoacoustic signal detection based on the differential duffing oscillator." International Journal of Modern Physics B 32, no. 09 (April 5, 2018): 1850103. http://dx.doi.org/10.1142/s0217979218501035.

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In view of photoacoustic spectroscopy theory, the relationship between weak photoacoustic signal and gas concentration is described. The studies, on the principle of Duffing oscillator for identifying state transition as well as determining the threshold value, have proven the feasibility of applying the Duffing oscillator in weak signal detection. An improved differential Duffing oscillator is proposed to identify weak signals with any frequency and ameliorate the signal-to-noise ratio. The analytical methods and numerical experiments of the novel model are introduced in detail to confirm its superiority. Then the signal detection system of weak photoacoustic based on differential Duffing oscillator is constructed, it is the first time that the weak signal detection method with differential Duffing oscillator is applied triumphantly in photoacoustic spectroscopy gas monitoring technology.

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Joshi, Pratibha, and Maheshwar Pathak. "Numerical Approximation of Nonlinear Duffing Oscillator Using a Coupled Approach." Mathematical Modelling of Engineering Problems 9, no. 3 (June 30, 2022): 715–20. http://dx.doi.org/10.18280/mmep.090318.

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Duffing equation can describe many important nonlinear physical systems. In this paper a coupled approach based on quasilinearization and Bessel polynomial collocation method has been suggested to solve nonlinear duffing oscillator equation. The nonlinearity in duffing oscillator can be of various variety. This approach is very efficient, stable and reliable to deal with any kind of nonlinearity. Numerical examples demonstrate the validity and applicability of the approach on various types of nonlinear duffing oscillator equation.

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Wu, Aiping, Saleh M. Mwachaka, Yanliang Pei, and Qingqing Fu. "A Novel Weak Signal Detection Method of Electromagnetic LWD Based on a Duffing Oscillator." Journal of Sensors 2018 (June 21, 2018): 1–14. http://dx.doi.org/10.1155/2018/5847081.

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The logging while drilling (LWD) electromagnetic weak signal detection model of a Duffing oscillator based on a one-dimensional nonlinear oscillator is established. The influences of noise on Duffing oscillator dynamic behavior of periodic driving force are discussed and evaluate the oscillator weak signal detection mechanism based on a phase change trajectory diagram. The improved Duffing oscillator is designed and applied to detect the electromagnetic logging signal resistivity at the drill bit using the time-scale transformation method. The simulation results show that the nonlinear dynamic characteristics of the Duffing oscillator are very noticeable, the Duffing circuit is very sensitive to detect the tested signal, and it has a reasonable level of immunity to noise. The smaller the amplitude of the tested signal, the more sensitive the circuit is to the signal, the better the antinoise system performance, and the lower the signal-to-noise ratio (SNR).

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LUO, ALBERT C. J., and FUHONG MIN. "THE MECHANISM OF A CONTROLLED PENDULUM SYNCHRONIZING WITH PERIODIC MOTIONS IN A PERIODICALLY FORCED, DAMPED DUFFING OSCILLATOR." International Journal of Bifurcation and Chaos 21, no. 07 (July 2011): 1813–29. http://dx.doi.org/10.1142/s0218127411029495.

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In this paper, the analytical conditions for the controlled pendulum synchronizing with periodic motions in the Duffing oscillator are developed using the theory of discontinuous dynamical systems. From the analytical conditions, the synchronization invariant domain is obtained. The partial and full synchronizations of the controlled pendulum with periodic motions in the Duffing oscillator are discussed. The control parameter map for the synchronization is achieved from the analytical conditions, and numerical illustrations of the partial and full synchronizations are carried out to illustrate the analytical conditions. This synchronization is different from the controlled Duffing oscillator synchronizing with chaotic motion in the periodically excited pendulum. Because the periodically forced, damped Duffing oscillator possesses periodic and chaotic motions, further investigation on the controlled pendulum synchronizing with complicated periodic and chaotic motions in the Duffing oscillator will be accomplished in sequel.

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Illahi, Ramadian Ridho, Marzuki Marzuki, and Lalu Sahrul Hudha. "Solution of The Duffing Equation Using Exponential Time Differencing Method." EIGEN MATHEMATICS JOURNAL 7, no. 1 (February 15, 2024): 112–14. http://dx.doi.org/10.29303/emj.v7i1.195.

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To describe the spring stiffening effect that occurs in physics and engineering problems, Georg Duffing added the cubic stiffness term to the linear harmonic oscillator equation and is now known as the Duffing oscillator. Despite its simplicity, its dynamic behavior is very diverse. In this research, the Exponential Time Difference method is introduced to solve the Duffing oscillator numerically. To formulate the ETD method, we were using the integration factors. It is a function which, when multiplied by an ordinary differential equation, produces a differential equation that can be integrated. This method is an effective numerical method for solving complex differential equations, especially equations that have strong non-linearity The ETD method delivers highly accurate numerical solutions for the Duffing oscillator, with minimal discrepancy from the analytical results. Through parameter variation, the ETD method's applicability extends to diverse Duffing oscillator configurations.

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Tsiganov, A. V. "Duffing Oscillator and Elliptic Curve Cryptography." Nelineinaya Dinamika 14, no. 2 (2018): 235–41. http://dx.doi.org/10.20537/nd180207.

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Kudryashov, Nikolay A. "The generalized Duffing oscillator." Communications in Nonlinear Science and Numerical Simulation 93 (February 2021): 105526. http://dx.doi.org/10.1016/j.cnsns.2020.105526.

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Saeed, Umer, and Mujeeb ur Rehman. "Haar Wavelet Operational Matrix Method for Fractional Oscillation Equations." International Journal of Mathematics and Mathematical Sciences 2014 (2014): 1–8. http://dx.doi.org/10.1155/2014/174819.

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We utilized the Haar wavelet operational matrix method for fractional order nonlinear oscillation equations and find the solutions of fractional order force-free and forced Duffing-Van der Pol oscillator and higher order fractional Duffing equation on large intervals. The results are compared with the results obtained by the other technique and with exact solution.

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Chen, Yang-Yang, Shu-Hui Chen, and Wei-Wei Wang. "Novel Hyperbolic Homoclinic Solutions of the Helmholtz-Duffing Oscillators." Shock and Vibration 2016 (2016): 1–10. http://dx.doi.org/10.1155/2016/9471423.

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The exact and explicit homoclinic solution of the undamped Helmholtz-Duffing oscillator is derived by a presented hyperbolic function balance procedure. The homoclinic solution of the self-excited Helmholtz-Duffing oscillator can also be obtained by an extended hyperbolic perturbation method. The application of the present homoclinic solutions to the chaos prediction of the nonautonomous Helmholtz-Duffing oscillator is performed. Effectiveness and advantage of the present solutions are shown by comparisons.

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CHUDZIK, A., P. PERLIKOWSKI, A. STEFANSKI, and T. KAPITANIAK. "MULTISTABILITY AND RARE ATTRACTORS IN VAN DER POL–DUFFING OSCILLATOR." International Journal of Bifurcation and Chaos 21, no. 07 (July 2011): 1907–12. http://dx.doi.org/10.1142/s0218127411029513.

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We discuss the mechanism leading to the multistability in the externally excited van der Pol–Duffing oscillator. It has been shown that the mechanism (the sequence of bifurcations) leading to the phase multistability in coupled oscillators is the same as the mechanism leading to the bistability in the single oscillator.

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Elías-Zúñiga, Alex, and Oscar Martínez-Romero. "Equivalent Mathematical Representation of Second-Order Damped, Driven Nonlinear Oscillators." Mathematical Problems in Engineering 2013 (2013): 1–11. http://dx.doi.org/10.1155/2013/670845.

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The aim of this paper focuses on applying a nonlinearization method to transform forced, damped nonlinear equations of motion of oscillatory systems into the well-known forced, damped Duffing equation. The accuracy obtained from the derived equivalent equations of motion is evaluated by studying the amplitude-time, the phase portraits, and the continuous wavelet transform diagrams of the cubic-quintic Duffing equation, the generalized pendulum equation, the power-form elastic term oscillator, the Duffing equation with linear and cubic damped terms, and the pendulum equation with a cubic damped term.

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Zhao, Zhike, Xin Wang, and Xiaoguang Zhang. "Fault Diagnosis of Broken Rotor Bars in Squirrel-Cage Induction Motor of Hoister Based on Duffing Oscillator and Multifractal Dimension." Advances in Mechanical Engineering 6 (January 1, 2014): 849670. http://dx.doi.org/10.1155/2014/849670.

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This paper is to propose a novel fault diagnosis method for broken rotor bars in squirrel-cage induction motor of hoister, which is based on duffing oscillator and multifractal dimension. Firstly, based on the analysis of the structure and performance of modified duffing oscillator, the end of transitional slope from chaotic area to large-scale cycle area is selected as the optimal critical threshold of duffing oscillator by bifurcation diagrams and Lyapunov exponent. Secondly, the phase transformation duffing oscillator from chaos to intermittent chaos is sensitive to the signals, whose frequency difference is quite weak from the reference signal. The spectrums of the largest Lyapunov exponents and bifurcation diagrams of the duffing oscillator are utilized to analyze the variance in different parameters of frequency. Finally, this paper is to analyze the characteristics of both single fractal (box-counting dimension) and multifractal and make a comparison between them. Multifractal detrended fluctuation analysis is applied to detect extra frequency component of current signal. Experimental results reveal that the method is effective for early detection of broken rotor bars in squirrel-cage induction motor of hoister.

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Han, R. P. S., and A. C. J. Luo. "Resonant Layers in Nonlinear Dynamics." Journal of Applied Mechanics 65, no. 3 (September 1, 1998): 727–36. http://dx.doi.org/10.1115/1.2789117.

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A new method based on an incremental energy approach and the standard mapping technique is proposed for the study of resonant layers in nonlinear dynamics. To demonstrate the procedure, the method is applied to four types of Duffing oscillators. The appearance, disappearance and accumulated disappearance strengths of the resonant layers for each type of oscillator are derived. A quantitative check of the appearance strength is performed by computing its value using three independent methods: Chirikov overlap criterion, renormalization group technique, and numerical simulations. It is also observed that for the case of the twin-well Duffing oscillator, its perturbed left and right wells are asymmetric.

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Elías-Zúñiga, Alex, Oscar Martínez-Romero, and René K. Córdoba-Díaz. "Approximate Solution for the Duffing-Harmonic Oscillator by the Enhanced Cubication Method." Mathematical Problems in Engineering 2012 (2012): 1–12. http://dx.doi.org/10.1155/2012/618750.

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The cubication and the equivalent nonlinearization methods are used to replace the original Duffing-harmonic oscillator by an approximate Duffing equation in which the coefficients for the linear and cubic terms depend on the initial oscillation amplitude. It is shown that this procedure leads to angular frequency values with a maximum relative error of 0.055%. This value is 21% lower than the relative errors attained by previously developed approximate solutions.

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Luo, Albert C. J., and Dennis M. O'Connor. "On Periodic Motions in a Parametric Hardening Duffing Oscillator." International Journal of Bifurcation and Chaos 24, no. 01 (January 2014): 1430004. http://dx.doi.org/10.1142/s0218127414300043.

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In this paper, analytical solutions for periodic motions in a parametric hardening Duffing oscillator are presented using the finite Fourier series expression, and the corresponding stability and bifurcation analysis for such periodic motions are carried out. The frequency-amplitude characteristics of asymmetric period-1 and symmetric period-2 motions are discussed. The hardening Mathieu–Duffing oscillator is also numerically simulated to verify the approximate analytical solutions of periodic motions. Period-1 asymmetric and period-2 symmetric motions are illustrated for a better understanding of periodic motions in the hardening Mathieu–Duffing oscillator.

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Fathi, Mojtaba, Ali Bakhshinejad, Ahmadreza Baghaie, and Roshan D’Souza. "Dynamic Denoising and Gappy Data Reconstruction Based on Dynamic Mode Decomposition and Discrete Cosine Transform." Applied Sciences 8, no. 9 (September 1, 2018): 1515. http://dx.doi.org/10.3390/app8091515.

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Dynamic Mode Decomposition (DMD) is a data-driven method to analyze the dynamics, first applied to fluid dynamics. It extracts modes and their corresponding eigenvalues, where the modes are spatial fields that identify coherent structures in the flow and the eigenvalues describe the temporal growth/decay rates and oscillation frequencies for each mode. The recently introduced compressed sensing DMD (csDMD) reduces computation times and also has the ability to deal with sub-sampled datasets. In this paper, we present a similar technique based on discrete cosine transform to reconstruct the fully-sampled dataset (as opposed to DMD modes as in csDMD) from sub-sampled noisy and gappy data using l 1 minimization. The proposed method was benchmarked against csDMD in terms of denoising and gap-filling using three datasets. The first was the 2-D time-resolved plot of a double gyre oscillator which has about nine oscillatory modes. The second dataset was derived from a Duffing oscillator. This dataset has several modes associated with complex eigenvalues which makes them oscillatory. The third dataset was taken from the 2-D simulation of a wake behind a cylinder at Re = 100 and was used for investigating the effect of changing various parameters on reconstruction error. The Duffing and 2-D wake datasets were tested in presence of noise and rectangular gaps. While the performance for the double-gyre dataset is comparable to csDMD, the proposed method performs substantially better (lower reconstruction error) for the dataset derived from the Duffing equation and also, the 2-D wake dataset according to the defined reconstruction error metrics.

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Zeng, Fang Ling, Yan Yan Hu, Xiao Juan Geng, and Shi Gui Zhang. "A New Method of Chaotic Detection for Weak Sinusoidal Signal Frequency." Advanced Materials Research 756-759 (September 2013): 265–70. http://dx.doi.org/10.4028/www.scientific.net/amr.756-759.265.

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This paper describes the intermittent chaos detection principle, combined with differential form the double Duffing oscillator difference system to the weak signal frequency detection, gives the detailed steps of the method and flow chart. Use the feature that Duffing oscillators with specific reference frequency are insensible to other frequency signals interference, only sensitive to a small part of the nearby specific frequency signal, it comes to a conclusion that small amplitude periodic motion and intermittent chaos motion may appear at the same time in the detection system, which can be used to detect complex frequency signals. At last, takes the single frequency signal and the complex frequency signal detection for simulation, the results show that, compared with traditional Duffing oscillator array method, this method not only reduces the array number, and improved frequency detection bandwidth without affecting the detection accuracy, reduces the calculation amount, so its much better than the traditional method for frequency detection.

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RASHTCHI, VAHID, and MOHSEN NOURAZAR. "DETECTING THE STATE OF THE DUFFING OSCILLATOR BY PHASE SPACE TRAJECTORY AUTOCORRELATION." International Journal of Bifurcation and Chaos 23, no. 04 (April 2013): 1350065. http://dx.doi.org/10.1142/s021812741350065x.

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Detecting the state of the Duffing oscillator, a type of well-known chaotic oscillator, deeply affects the accuracy of its application. Considering this, the present paper introduced a novel method for detecting the state of the Duffing oscillator. Binary outputs, simple calculation, high precision and fast response time were the main advantages of the phase space trajectory autocorrelation. Also, this study explained the largest Lyapunov exponent as well as a number of other methods commonly employed in detecting the state of the Duffing oscillator. The precision and effectiveness of the method introduced was compared with other well-known state detection methods such as the 0-1 test and the largest Lyapunov exponent.

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El-Tantawy, S. A., Alvaro H. Salas, and M. R. Alharthi. "On the Analytical Solutions of the Forced Damping Duffing Equation in the Form of Weierstrass Elliptic Function and its Applications." Mathematical Problems in Engineering 2021 (February 15, 2021): 1–9. http://dx.doi.org/10.1155/2021/6678102.

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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.

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He, Jianbin, and Jianping Cai. "Dynamic Analysis of Modified Duffing System via Intermittent External Force and Its Application." Applied Sciences 9, no. 21 (November 3, 2019): 4683. http://dx.doi.org/10.3390/app9214683.

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Over the past century, a tremendous amount of work on the Duffing system has been done with continuous external force, including analytical and numerical solution methods, and the dynamic behavior of physical systems. However, hows does the Duffing oscillator behave if the external force is intermittent? This paper investigates the Duffing oscillator with intermittent external force, and a modified Duffing chaotic system is proposed. Different from the continuous-control method, an intermittent external force of cosine function was designed to control the Duffing oscillator, such that the modified Duffing (MD) system could behave chaotically. The dynamic characteristics of MD system, such as the strange attractors, Lyapunov exponent spectra, and bifurcation diagram spectra were outlined with numerical simulations. Numerical results showed that there existed a positive Lyapunov exponent in some parameter intervals. Furthermore, by combining it with chaos scrambling and chaos XOR encryption, a chaos-based encryption algorithm was designed via the pseudorandom sequence generated from the MD. Finally, feasibility and validity were verified by simulation experiments of image encryption.

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Salas, Alvaro H., S. A. El-Tantawy, and Noufe H. Aljahdaly. "An Exact Solution to the Quadratic Damping Strong Nonlinearity Duffing Oscillator." Mathematical Problems in Engineering 2021 (January 18, 2021): 1–8. http://dx.doi.org/10.1155/2021/8875589.

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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.

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Liu, Xuan Chao, and Pei Pei Li. "Weak Signal Detection Based on the Nonlinear Dynamic Model." Applied Mechanics and Materials 157-158 (February 2012): 887–91. http://dx.doi.org/10.4028/www.scientific.net/amm.157-158.887.

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In order to obtain new ways of weak signal detection, we analyzed the motion states of nonlinear dynamic model Duffing oscillator in the case of different amplitude of input signals by solving the Duffing equation and expounded the basic principles of weak signal detection based on Duffing oscillator phase-change characteristics, further illustrated the relationship between signal detection accuracy and detection time by the experimental, researched the impact on signal detection coming from Gaussian white noise and also pointed out how to use intermittent chaos state to implement weak signal detection. The results showed that Duffing oscillator can be effectively detect the slight changes of input signal in the strong noises background, so as to achieve the purpose of weak signal detection. Compared with existing methods, it could greatly improve the detection results.

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Bota, Constantin, Bogdan Căruntu, and Olivia Bundău. "Approximate Periodic Solutions for Oscillatory Phenomena Modelled by Nonlinear Differential Equations." Mathematical Problems in Engineering 2014 (2014): 1–11. http://dx.doi.org/10.1155/2014/513473.

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We apply the Fourier-least squares method (FLSM) which allows us to find approximate periodic solutions for a very general class of nonlinear differential equations modelling oscillatory phenomena. We illustrate the accuracy of the method by using several significant examples of nonlinear problems including the cubic Duffing oscillator, the Van der Pol oscillator, and the Jerk equations. The results are compared to those obtained by other methods.

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Salas Salas, Alvaro Humberto, Jairo Ernesto Castillo Hernández, and Lorenzo Julio Martínez Hernández. "The Duffing Oscillator Equation and Its Applications in Physics." Mathematical Problems in Engineering 2021 (November 30, 2021): 1–13. http://dx.doi.org/10.1155/2021/9994967.

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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.

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Eze, S. C. "Analysis of fractional duffing oscillator." Revista Mexicana de Física 66, no. 2 Mar-Apr (March 1, 2020): 187. http://dx.doi.org/10.31349/revmexfis.66.187.

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In this contribution, a simple analytical method (which is an elegant combination of a well known methods; perturbation method and Laplace method) for solving non-linear and non-homogeneous fractional diﬀerential equations is pro- posed. In particular, the proposed method was used to analysed the fractional Duﬃng oscillator.The technique employed in this method can be used to analyse other nonlinear fractional diﬀerential equations, and can also be extended to non- linear partial fractional diﬀerential equations.The performance of this method is reliable, eﬀective and gives more general solution.

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Johannessen, Kim. "The Duffing oscillator with damping." European Journal of Physics 36, no. 6 (September 8, 2015): 065020. http://dx.doi.org/10.1088/0143-0807/36/6/065020.

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Bapat, C. N. "Duffing oscillator under periodic impulses." Journal of Sound and Vibration 179, no. 4 (January 1995): 725–32. http://dx.doi.org/10.1006/jsvi.1995.0047.

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Alhejaili, Weaam, Alvaro H. Salas, and Samir A. El-Tantawy. "On the Krýlov–Bogoliúbov-Mitropólsky and Multiple Scales Methods for Analyzing a Time Delay Duffing–Helmholtz Oscillator." Symmetry 15, no. 3 (March 13, 2023): 715. http://dx.doi.org/10.3390/sym15030715.

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This study is divided into two important axes; for the first one, a new symmetric analytical (approximate) solution to the Duffing–Helmholtz oscillatory equation in terms of elementary functions is derived. The obtained solution is compared with the numerical solution using 4th Range–Kutta (RK4) approach and with the exact analytical solution that is obtained using elliptic functions. As for the second axis, we consider the time-delayed version for the same oscillator taking the impact of both forcing and damping terms into consideration. Some analytical approximations for the time delayed Duffing–Helmholtz oscillator are derived using two different perturbation techniques, known as Krylov–Bogoliubov–Mitropolsky method (KBMM) and the multiple scales method (MSM). Moreover, these perturbed approximations are analyzed numerically and compared with the RK4 approximations.

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Durmaz, Seher, and Metin Orhan Kaya. "High-Order Energy Balance Method to Nonlinear Oscillators." Journal of Applied Mathematics 2012 (2012): 1–7. http://dx.doi.org/10.1155/2012/518684.

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Energy balance method (EBM) is extended for high-order nonlinear oscillators. To illustrate the effectiveness of the method, a cubic-quintic Duffing oscillator was chosen. The maximum relative errors of the frequencies of the oscillator read 1.25% and 0.6% for the first- and second-order approximation, respectively. The third-order approximation has an accuracy as high as 0.008%. Excellent agreement of the approximated frequencies and periodic solutions with the exact ones is demonstrated for several values of parameters of the oscillator.

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Hajdu, Flóra. "Parallel Numerical Creation of 2-parametric Bifurcation Diagram of Nonlinear Oscillators." Acta Technica Jaurinensis 11, no. 2 (July 23, 2018): 61–83. http://dx.doi.org/10.14513/actatechjaur.v11.n2.453.

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This paper presents the numerical creation of 2-parametric bifurcation diagrams of nonlinear oscillators with a simple iterative algorithm, which can be easily parallelized. The parallel algorithm was tested with two simple well-known nonlinear oscillators, the Van der Pol oscillator and the Duffing-Holmes oscillator. It was examined how the resolution (number of iterations) affects the speedup and the efficiency. The test results show that a relative good speed up with a good efficiency could be achieved even using a simple desktop.

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Big-Alabo, Akuro. "A simple cubication method for approximate solution of nonlinear Hamiltonian oscillators." International Journal of Mechanical Engineering Education 48, no. 3 (January 15, 2019): 241–54. http://dx.doi.org/10.1177/0306419018822489.

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A new cubication method is proposed for periodic solution of nonlinear Hamiltonian oscillators. The method is formulated based on quasi-static equilibrium of the original oscillator and the undamped cubic Duffing oscillator. The cubication constants derived from the present cubication method are always based on elementary functions and are simpler than the constants derived by other cubication methods. The present method was verified using three common examples of strongly nonlinear oscillators and was found to give reasonably accurate results. The method can be used to introduce nonlinear oscillators in relevant undergraduate physics and mechanics courses.

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Hao, D. N., and N. D. Anh. "Response of Duffing Oscillator with Time Delay Subjected to Combined Harmonic and Random Excitations." Mathematical Problems in Engineering 2017 (2017): 1–8. http://dx.doi.org/10.1155/2017/4907520.

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This paper aims to investigate the stationary probability density functions of the Duffing oscillator with time delay subjected to combined harmonic and white noise excitation by the method of stochastic averaging and equivalent linearization. By the transformation based on the fundamental matrix of the degenerate Duffing system, the paper shows that the displacement and the velocity with time delay in the Duffing oscillator can be computed approximately in non-time delay terms. Hence, the stochastic system with time delay is transformed into the corresponding stochastic non-time delay equation in Ito sense. The approximate stationary probability density function of the original system can be found by combining the stochastic averaging method, the equivalent linearization method, and the technique of auxiliary function. The response of Duffing oscillator is investigated. The analytical results are verified by numerical simulation results.

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Liu, Chein-Shan, Chung-Lun Kuo, and Chih-Wen Chang. "Energy-Preserving/Group-Preserving Schemes for Depicting Nonlinear Vibrations of Multi-Coupled Duffing Oscillators." Vibration 7, no. 1 (January 18, 2024): 98–128. http://dx.doi.org/10.3390/vibration7010006.

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In the paper, we first develop a novel automatically energy-preserving scheme (AEPS) for the undamped and unforced single and multi-coupled Duffing equations by recasting them to the Lie-type systems of ordinary differential equations. The AEPS can automatically preserve the energy to be a constant value in a long-term free vibration behavior. The analytical solution of a special Duffing–van der Pol equation is compared with that computed by the novel group-preserving scheme (GPS) which has fourth-order accuracy. The main novelty is that we constructed the quadratic forms of the energy equations, the Lie-algebras and Lie-groups for the multi-coupled Duffing oscillator system. Then, we extend the GPS to the damped and forced Duffing equations. The corresponding algorithms are developed, which are effective to depict the long term nonlinear vibration behaviors of the multi-coupled Duffing oscillators with an accuracy of O(h4) for a small time stepsize h.

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Dumont, Yves, and Jean M. S. Lubuma. "Non-standard finite-difference methods for vibro-impact problems." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 461, no. 2058 (May 27, 2005): 1927–50. http://dx.doi.org/10.1098/rspa.2004.1425.

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Impact oscillators are non-smooth systems with such complex behaviours that their numerical treatment by traditional methods is not always successful. We design non-standard finite-difference schemes in which the intrinsic qualitative parameters of the system—the restitution coefficient, the oscillation frequency and the structure of the nonlinear terms—are suitably incorporated. The schemes obtained are unconditionally stable and replicate a number of important physical properties of the involved oscillator system such as the conservation of energy between two consecutive impact times. Numerical examples, including the Duffing oscillator that develops a chaotic behaviour for some positions of the obstacle, are presented. It is observed that the cpu times of computation are of the same order for both the standard and the non-standard schemes.

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Rajaman, S., and S. Rajasekar. "Variation of Response Amplitude in Parametrically Driven Single Duffing Oscillator and Unidirectionally Coupled Duffing Oscillators." Journal of Applied Nonlinear Dynamics 1, no. 1 (March 2017): 121–29. http://dx.doi.org/10.5890/jand.2017.03.009.

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Piccardo, G., F. Tubino, and A. Luongo. "On the effect of mechanical non-linearities on vortex-induced lock-in vibrations." Mathematics and Mechanics of Solids 22, no. 10 (June 16, 2016): 1922–35. http://dx.doi.org/10.1177/1081286516649991.

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Vortex-induced vibrations at lock-in conditions are modeled through generalized van der Pol-Duffing oscillators endowed with frequency-dependent coefficients, taking inspiration from fluid-elastic models. Accordingly, it is found that the limit-cycle amplitude and the non-linear frequency are mutually dependent (feedback effect), differently from the classic oscillator behavior. Consequently, the mechanical non-linearities, which are often believed to be unimportant, do affect the amplitude of motion. Examples concerning an ideal one degree-of-freedom van der Pol-Duffing oscillator and a two degree-of-freedom model, coarsely representative of a tower building, confirm the importance of this approach also from a technical point of view. Thus, non-linear geometric terms and modal interaction (even in non-resonant cases) can lead to non-negligible modifications of purely aeroelastic problems.

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Journal articles on the topic 'Duffing Oscillator'

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