Nonlinear behaviors as well as the mechanism in a piecewise-linear dynamical system with two time scales

Bi, Qinsheng; Chen, Xiaoke; Kurths, Jüergen; Zhang, Zhengdi

doi:10.1007/s11071-016-2825-y

Cited by 31 publications

(8 citation statements)

References 29 publications

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“…Considering piecewise linear and nonlinear problems, a simple way for numerical solutions is to employ an explicit integrator directly, like in Refs. 4,21. These methods update the current step by known state variables, and then make the updated variables satisfy the piecewise characteristics.…”

Section: Introductionmentioning

confidence: 99%

“…When applied to linear systems, implicit methods require the factorization of the effective stiffness matrix; for nonlinear systems, an iterative solution approach is inevitable. A more comprehensive review of time integration methods can be found in Tamma et al 20 Considering piecewise linear and nonlinear problems, a simple way for numerical solutions is to employ an explicit integrator directly, like in [4,21]. These methods update the current step by known state variables, and then make the updated variables satisfy the piecewise characteristics.…”

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confidence: 99%

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A generalized approach for implicit time integration of piecewise linear/nonlinear systems

Zhang

Zanoni

et al. 2021

Int Journal of Mech Sys Dyn

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A generalized solution scheme using implicit time integrators for piecewise linear and nonlinear systems is developed. The piecewise linear characteristic has been welldiscussed in previous studies, in which the original problem has been transformed into linear complementarity problems (LCPs) and then solved via the Lemke algorithm for each time step. The proposed scheme, instead, uses the projection function to describe the discontinuity in the dynamics equations, and solves for each step the nonlinear equations obtained from the implicit integrator by the semismooth Newton iteration. Compared with the LCP-based scheme, the new scheme offers a more general choice by allowing other nonlinearities in the governing equations. To assess its performances, several illustrative examples are solved. The numerical solutions demonstrate that the new scheme can not only predict satisfactory results for piecewise nonlinear systems, but also exhibits substantial efficiency advantages over the LCP-based scheme when applied to piecewise linear systems.

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Section: Introductionmentioning

confidence: 99%

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confidence: 99%

A generalized approach for implicit time integration of piecewise linear/nonlinear systems

Zhang

Zanoni

et al. 2021

Int Journal of Mech Sys Dyn

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“…Although w is not a real state variable, it can be regarded as a slow-varying parameter, which changes according to a periodic function, whereas in (1), the slow-varying y changes according to an ordinary differential equation. Upon the approach, the equilibrium branches as well as the bifurcations with the variation of w can be derived, which can be used to account for the mechanism of the bursting attractors in periodic excited systems, such as the fold/Hopf Bi et al (2016a) and Hopf/Hopf bursting oscillations Bi et al (2016b).…”

Section: Introductionmentioning

confidence: 99%

“…Upon the approach, the equilibrium branches as well as the bifurcations with the variation of w can be derived, which can be used to account for the mechanism of the bursting attractors in periodic excited systems, such as the fold/Hopf Bi et al (2016a) and Hopf/Hopf bursting oscillations Bi et al (2016b).…”

Section: Introductionmentioning

confidence: 99%

Bursting oscillations in a slow-varying periodically excited vector field with Bogdanov–Takens bifurcation

2021

Journal of Vibration and Control

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The main purpose of the article is to classify all the possible bursting oscillations in a vector field with Bogdanov–Takens bifurcation at the origin. Based on the universal unfolding of the normal form of the vector field, the topological structure in the neighborhood of the bifurcation point on the unfolding parameters is presented. Replacing one of the unfolding parameters by a slow-varying periodic exciting term, the coupling of two scales in frequency domain involves the vector field, which may lead to the bursting oscillations. According to the bifurcation analysis, we focus on three typical cases to investigate the dynamical evolution with the increase of the exciting amplitude. By introducing the transformed phase portrait, the mechanism of bursting oscillations can be presented. Three types of bifurcations, that is, fold, Hopf, and saddle on the limit cycle bifurcations may cause the alternations of the trajectory between the quiescent states and the spiking states, different combinations of which may result in different bursting attractors. Furthermore, the inertia of the movement may result in the delay effect of the bifurcation, which may lead to the disappearance of the bifurcation influence and the corresponding spiking oscillations.

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“…Multiple time scales problems can be observed in many real systems, such as catalytic reactions in chemical systems [1,2], electrophysiological experiments [3][4][5][6], and circuit systems [7,8]. The coupling effect of these different fastslow time scales generally results in a system that exhibits periodic motion characterized by a combination of relatively large amplitude and nearly harmonic small amplitude oscillations [9,10], conventionally denoted by with and corresponding to the numbers of the large and small amplitude oscillations, respectively [11].…”

Section: Introductionmentioning

confidence: 99%

Bursting Oscillations in Shimizu‐Morioka System with Slow‐Varying Periodic Excitation

Cao

2018

Shock and Vibration

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The coupling effect of two different frequency scales between the exciting frequency and the natural frequency of the ShimizuMorioka system with slow-varying periodic excitation is investigated. First, based on the analysis of the equilibrium states, homoclinic bifurcation, fold bifurcation, and supercritical Hopf bifurcation are observed in the system under a certain parameter condition. When the exciting frequency is much smaller than the natural frequency, we can regard the periodic excitation as a slowvarying parameter. Second, complicated dynamic behaviors are analyzed when the slow-varying parameter passes through different bifurcation points, of which the mechanisms of four different bursting patterns, namely, symmetric "homoclinic/homoclinic" bursting oscillation, symmetric "fold/Hopf" bursting oscillation, symmetric "fold/fold" bursting oscillation, and symmetric "Hopf/Hopf" bursting oscillation via "fold/fold" hysteresis loop, are revealed with different values of the parameter by means of the transformed phase portrait. Finally, we can find that the time interval between two symmetric adjacent spikes of bursting oscillations exhibits dependency on the periodic excitation frequency.

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Nonlinear behaviors as well as the mechanism in a piecewise-linear dynamical system with two time scales

Cited by 31 publications

References 29 publications

A generalized approach for implicit time integration of piecewise linear/nonlinear systems

A generalized approach for implicit time integration of piecewise linear/nonlinear systems

Bursting oscillations in a slow-varying periodically excited vector field with Bogdanov–Takens bifurcation

Bursting Oscillations in Shimizu‐Morioka System with Slow‐Varying Periodic Excitation

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