A novel smooth and discontinuous oscillator with strong irrational nonlinearities

Han, Yanwei; Cao, Qingjie; Chen, Yushu; Wiercigroch, Marian

doi:10.1007/s11433-012-4880-9

Cited by 46 publications

(19 citation statements)

References 28 publications

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“…A novel model which comprises a rotating pendulum linked by an oblique spring pinned to its rigid support was presented [17], which provides a cylindrical dynamical system with both smooth and discontinuous regimes depending on the value of a system parameter. A novel SD oscillator with strong irrational non-linearities was proposed in [18], and the chaotic thresholds for this oscillator were studied using the Melnikov method.…”

Section: Introductionmentioning

confidence: 99%

“…The motivation of this paper is to study the limiting discontinuous case of a rig-coupled SD oscillator introduced in [18] and model it as a piecewise linear system. The rig-coupled SD system is based upon an SD oscillator [3] which derived from an arch model.…”

Section: Introductionmentioning

confidence: 99%

“…

a b s t r a c tIn this paper we investigate the bifurcations and the chaos of a piecewise linear discontinuous (PWLD) system based upon a rig-coupled SD oscillator, which can be smooth or discontinuous (SD) depending on the value of a system parameter, proposed in [18], showing the equilibrium bifurcations and the transitions between single, double and triple well dynamics for smooth regions. All solutions of the perturbed PWLD system, including equilibria, periodic orbits and homoclinic-like and heteroclinic-like orbits, are obtained and also the chaotic solutions are given analytically for this system.

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confidence: 99%

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Chaotic thresholds for the piecewise linear discontinuous system with multiple well potentials

Han

Cao

Chen

et al. 2015

International Journal of Non-Linear Mechanics

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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confidence: 99%

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Chaotic thresholds for the piecewise linear discontinuous system with multiple well potentials

Han

Cao

Chen

et al. 2015

International Journal of Non-Linear Mechanics

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“…The motivations and contributions of this paper are (1) to present a typical rotating machinery model with irrational nonlinearity [15][16][17] which consists of a rotating disk linked by a pair of springs with the fixed ends, (2) to provide a cylindrical dynamical system which exhibits both smooth and discontinuous dynamics depending on a smooth parameter, (3) to develop the nonlinear analysis techniques to remove the barrier raised from the irrationality avoiding Taylar expansion, (4) to propose an extended Melnikov method to investigate the chaotic threshold analytical for the heteroclinic-like orbits in discontinuous system. This paper is organized as follows.…”

Section: Introductionmentioning

confidence: 99%

A rotating disk linked by a pair of springs

2014

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In this paper, we propose a novel model which consists of a rotating disk linked by a pair of springs exhibiting both smooth and discontinuous dynamics. This model provides a cylindrical dynamical system with irrational restoring force due to the geometry configuration. The dynamic behaviors of the unperturbed system are similar to the conventional pendulum coupled with SD oscillator of the homoclinic orbits of the first and second type. A new-type heteroclinic orbit connecting a standard saddle equilibrium and a nonstandard saddle-like equilibrium is presented and investigated analytically in the discontinuous case. Furthermore, the conventional and the extended Melnikov methods are employed to detect the chaotic thresholds for the homoclinic orbits of the first and second type and the heteroclinic-like orbits under the perturbation of both viscous damping and external harmonic forcing for both smooth and discontinuous cases, respectively. The results presented herein this paper demonstrate the predicated periodic solutions and the chaotic attractors of SD type and pendulum type.

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“…The motivations and the contributions of this paper are (i) to provide a new and more effective mathematical model for flight mechanism of dipteran flight motor with irrational nonlinearity which currently attracted much attention, see [22][23][24][25][26][27][28][29][30] for example, (ii) to develop the nonlinear analysis techniques to remove the barrier raised from the irrationality avoiding Taylor expansion to a truncated Duffing system, and (iii) to bridge the gap between the nonlinear dynamics and the biology through the detailed analysis of the novel system to gain a deeper understanding of dipteran flight mechanism.…”

Section: Introductionmentioning

confidence: 99%

A novel model of dipteran flight mechanism

Cao

Xiong

Wiercigroch

2013

Int. J. Dynam. Control

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In this paper, a novel model for the mechanism of dipteran flight motor is proposed and a flight dynamics is explored by nonlinear dynamics techniques. This model comprises a lumped mass, a pair of inclined rigid bars and a pair of horizontal springs. Even the springs are linear, this system is irrationally nonlinear which causes difficulty for classical nonlinear analysis. We investigate the original irrational equation by direct numerical simulation avoiding Taylor's expansion to retain the intrinsic character of dipteran flight. This enables us to reveal the bounded characteristics of depteran flight including "click", bursting and the "complex" flight patterns. Equilibrium analysis demonstrates the "click" mechanism and the "non-click" condition for the unperturbed system. While for the perturbed system, Poincaré section and basin analysis are carried out to demonstrate the bursting behaviour, transitions between different flight modes and their coexistence. The results obtained herein reveal that the complex bifurcations of equilibria, periodic behaviours and the chaotic motions of the presented system associate respectively with "resting", "calm" and "complex" flight. All the results related to the perturbed system are obtained by the fourth order Runge-Kutta method which ensures the accuracy of the computation. This study provides an additional

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A novel smooth and discontinuous oscillator with strong irrational nonlinearities

Cited by 46 publications

References 28 publications

Chaotic thresholds for the piecewise linear discontinuous system with multiple well potentials

Chaotic thresholds for the piecewise linear discontinuous system with multiple well potentials

A rotating disk linked by a pair of springs

A novel model of dipteran flight mechanism

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