Degenerate grazing bifurcations in a three-degree-of-freedom impact oscillator

Yin, Shu; Ji, Jinchen; Deng, Shuning; Wen, Guilin

doi:10.1007/s11071-019-04995-7

Cited by 19 publications

(4 citation statements)

References 31 publications

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“…The approximation of the history of q by u i on the evenly spaced grid of τ i on [0, 1] with order M = 2 is a special case of the pseudospectral approximation of DDEs by ODEs, as used for bifurcation analysis in MATCONT by [48]. The spectral approximation from [48] has no advantage over the low-order approximation (19) for our problem, because the history segment is only differentiable once with Lipschitz continuous derivative, whenever the contact threshold x = e has been crossed during the previous time interval of length τ d . Thus, the proposed second-order approximation has the most suitable order M.…”

Section: Defining Systems For Continuationmentioning

confidence: 99%

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Controlling grazing-induced multistability in a piecewise-smooth impacting system via the time-delayed feedback control

et al. 2021

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Grazing events may create coexisting attractors and cause complex dynamics in piecewise-smooth dynamical systems. This paper studies the control of grazing-induced multistability in a soft impacting oscillator by using the time-delayed feedback control. The control switches from one of the coexisting attractors to a desired one to suppress complex dynamics near grazing events. We use path-following (continuation) techniques for non-smooth dynamical systems to investigate robustness of the controller and the parameter dependence of the controlled system. In particular, several newly developed computational methods are used, including a numerical method for analysing non-smooth delay equations and a method for calculating the Lyapunov exponents and the grazing point estimation. Numerical simulations demonstrate that the delayed feedback controller is effective, and a proper selection of the control gain and delay time can simplify the complex dynamics of the system near grazing.

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Section: Defining Systems For Continuationmentioning

confidence: 99%

“…When an attractor encounters a grazing event, it may change qualitatively (see, e.g. [18][19][20]), which is called a discontinuity-induced (or, short, discontinuous or non-smooth) bifurcation. Nordmark [21] and Pavlovskaia et al [22] analyse what dynamics occurs near grazing events.…”

Section: Introductionmentioning

confidence: 99%

Controlling grazing-induced multistability in a piecewise-smooth impacting system via the time-delayed feedback control

et al. 2021

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“…As a special bifurcation of nonsmooth systems, the grazing bifurcation is a qualitative transition between impact and nonimpact motion [31][32][33][34][35][36][37][38][39][40][41][42][43][44][45][46]. Nordmark [31,32] studied the dynamic behavior near the grazing trajectory and a nonlinear mapping is obtained.…”

Section: Introductionmentioning

confidence: 99%

“…e existence condition and stability of the grazing periodic orbit are derived in a two degree-of-freedom vibroimpact system with one-sided constraint in [36]. Yin et al [37][38][39][40][41][42] analyzed and measured the degenerate grazing bifurcations of the impact systems.…”

Section: Introductionmentioning

confidence: 99%

Dynamic Behaviors of a Two‐Degree‐of‐Freedom Impact Oscillator with Two‐Sided Constraints

Meng

2021

Shock and Vibration

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The dynamic model of a vibroimpact system subjected to harmonic excitation with symmetric elastic constraints is investigated with analytical and numerical methods. The codimension-one bifurcation diagrams with respect to frequency of the excitation are obtained by means of the continuation technique, and the different types of bifurcations are detected, such as grazing bifurcation, saddle-node bifurcation, and period-doubling bifurcation, which predicts the complexity of the system considered. Based on the grazing phenomenon obtained, the zero-time-discontinuity mapping is extended from the single constraint system presented in the literature to the two-sided elastic constraint system discussed in this paper. The Poincare mapping of double grazing periodic motion is derived, and this compound mapping is applied to obtain the existence conditions of codimension-two grazing bifurcation point of the system. According to the deduced theoretical result, the grazing curve and the codimension-two grazing bifurcation points are validated by numerical simulation. Finally, various types of periodic-impact motions near the codimension-two grazing bifurcation point are illustrated through the unfolding diagram and phase diagrams.

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