Chaotic threshold for non-smooth system with multiple impulse effect

Tian, Ruilan; Zhou, Yafu; Wang, Yanzhao; Feng, Wenjie; Yang, Xinwei

doi:10.1007/s11071-016-2800-7

Cited by 26 publications

(5 citation statements)

References 22 publications

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“…due to F þ (p 01 ) = (0, 0) T . Using the Hamiltonian function to measure the distance between the perturbed stable and unstable manifolds yields [11][12][13][14]…”

Section: Non-smooth Homoclinic Orbits Bifurcationmentioning

confidence: 99%

“…Previously, the criteria for chaotic motion have been constructed in some non-smooth systems. [4][5][6][7][8][9][10][11][12][13][14] A lot of effort will be made to extend the Melnikov method to this kind of non-smooth system. Homoclinic bifurcation is detected here and heteroclinic bifurcation will be carried out in a separate paper.…”

Section: Non-smooth Homoclinic Orbits Bifurcationmentioning

confidence: 99%

“…[1][2][3] Melnikov's method, a tool to obtain analytical results, has been extended to piecewise-smooth dynamical systems by geometrical or analytical technique. [4][5][6][7][8][9][10][11][12][13][14] It is important to note that the particular piecewise-smooth dynamical system was investigated in Tian et al [11][12][13] and Niziol and Swiatoniowski. 14 Although the known works have given some good ideas, these results were limited in terms of the theoretical results, which encourage continued research on the practical application.…”

Section: Introductionmentioning

confidence: 99%

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Bifurcation and chaos for piecewise nonlinear roll system of rolling mill

Tian

Feng

et al. 2017

Advances in Mechanical Engineering

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A non-smooth cold roll system of rolling mill is studied to reveal the bifurcation of the piecewise-smooth and discontinuous system. To examine the influence of the parameters on the dynamics, the bifurcation diagram is constructed when it is unperturbed. Hamilton phase diagrams of the non-smooth system are detected, which differ significantly from the ones obtained in the smooth system. Non-smooth homoclinic, heteroclinic, and periodic orbits are determined, which depend on the classical heteroclinic periodic orbits, periodic orbits, and a necessary condition. A extended Melnikov function is employed to obtain the criteria for the non-smooth homoclinic bifurcation in this class of piecewise-smooth and discontinuous system, which implies that the existence of homoclinic bifurcation arises from the breaking of homoclinic orbits under the perturbation of damping. The results reveal that this kind of non-smooth factor has little influence on the chaotic threshold apart from calculating piecewise integrals. The efficiency of the criteria for non-smooth homoclinic bifurcation mentioned above is verified by the phase portraits, Poincaré section, and bifurcation diagrams, which laid a theoretical foundation for parameter design, stable operation, and fault diagnosis of rolling mills.

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“…due to F þ (p 01 ) = (0, 0) T . Using the Hamiltonian function to measure the distance between the perturbed stable and unstable manifolds yields [11][12][13][14]…”

Section: Non-smooth Homoclinic Orbits Bifurcationmentioning

confidence: 99%

Section: Non-smooth Homoclinic Orbits Bifurcationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Bifurcation and chaos for piecewise nonlinear roll system of rolling mill

Tian

Feng

et al. 2017

Advances in Mechanical Engineering

Self Cite

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“…From the perspective of engineering applications, some scholars have paid attention to vibro-impact bistable oscillators by considering the collision constraint, such as the vibro-impact inverted pendulum [3,4,5], the vibroimpact SD oscillator [6,7,8], etc. The basic idea of global vibro-impact dynamics is to extend the non-smooth Melnikov method, which was first proposed by Melnikov [9] and developed in Guckenheimer & Holmes [10] and Wiggins [11] for smooth dynamical systems and extended by Kunze [12], Shi et al [13], Battelli and Feckan [14,15,16,17], Tian et al [18,19,20], and Li et al [21,22,23,24,25,26,27] to non-smooth dynamical systems. In recent times, an increasing number of academics have developed a keen interest in these models.…”

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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“…Impact pendulum system is a typical non-smooth dynamic system. In low-dimensional systems most studies focused on the periodic solution, bifurcation, chaos, and so on [8][9][10][11][12]. For non-smooth high-dimensional system, numerical simulation was applied to detect the dynamics [13][14][15][16][17][18].…”

Section: Introductionmentioning

confidence: 99%

Periodic Solution of a Non-Smooth Double Pendulum with Unilateral Rigid Constrain

2019

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In this paper, a double pendulum model is presented with unilateral rigid constraint under harmonic excitation, which leads to be an asymmetric and non-smooth system. By introducing impact recovery matrix, modal analysis, and matrix theory, the analytical expressions of the periodic solutions for unilateral double-collision will be discussed in high-dimensional non-smooth asymmetric system. Firstly, the impact laws are classified in order to detect the existence of periodic solutions of the system. The impact recovery matrix is introduced to transform the impact laws of high-dimensional system into matrix. Furthermore, by use of modal analysis and matrix theory, an invertible transformation is constructed to obtain the parameter conditions for the existence of the impact periodic solution, which simplifies the calculation and can be easily extended to high-dimensional non-smooth system. Hence, the range of physical parameters and the restitution coefficients is calculated theoretically and non-smooth analytic expression of the periodic solution is given, which provides ideas for the study of approximate analytical solutions of high-dimensional non-smooth system. Finally, numerical simulation is carried out to obtain the impact periodic solution of the system with small angle motion.

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Chaotic threshold for non-smooth system with multiple impulse effect

Cited by 26 publications

References 22 publications

Bifurcation and chaos for piecewise nonlinear roll system of rolling mill

Bifurcation and chaos for piecewise nonlinear roll system of rolling mill

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

Periodic Solution of a Non-Smooth Double Pendulum with Unilateral Rigid Constrain

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