The limit case response of the archetypal oscillator for smooth and discontinuous dynamics

Cao, Qingjie; Wiercigroch, Marian; Pavlovskaia, Ekaterina; Grebogi, Celso; Thompson, J. M. T.

doi:10.1016/j.ijnonlinmec.2008.01.003

Cited by 95 publications

(42 citation statements)

References 31 publications

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“…1 Schematics of the snap-through mechanism Therefore, both stable equilibriums are symmetrical about the horizontal unstable equilibrium as the previous works [2][3][4][5][6][7][8] found. Introduce the dimensionless displacement, the dimensionless time and a dimensionless control parameter…”

Section: Dynamical Equationsupporting

confidence: 56%

“…In the previous works [2][3][4][5][6][7][8], the term associated with the weight of the mass, namely mg, was neglected. That is,…”

Section: Dynamical Equationmentioning

confidence: 99%

“…In above-mentioned works [2][3][4][5][6][7][8], it was concluded that the snap-through mechanism possesses two stable equilibriums symmetrical to a horizontal configuration as an unstable equilibrium. The following analysis will demonstrate that the conclusion holds only for exceptionally small mass.…”

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confidence: 99%

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Equilibriums and their stabilities of the snap-through mechanism

Chen

2015

Arch Appl Mech

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A snap-through mechanism is investigated with the focus on its equilibriums and their stabilities. The snap-through mechanism consists of two inclined linear springs connected to a damper and a mass. The influence of the gravitation effects of the mass on the equilibriums, which has been neglected in all previous works, is taken into account in the investigation. The equilibrium equation is derived from the dynamical equation. The Lyapunov linearized stability theory is applied to determine the stability of the equilibrium. A dimensionless control parameter is introduced. For the sufficient small control parameter or the sufficient larger inclination angle, there are four equilibriums, two stable and two unstable. If the control parameter is larger than the critical value or the absolute value of the angle is smaller than a critical value, a stable and an unstable equilibriums disappear via a saddle-node bifurcation.

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Section: Dynamical Equationsupporting

confidence: 56%

“…In the previous works [2][3][4][5][6][7][8], the term associated with the weight of the mass, namely mg, was neglected. That is,…”

Section: Dynamical Equationmentioning

confidence: 99%

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Equilibriums and their stabilities of the snap-through mechanism

Chen

2015

Arch Appl Mech

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“…The efficiency of the proposed method have been presented by using numerical simulations, which clearly demonstrates the predicated periodic solutions and chaotic attractors. The future study on the complicated nonlinear dynamics of this cylindrical pendulum is being carried out by the current authors in two aspects: the first is the chaotic behaviors of discontinuous regime (Cao et al, 2008) and the second is the global bifurcations (Han M.A., 1999).…”

Section: Discussionmentioning

confidence: 99%

Dynamic Behavior Analysis of a Rotating Smooth and Discontinuous Oscillator with Irrational Nonlinearity

Han¹,

Liu²

2018

MAS

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This paper focuses on a novel rotating mechanical model which provides a cylindrical example of transition from smooth to discontinuous dynamics. The remarkable feature of the proposed system is a cylindrical dynamical system with strongly irrational nonlinearity exhibiting both smooth and discontinuous characteristics due to the geometry configuration. By using nonlinear dynamical technique, the unperturbed dynamics of the proposed system are studied including the irrational restoring force, stability of equilibria, Hamiltonian function and phase portraits. Note that a pair of double heteroclinic-like orbits connecting two non-standard saddle points are proposed in discontinuous case. For the perturbed system, we introduce a cylindrical approximate system for which the analytical solutions can be obtained successfully to reflect the nature of the original system without barrier of the irrationalities. Melnikov method is employed to detect the chaotic thresholds for the double heteroclinic orbits under the perturbation of viscous damping and external harmonic forcing in smooth regime. Finally, numerical simulations show the efficiency of the proposed method and demonstrate the predicated periodic solution and chaotic attractors. It is found that a good degree of correlation is demonstrated in the bifurcation diagram, the phase portraits of periodic solution, the chaotic attractor' structures and the Lyapunov characteristics between the original system and approximate system.

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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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The limit case response of the archetypal oscillator for smooth and discontinuous dynamics

Cited by 95 publications

References 31 publications

Equilibriums and their stabilities of the snap-through mechanism

Equilibriums and their stabilities of the snap-through mechanism

Dynamic Behavior Analysis of a Rotating Smooth and Discontinuous Oscillator with Irrational Nonlinearity

A novel model of dipteran flight mechanism

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