Chaotic threshold for a class of impulsive differential system

Tian, Ruilan; Zhou, Yafu; Zhang, BaoLing; Yang, Xinwei

doi:10.1007/s11071-015-2477-3

Cited by 33 publications

(14 citation statements)

References 30 publications

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“…In this section, we will study the condition for the emergence of chaotic motion in Equation (5) by applying Melnikov's method, which has been successfully applied to the analysis of chaos in many nonlinear systems [18][19][20][21].…”

Section: Melnikov Function Of the Smib Power Systemmentioning

confidence: 99%

Chaotic Threshold for a Class of Power System Model

Wang

Song

2019

Shock and Vibration

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This paper deals with the bifurcation and chaotic dynamic characteristic of a single-machine infinite-bus (SMIB) power system under two kinds of harmonic excitation disturbance, which are induced by the external periodic load and the outer mechanical disturbance. By applying Melnikov’s method, the threshold value for the occurrence of chaotic motion is provided. In addition, the chaotic boundary surface is given. The efficiency of the criteria for chaotic motion obtained in this paper is verified by bifurcation diagram, phase portraits, Poincaré section, and frequency spectrum. The results obtained in this paper will provide a better understanding of the nonlinear dynamic behaviors for this class of SMIB power system subjected to two kinds of harmonic excitation components.

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Section: Melnikov Function Of the Smib Power Systemmentioning

confidence: 99%

Chaotic Threshold for a Class of Power System Model

Wang

Song

2019

Shock and Vibration

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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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“…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 a class of impulsive differential system

Cited by 33 publications

References 30 publications

Chaotic Threshold for a Class of Power System Model

Chaotic Threshold for a Class of Power System Model

Bifurcation and chaos for piecewise nonlinear roll system of rolling mill

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

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