Nonlinear dynamics of a $$\varvec{\phi ^6}-$$ ϕ 6 - modified Duffing oscillator: resonant oscillations and transition to chaos

Miwadinou, C. H.; Hinvi, L. A.; Monwanou, A. V.; Orou, J. B. Chabi

doi:10.1007/s11071-016-3232-0

Cited by 38 publications

(19 citation statements)

References 19 publications

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“…For this study, the parameters as defined in Table 2 were investigated. Inserting the values of parameters with single values into the general form given by equation 1gives: Three excitation frequencies [22]. ( , , ̇) (0, −1, 0), (0,0,0), (0,1,0) Equilibrium points of Equation 2, used as initial conditions Table 3 lists some constants and expressions required for the simulation of the Duffing equation.…”

Section: Harmonically Excited Duffing Oscillatormentioning

confidence: 99%

Investigation of Selected Versions of Fourth Order Runge-Kutta Algorithms as Simulation Tools for Harmonically Excited Duffing Oscillator

Salau

Adeleke

2018

ACRI

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This study aim is to investigate the properties of selected fourth order Runge-Kutta algorithms. Fiftyfive versions of fourth order Runge-Kutta (RK_1, RK_2, RK_3 …, RK_55) methods; inclusive of the classical fourth order RK version, were selected. Thereafter, these versions were used to simulate, with a constant and adaptive step-size algorithm, the dynamics of the harmonically excited Duffing Oscillator over a range of parameters and initial conditions. The simulation was carried out with a FORTRAN program developed and validated by comparing the program generated Poincaré section with literature standard. The number of successive steps taken between start and end of simulation periods was recorded for each simulation run. A total of 91809 simulations were run. The number of successive steps taken between start and end of simulation periods show that significant variations exists among different versions of the same Runge-Kutta order used for seeking solution of Duffing oscillator dynamics. Ranking results by the number of successive steps showed that RK_55 is not the fastest version available, despite its popularity, as other versions including RK_17, RK_2, RK_14, RK_20, and RK_8 outperformed it. Furthermore, the version performance was observed to be highly dependent on the excitation frequency, but not on initial conditions.

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Section: Harmonically Excited Duffing Oscillatormentioning

confidence: 99%

Investigation of Selected Versions of Fourth Order Runge-Kutta Algorithms as Simulation Tools for Harmonically Excited Duffing Oscillator

Salau

Adeleke

2018

ACRI

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“…From tools that are used in the analysis of nonlinear systems, the most general tools are the frequency response curve 4 , 5 , 11 , 15 , 17 – 21 , 27 , 33 , 34 , 38 , the backbone curve 11 , 12 , 19 and, when the numerical approach is used, time histories 16 , 35 , 43 . In the group of more sophisticated tools, however, more specific techniques can also be mentioned: phase portraits 7 , 8 , 35 – 37 , bifurcation diagrams 8 , 38 , 39 , basins of attraction 40 and Poincaré maps 8 , 35 , 41 , 42 .…”

Section: Introductionmentioning

confidence: 99%

Duffing-type oscillator under harmonic excitation with a variable value of excitation amplitude and time-dependent external disturbances

Wawrzyński

2021

Sci Rep

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For more complex nonlinear systems, where the amplitude of excitation can vary in time or where time-dependent external disturbances appear, an analysis based on the frequency response curve may be insufficient. In this paper, a new tool to analyze nonlinear dynamical systems is proposed as an extension to the frequency response curve. A new tool can be defined as the chart of bistability areas and area of unstable solutions of the analyzed system. In the paper, this tool is discussed on the basis of the classic Duffing equation. The numerical approach was used, and two systems were tested. Both systems are softening, but the values of the coefficient of nonlinearity are significantly different. Relationships between both considered systems are presented, and problems of the nonlinearity coefficient and damping influence are discussed.

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“…e search for these phenomena for the new Helmholtz model therefore seems very important. e determination of the states of resonance in a nonlinear dynamic system makes it possible to predict energy exchanges by making the energy proportional to the square of the amplitude of the vibrations [18][19][20][21]. For example, in mechanical systems, a sudden increase in the resonance energy can cause damage to the mechanical system, while antiresonance systems can be used to store energy.…”

Section: Introductionmentioning

confidence: 99%

Nonlinear Dynamics of the Quadratic-Damping Helmholtz Oscillator

et al. 2020

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In this paper, the Helmholtz equation with quadratic damping themes is used for modeling the dynamics of a simple prey-predator system also called a simple Lotka–Volterra system. From the Helmholtz equation with quadratic damping themes obtained after modeling, the equilibrium points have been found, and their stability has been analyzed. Subsequently, the harmonic oscillations have been studied by the harmonic balance method, and the phenomena of resonance and hysteresis are observed. The primary and secondary resonances have been researched by the multiple-scale method, and the conditions of stability of the amplitudes of oscillations are determined. Chaos is detected analytically by the Melnikov method and numerically using the basin of attraction, the bifurcation diagram, the Lyapunov exponent, the phase portrait, and the Poincaré section. The effects of all the parameters of the system are analyzed in detail, and special emphasis is placed on the new parameters. Through this analysis, the complex phenomena such as hysteresis, bistability, amplitude jump, resonances, and chaos have been obtained. The control of the parameters and the necessary conditions to control the aforementioned phenomena have been found.

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Nonlinear dynamics of a $$\varvec{\phi ^6}-$$ ϕ 6 - modified Duffing oscillator: resonant oscillations and transition to chaos

Cited by 38 publications

References 19 publications

Investigation of Selected Versions of Fourth Order Runge-Kutta Algorithms as Simulation Tools for Harmonically Excited Duffing Oscillator

Investigation of Selected Versions of Fourth Order Runge-Kutta Algorithms as Simulation Tools for Harmonically Excited Duffing Oscillator

Duffing-type oscillator under harmonic excitation with a variable value of excitation amplitude and time-dependent external disturbances

Nonlinear Dynamics of the Quadratic-Damping Helmholtz Oscillator

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