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Bifurcation and chaos analysis for aeroelastic airfoil with freeplay structural nonlinearity in pitch

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Chinese Physics B 2010年第3期19卷 217-226页

作者：赵德敏张琪昌 Department of Mechanics School of Mechanical EngineeringTianjin University State Key Laboratory of Engines Tianjin University

The dynamics character of a two degree-of-freedom aeroelastic airfoil with combined freeplay and cubic stiffness nonlinearities in pitch submitted to supersonic and hypersonic flow has been gaining significant attention. The poincare mapping method and Floquet theory are adopted to analyse the limit cycle oscillation flutter and chaotic motion of this system. The result shows that the limit cycle oscillation flutter can be accurately predicted by the Floquet multiplier. The phase trajectories of both the pitch and plunge motion are obtained and the results show that the plunge motion is much more complex than the pitch motion. It is also proved that initial conditions have important influences on the dynamics character of the airfoil system. In a certain range of airspeed and with the same system parameters, the stable limit cycle oscillation, chaotic and multi-periodic motions can be detected under different initial conditions. The figure of the poincare section also approves the previous conclusion.

关键词： airfoil flutter bifurcation and chaos freeplay nonlinearity poincare map

Optimal and Memristor-Based Control of A Nonlinear Fractional Tumor-Immune Model

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Computers, Materials & Continua 2021年第6期67卷 3463-3486页

作者： Amr M.S.Mahdy Mahmoud Higazy Mohamed S.Mohamed Department of Mathematics College of ScienceTaif UniversityTaif21944Saudi Arabia Department of Mathematics Faculty of ScienceZagazig UniversityZagazig44519Egypt Department of Physics and Engineering Mathematics Faculty of Electronic engineeringMenoufia UniversityMenouf32952Egypt Department of Mathematics Faculty of ScienceAl-Azher UniversityNasr City11884Egypt

In this article,the reduced differential transform method is introduced to solve the nonlinear fractional model of Tumor-Immune.The fractional derivatives are described in the Caputo sense.The solutions derived using this method are easy and very accurate.The model is given by its signal flow diagram.Moreover,a simulation of the system by the Simulink of MATLAB is given.The disease-free equilibrium and stability of the equilibrium point are calculated.Formulation of a fractional optimal control for the cancer model is calculated.In addition,to control the system,we propose a novel modification of its model.This modification is based on converting the model to a memristive one,which is a first time in the literature that such idea is used to control this type of diseases.Also,we study the system’s stability via the Lyapunov exponents and poincare maps before and after control.Fractional order differential equations(FDEs)are commonly utilized to model systems that have memory,and exist in several physical phenomena,models in thermoelasticity field,and biological paradigms.FDEs have been utilized to model the realistic biphasic decline manner of elastic systems and infection of diseases with a slower rate of change.FDEs are more useful than integer-order in modeling sophisticated models that contain physical phenomena.

关键词： RDTM tumor-immune optimal control caputo derivative signal flow simulink disease-free equilibrium stability memristive lyapunov exponents poincare map

Topological horseshoe in nonlinear Bloch system

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Chinese Physics B 2010年第12期19卷 105-108页

作者：樊庆菊 Department of Control Science and Engineering Huazhong University of Science and Technology Department of Statistics School of ScienceWuhan University of Technology

This paper demonstrates rigorous chaotic dynamics in nonlinear Bloch system by virtue of topological horseshoe and numerical method. It considers a properly chosen cross section and the corresponding poincare map, and shows the existence of horseshoe in the poincare map. In this way, a rigorous verification of chaos in the nonlinear Bloch system is presented.

关键词： Bloch equation chaos topological horseshoe poincare map

Lyapunov exponent calculation of a two-degree-of-freedom vibro-impact system with symmetrical rigid stops

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Chinese Physics B 2011年第4期20卷 123-131页

作者：李群宏谭洁燕 College of Mathematics and Information Science Guangxi University

A two-degree-of-freedom vibro-impact system having symmetrical rigid stops and subjected to periodic excitation is investigated in this paper. By introducing local maps between different stages of motion in the whole impact process, the poincare map of the system is constructed. Using the poincare map and the Gram Schmidt orthonormalization, a method of calculating the spectrum of Lyapunov exponents of the above vibro-impact system is presented. Then the phase portraits of periodic and chaotic attractors for the system and the corresponding convergence diagrams of the spectrum of Lyapunov exponents are given out through the numerical simulations. To further identify the validity of the aforementioned computation method, the bifurcation diagram of the system with respect to the bifurcation parameter and the corresponding largest Lyapunov exponents are shown.

关键词： vibro-impact system poincare map Gram-Schmidt orthonormalization Lyapunov exponent

Novel four-dimensional autonomous chaotic system generating one-,two-,three- and four-wing attractors

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Chinese Physics B 2011年第11期20卷 151-158页

作者：余飞王春华尹晋文徐浩 College of Information Science and Engineering Hunan University

In this paper, we propose a novel four-dimensional autonomous chaotic system. Of particular interest is that this novel system can generate one-, two, three- and four-wing chaotic attractors with the variation of a single parameter, and the multi-wing type of the chaotic attractors can be displayed in all directions. The system is simple with a large positive Lyapunov exponent and can exhibit some interesting and complicated dynamical behaviours. Basic dynamical properties of the four-dimensional chaotic system, such as equilibrium points, the poincare map, the bifurcation diagram and the Lyapunov exponents are investigated by using either theoretical analysis or numerical method. Finally, a circuit is designed for the implementation of the multi-wing chaotic attractors. The electronic workbench observations axe in good agreement with the numerical simulation results.

关键词： multi-wing chaotic attractors four-dimensional chaotic system poincare map bifurcation diagram

Impact of lubricant traction coefficient on cage's dynamic characteristics in high-speed angular contact ball bearing

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Chinese Journal of Aeronautics 2017年第2期30卷 827-835页

作者： Zhang Wenhu Deng Sier Chen Guoding Cui Yongcun School of Mechatronics Engineering Northwestern Polytechnical University Xi'an 710071 China School of Mechatronics Engineering Henan University of Science and Technology Luoyang 471003 China Collaborative Innovation Center of Major Machine Manufacturing in Liaoning Dalian 116024 China

In this paper,the formulas of elasto-hydrodynamic traction coefficients of three Chinese aviation lubricating oils,4109,4106 and 4050,were obtained by a great number of elastohydrodynamic traction tests.The nonlinear dynamics differential equations of high-speed angular contact ball bearing were built on the basis of dynamic theory of rolling bearings and solved by Gear Stiff（GSTIFF） integer algorithm with variable step.The impact of lubricant traction coefficient on cage＇s dynamic characteristics in high-speed angular contact ball bearing was investigated,and poincare map was used to analyze the impact of three types of aviation lubricating oils on the dynamic response of cage＇s mass center.And then,the period of dynamic response of cage＇s mass center and the slip ratio of cage were used to assess the stability of cage under various working conditions.The results of this paper provide the theoretical basis for the selection and application of aviation lubricating oil.

关键词： Dynamic characteristic High-speed angular contact ball bearing Lubricant traction coefficient poincare map Stability

Representing dynamics in the eccentric Hill system using a neural network architecture

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Astrodynamics 2019年第4期3卷 301-324页

作者： Stijn De Smet Daniel JScheeres Jeffrey SParker Department of Aerospace Engineering Sciences The University of Colorado BoulderBoulderCO 80309USA Aduanced Space BoulderCo 80301USA

This paper demonstrates how artificial neural networks can be used to alleviate common problems encountered when creating a large database of Poincar´e map responses.A general architecture is developed using a combination of regression and classification feedforward neural networks.This allows one to predict the response of the Poincar´e map,as well as to identify anomalies,such as impact or escape.Furthermore,this paper demonstrates how an artificial neural network can be used to predict the error between a more complex and a simpler dynamical system.As an example application,the developed architecture is implemented on the Sun-Mars eccentric Hill system.Error statistics of the entire architecture are computed for both one Poincar´e map and for iterated maps.The neural networks are then applied to study the long-term impact and escape stability of trajectories in this system.

关键词： periapse poincare map artificial neural networks eccentric Hill system

Bifurcations of Fine 3-point-loop in Higher Dimensional Space

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Acta Mathematica Sinica,English Series 2005年第1期21卷 39-52页

作者： YinLaiJIN DeMingZHU DepartmentofMathematics EastChinaNormalUniversityShanghai200062P.R.China DepartmentofMathematics LinyiTeachers'UniversityLinyi276005P.R.China//DepartmentofMathematicsEastChinaNormalUniversityShanghai200062P.R.China

By using the linear independent fundamental solutions of the linearvariational equation along the heteroclinic loop to establish a suitable local coordinate system insome small tubular neighborhood of the heteroclinic loop, the Poincaré map is constructed to studythe bifurcation problems of a fine 3–point loop in higher dimensional space. Under some transversalconditions and the non–twisted condition, the existence, coexistence and incoexistence of2–point–loop, 1–homoclinic orbit, simple 1–periodic orbit and 2–fold 1–periodic orbit, and thenumber of 1–periodic orbits are studied. Moreover, the bifurcation surfaces and existence regionsare given. Lastly, the above bifurcation results are applied to a planar system and an insidestability criterion is obtained.

关键词： Local coordinates poincare map 1–homoclinic orbit 1–periodic orbit Bifurcation surface

Homoclinic and Periodic Orbits Arising Near the Heteroclinic Cycle Connecting Saddle-focus and Saddle Under Reversible Condition

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Acta Mathematica Sinica,English Series 2007年第8期23卷 1495-1504页

作者： De Ming ZHU Ying SUN Department of Mathematics East China Normal University Shanghai 200062 P. R. China School of Science University of Ji'nan Ji'nan 250022 P. R. China

In this paper, we study the dynamical behavior for a 4-dimensional reversible system near its heteroclinic loop connecting a saddle-focus and a saddle. The existence of infinitely many reversible 1-homoclinic orbits to the saddle and 2-homoclinic orbits to the saddle-focus is shown. And it is also proved that, corresponding to each 1-homoclinic （resp. 2-homoclinic） orbit F, there is a spiral segment such that the associated orbits starting from the segment are all reversible 1-periodic （resp. 2-periodic） and accumulate onto F. Moreover, each 2-homoclinic orbit may be also accumulated by a sequence of reversible 4-homoclinic orbits.

关键词： reversible condition homoclinic orbit poincare map periodic orbit saddle-focus