Analysis of the High Order Terms on Periodic Solution Bifurcations of the Generalized Duffing Systems

2013 ◽  
Vol 718-720 ◽  
pp. 1705-1710
Author(s):  
Shi Dong Chen ◽  
Zhi Qiang Wu
Keyword(s):  
Periodic Solution ◽  
Order Term ◽  
Singularity Theory ◽  
Multiple Scale ◽  
High Order ◽  
Duffing Equation ◽  
High Order Term ◽  
Bifurcation Equation ◽  
Bifurcation Diagrams

This paper focuses on the effectsof the high order term in Duffing equation. Firstly the averaging equation andthe bifurcation equation are deduced through the multiple scale method.Secondly, the transition sets and several different bifurcation diagrams areobtained based on the singularity theory. The result shows that the high order term induces richer bifurcationcharacteristics.

scholarly journals Arithmetic expressions optimisation using dual polarity property

2003 ◽  
Vol 1 (1) ◽  
pp. 71-80 ◽  
Author(s):  
Dragan Jankovic ◽  
Radomir Stankovic ◽  
Claudio Moraga
Keyword(s):  
Order Term ◽  
High Order ◽  
High Order Term ◽  
Experimental Results ◽  
Simple Relationship ◽  
Dual Polarity ◽  
The Given ◽  
Arithmetic Expressions

A method for optimisation of fixed polarity arithmetic expressions (FPAEs) based on dual polarity is proposed. The method exploits a simple relationship between two FPAEs for dual polarities. It starts from the zero polarity FPAE of the given function and calculates all FPAEs using the dual polarity route. Using one-bit check carries out conversion from one FPAE to another. Each term in an FPAE is processed by the proposed processing rule. Terms, which differ in a single position, can be substituted by a high order term (cube). Experimental results show efficiency of proposed method.

A high order term-by-term stabilization solver for incompressible flow problems

2012 ◽  
Vol 33 (3) ◽  
pp. 974-1007 ◽  
Author(s):  
T. C. Rebollo ◽  
M. G. Marmol ◽  
V. Girault ◽  
I. S. Munoz
Keyword(s):  
Incompressible Flow ◽  
Order Term ◽  
High Order ◽  
High Order Term ◽  
Flow Problems

Theoretical Analysis and Engineering Implications for the High Order Term Solutions of the Mode II Type Creep Crack

2017 ◽  
Author(s):  
Yanwei Dai ◽  
Yinghua Liu ◽  
Yuh J. Chao
Keyword(s):  
Theoretical Analysis ◽  
Plane Strain ◽  
Order Term ◽  
Crack Tip ◽  
High Order ◽  
High Order Term ◽  
Mode I ◽  
Mode Ii ◽  
Mode I Crack ◽  
Creep Crack

The high order term (HOT) solutions of crack tip field should be taken into account in engineering applications due to their potential influences on the fracture toughness as well as the crack growth rate. Towards this end, the high order asymptotic analysis of mode II crack in a power-law creeping material under the plane strain condition is presented theoretically in this paper. By comparing among the three order term solution, HRR field and finite element calculations, it can be found that the proposed three order term solution is able to characterize the full field of mode II creep crack tip accurately. The high order term solution of mode II creep crack relies on the creep exponent, which is different from that of mode I crack case. There are two independent terms among these three terms of asymptotic solution for mode II creep crack, which indicates that there exists the constraint effect for the plane strain mode II creep crack though the effect of high order term may not be as significant as the mode I crack case. Based on the theoretical analysis, a modified time-dependent failure assessment diagram (TDFAD) for mode II creep crack is proposed by considering the high order term solution.

Reducibility of a class of nonlinear quasi-periodic systems with Liouvillean basic frequencies

2020 ◽  
pp. 1-38
Author(s):  
DONGFENG ZHANG ◽  
JUNXIANG XU
Keyword(s):  
Periodic Solution ◽  
Small Parameter ◽  
Small Perturbation ◽  
Periodic System ◽  
Order Term ◽  
High Order ◽  
Periodic Systems ◽  
Elliptic Type ◽  
Constant Matrix ◽  
Image Position

In this paper we consider the following nonlinear quasi-periodic system: $$\begin{eqnarray}{\dot{x}}=(A+\unicode[STIX]{x1D716}P(t,\unicode[STIX]{x1D716}))x+\unicode[STIX]{x1D716}g(t,\unicode[STIX]{x1D716})+h(x,t,\unicode[STIX]{x1D716}),\quad x\in \mathbb{R}^{d},\end{eqnarray}$$ where $A$ is a $d\times d$ constant matrix of elliptic type,  $\unicode[STIX]{x1D716}g(t,\unicode[STIX]{x1D716})$ is a small perturbation with $\unicode[STIX]{x1D716}$ as a small parameter, $h(x,t,\unicode[STIX]{x1D716})=O(x^{2})$ as $x\rightarrow 0$ , and $P,g$ and $h$ are all analytic quasi-periodic in $t$ with basic frequencies $\unicode[STIX]{x1D714}=(1,\unicode[STIX]{x1D6FC})$ , where $\unicode[STIX]{x1D6FC}$ is irrational. It is proved that for most sufficiently small $\unicode[STIX]{x1D716}$ , the system is reducible to the following form: $$\begin{eqnarray}{\dot{x}}=(A+B_{\ast }(t))x+h_{\ast }(x,t,\unicode[STIX]{x1D716}),\quad x\in \mathbb{R}^{d},\end{eqnarray}$$ where $h_{\ast }(x,t,\unicode[STIX]{x1D716})=O(x^{2})~(x\rightarrow 0)$ is a high-order term. Therefore, the system has a quasi-periodic solution with basic frequencies $\unicode[STIX]{x1D714}=(1,\unicode[STIX]{x1D6FC})$ , such that it goes to zero when $\unicode[STIX]{x1D716}$ does.

Harmonic Resonance Frequency Factor of Spur Gear System and Oscillation Reduction Approaches

2007 ◽  
Author(s):  
Jianping Wang ◽  
Pengfei Li ◽  
Ziying Wu ◽  
Minghong Zhang
Keyword(s):  
Parametric Excitation ◽  
Linear Time ◽  
Primary Resonance ◽  
Frequency Factor ◽  
Multiple Scale ◽  
Transmission Error ◽  
Spur Gear ◽  
High Order ◽  
Time Varying ◽  
Harmonic Resonance

In this study, a non-linear time-varying dynamic model of a spur gear pair system is used to investigate the dynamic behavior of the system by means of multiple scale approach. Both time-varying stiffness, transmission error and tooth backlash clearance of the system are taken into account in the model. The mesh stiffness fluctuation is developed as high order Fourier series and tooth backlash clearance is fitted by high order polynomial function. The frequency factors of the system are investigated and the frequency-response equations at the case of internal and external excitation, parametric excitation and combined excitation are obtained. The peak value of the amplitude of the primary resonance, super and sub harmonic resonance and combination harmonic under internal, external and parametric excitation are researched. The approaches of vibration reduction are investigated. Finally an example is investigated using the presented process and the results indicate the sensitivity and correctness of the presented analysis approaches.

scholarly journals Existence and global stability of periodic solution for delayed discrete high-order Hopfield-type neural networks

2005 ◽  
Vol 2005 (3) ◽  
pp. 281-297 ◽  
Author(s):  
Hong Xiang ◽  
Ke-Ming Yan ◽  
Bai-Yan Wang
Keyword(s):  
Neural Networks ◽  
Periodic Solution ◽  
Global Stability ◽  
A Priori Estimates ◽  
A Priori ◽  
Sufficient Conditions ◽  
Coincidence Degree ◽  
Degree Theory ◽  
High Order ◽  
Priori Estimates

By using coincidence degree theory as well as a priori estimates and Lyapunov functional, we study the existence and global stability of periodic solution for discrete delayed high-order Hopfield-type neural networks. We obtain some easily verifiable sufficient conditions to ensure that there exists a unique periodic solution, and all theirs solutions converge to such a periodic solution.

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