Heteroclinic Bifurcation Behaviors of a Duffing Oscillator with Delayed Feedback

Wen, Shaofang; Chen, Jufeng; Guo, Shuaicheng

doi:10.1155/2018/7213606

Cited by 8 publications

(5 citation statements)

References 39 publications

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“…Shen et al [32] investigated the bifurcation and chaotic behaviors of a Duffing oscillator with delayed displacement and velocity feedbacks under harmonic excitation and established the analytically necessary condition for the chaos in the sense of Smale horseshoes based on Melnikov method. Wen et al [33] studied the heteroclinic bifurcation and chaos of a Duffing oscillator with forcing excitation under both delayed displacement feedback and delayed velocity feedback by Melnikov method.…”

Section: Introductionmentioning

confidence: 99%

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Orbits and chaos of a two‐fluid magnetized plasma oscillator subjected to parametric excitation with square delayed feedback

Hu,

Zhou

2023

Z Angew Math Mech

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In this manuscript, a two‐fluid magnetized plasma oscillator subjected to parametric excitation with square delayed feedback is studied both analytically and numerically. The Hamilton systems of triple‐well and narrow single‐well are discussed in detail. The scenarios of phase portraits and equilibria are given. Homoclinic and heteroclinic orbits are strictly derived. With the Melnikov method, the critical value of chaos arising from homoclinic or heteroclinic intersections is derived analytically. We have discovered some interesting dynamic phenomena, such as uncontrollable time delays with which chaos always occurs for this system. The influence of time‐delay on the chaotic property is also studied rigorously. On the basis of theoretical analysis, some numerical simulations including time histories, phase portraits, bifurcation diagrams, Poincaré sections, Lyapunov exponential spectrums and attractor basins are given. Numerical simulations are consistent with theoretical results.

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Section: Introductionmentioning

confidence: 99%

“…Wen et al. [33] studied the heteroclinic bifurcation and chaos of a Duffing oscillator with forcing excitation under both delayed displacement feedback and delayed velocity feedback by Melnikov method.…”

Section: Introductionmentioning

confidence: 99%

Orbits and chaos of a two‐fluid magnetized plasma oscillator subjected to parametric excitation with square delayed feedback

Hu,

Zhou

2023

Z Angew Math Mech

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“…Yang et al [27] analyzed the resonance phenomena of fractional Duffing systems based on linear time-delay feedback with direct separation of slow and fast motion. Shen et al [28,29] investigated the bifurcation and chaotic behaviors of a Duffing oscillator with delayed displacement and velocity feedbacks by Melnikov's theorem. Mesbahia.…”

Section: Introductionmentioning

confidence: 99%

Bifurcation and Chaotic Behavior of Duffing System with Fractional-Order Derivative and Time Delay

Wang,

Xing

et al. 2023

Fractal Fract

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In this paper, the abundant nonlinear dynamical behaviors of a fractional-order time-delayed Duffing system under harmonic excitation are studied. By constructing Melnikov function, the necessary conditions of chaotic motion in horseshoe shape are detected, and the chaos threshold curve is obtained by comparing the results obtained through the Melnikov theory and numerical iterative algorithm. The results show that the trend of change is the same, which confirms the accuracy of the chaos threshold curve. It could be found that when the excitation frequency ω is larger than a certain value, the Melnikov theory is not valid for these values. Furthermore, by numerical simulation, some numerical results are obtained, including phase portraits, the largest Lyapunov exponents, and the bifurcation diagrams, Poincare maps, time histories, and frequency spectrograms at some typical points. These numerical simulation results show that the system exhibits some new complex dynamical behaviors, including entry into the state of chaotic motion from single period to period-doubling bifurcation and chaotic motion and periodic motion alternating under the necessary condition of chaotic occurrence. In addition, the effects of time delay, fractional-order coefficient, fractional order, linear viscous damping coefficient, and linear stiffness coefficient on the chaotic threshold curve are discussed, respectively. Those results reveal that there exist abundant nonlinear dynamic behaviors in this fractional-order system, and by adjusting these parameters reasonably, the system could be transformed from chaotic motion to non-chaotic motion.

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“…Luo et al [28] investigated the bifurcation and chaos of transverse vibration of viscoelastic radial transmission structure under nonlinear parametric excitation by the numerical method. For a Duffing oscillator with delayed displacement and velocity feedbacks under harmonic excitation, the necessary conditions for generating homoclinic orbit and heteroclinic orbit were, respectively, investigated by Shen et al [29] and Wen et al [30]. Li et al [31] applied the Melnikov method to globally analyze the Duffing oscillator of the simultaneous primary and super-harmonic resonance.…”

Section: Introductionmentioning

confidence: 99%

Chaotic Threshold of a Nonlinear Zener Systems Based on the Melnikov Method

Fan

Shen

Wen

2022

Mathematical Problems in Engineering

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Regarding a nonlinear Zener model with a viscoelastic Maxwell element as the research object, the complicated dynamic behaviors such as homoclinic bifurcation and chaos under harmonic excitation are investigated. At first, the analytically necessary condition for chaos in the sense of Smale horseshoe is derived based on the Melnikov method. Then, the system parameters that meet the analytical condition and the main resonance condition are selected for the numerical simulation. From the bifurcation diagrams and the largest Lyapunov exponents, it is found that the homoclinic orbit breaks, and the system goes to chaos in a crisis way when the excitation amplitude passes the first threshold. The system enters another new chaotic state in the form of period-doubling bifurcation with the increase of the excitation amplitude. At last, the effects of nonlinear term, stiffness coefficient and damping coefficient of Maxwell element on the analytically necessary condition for chaos are analyzed, respectively, and the correctness of the analytical result is proved by numerical simulation. The research result shows that the critical excitation amplitude decreases with the increase of nonlinear term. In addition, the chaotic threshold increases first and then tends to remain unchanged with the raise of stiffness coefficient. The chaotic threshold increases first and then decreases with the enhancement of damping. These results provide a theoretical basis for the research of nonlinear viscoelastic system in the future.

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Heteroclinic Bifurcation Behaviors of a Duffing Oscillator with Delayed Feedback

Cited by 8 publications

References 39 publications

Orbits and chaos of a two‐fluid magnetized plasma oscillator subjected to parametric excitation with square delayed feedback

Orbits and chaos of a two‐fluid magnetized plasma oscillator subjected to parametric excitation with square delayed feedback

Bifurcation and Chaotic Behavior of Duffing System with Fractional-Order Derivative and Time Delay

Chaotic Threshold of a Nonlinear Zener Systems Based on the Melnikov Method

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