Dichotomous-noise-induced chaos in a generalized Duffing-type oscillator with fractional-order deflection

Lei, Youming; Fu, Rui; Yang, Yong; Wang, Yanyan

doi:10.1016/j.jsv.2015.10.031

Cited by 14 publications

(5 citation statements)

References 44 publications

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“…Using the mean square criterion of random Melnikov processes, Lin and Scs [33] studied the chaotic behavior of Duffing systems under harmonic and additive noise excitations. In recent years, this criterion has been further developed and applied in the gaussian white noise excitation [34], the bounded noise excitation [35,36], the dichotomous noise excitation [37,38], the trichotomous noise [39] and so on. However, there is relatively little research on non-smooth dynamic systems under random excitation.…”

Section: Introductionmentioning

confidence: 99%

Homoclinic bifurcations and chaotic dynamics in a bistable vibro-impact SD oscillator under Gaussian white noise

Jia,

Li,

Kou

et al. 2024

Preprint

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This paper studies the effect of Gaussian white noise on homoclinic bifurcations and chaotic dynamics of a bistable vibro-impact SD oscillator. Firstly, the SD oscillator is reproduced and generalized by installing a slider on a fixed rod so that the slider is connected by a pair of linear springs initially pre-compressed in the vertical direction to achieve bistble vibration characteristic, two screw nuts are installed on the rod as two adjustable bilateral rigid constraints to generate vibro-impact. A discontinuous dynamical equation with a map defined on switching boundaries to represent velocity loss during each collision is derived to describe the vibration pattern of the bistable vibro-impact SD oscillator through studying the persistence of the unique unperturbed non-smooth homoclinic structure. Secondly, the general framework of random Melnikov process for a class of bistable vibro-impact systems under Gaussian white noise is simply derived and employed through the corresponding Melnikov function to obtain the necessary parameter thresholds for homoclinic tangency and possible chaos of the bistable vibro-impact SD oscillator. Thirdly, the effectiveness of a semi-analytical prediction by the Melnikov function is verified through the lagest Lyapunov exponent, bifurcation series, and 0 − 1 test. In addition, the sensitivity to the initial values of chaos is verified by the fractal of attractor basins, and the influence of the Gaussian white noise on periodic and chaotic structures is studied through Poincaré mapping to show rich dynamical geometric structures.

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Section: Introductionmentioning

confidence: 99%

Homoclinic bifurcations and chaotic dynamics in a bistable vibro-impact SD oscillator under Gaussian white noise

Jia,

Li,

Kou

et al. 2024

Preprint

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“…Trichotomous noise, a particular case of non-Gaussian colored noise, [12] can be regarded as a stochastic telegraph process like dichotomous noise, whose effect on dynamic systems has been analyzed by many researchers. [13][14][15][16][17][18] However, trichotomous noise can represent real noise better than dichotomous noise in actual applications because the former can degenerate into the latter and is more flexible to model random fluctuations in nature. [19] Besides, trichotomous noise can be directly expressed as a physical condition such as the heat transfer between three states or configurations.…”

Section: Introductionmentioning

confidence: 99%

“…For the stochastic chaos, several studies have focused on systems induced by Gaussian noise [34,35] or Poisson noise. [36,37] Lei et al [17] studied the chaos in a generalized Duffing-type oscillator with a fractional-order deflection subjected to dichotomous noise excitation. They showed that for such an oscillator, the threshold of chaos can be enhanced and then chaotic behaviors can be controlled while appropriately changing the transition rate of dichotomous noise.…”

Section: Introductionmentioning

confidence: 99%

Homoclinic and heteroclinic chaos in nonlinear systems driven by trichotomous noise

Lei¹,

Zhang²

2017

Chinese Phys. B

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The homoclinic and heteroclinic chaos in nonlinear systems subjected to trichotomous noise excitation are studied. The Duffing system and the Josephson-junction system are taken for example to calculate the corresponding amplitude thresholds for the onset of chaos on the basis of the stochastic Melnikov process with the mean-square criterion. It is shown that the amplitude threshold for the onset of chaos can be adjusted by changing the internal parameters of trichotomous noise, thereby inducing or suppressing chaotic behaviors in the two systems driven by trichotomous noise. The effects of trichotomous noise on the systems are verified by vanishing the mean largest Lyapunov exponent and demonstrated by phase diagrams and time histories.

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“…Second, the phenomena can be observed during nonlinear oscillations. Finally, the topic of chaotic response is relevant [6][7][8][9] .To the issue of nonlinear oscillations, the Duffing equation, called the Duffing oscillator, is inseparably connected. Moreover, the Duffing oscillator is regarded as one of the prototypes for systems of nonlinear dynamics 10 .…”

mentioning

confidence: 99%

“…Second, the phenomena can be observed during nonlinear oscillations. Finally, the topic of chaotic response is relevant [6][7][8][9] .…”

mentioning

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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Dichotomous-noise-induced chaos in a generalized Duffing-type oscillator with fractional-order deflection

Cited by 14 publications

References 44 publications

Homoclinic bifurcations and chaotic dynamics in a bistable vibro-impact SD oscillator under Gaussian white noise

Homoclinic bifurcations and chaotic dynamics in a bistable vibro-impact SD oscillator under Gaussian white noise

Homoclinic and heteroclinic chaos in nonlinear systems driven by trichotomous noise

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

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