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Lunar landing trajectory design based on invariant manifold

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Applied Mathematics and Mechanics(English Edition) 2007年第2期28卷 201-207页

作者：龚胜平李俊峰宝音贺西高云峰 School of Aerospace Tsinghua University

The low-energy lunar landing trajectory design using the invariant manifolds of restricted three-body problem is studied. Considering angle between the ecliptic plane and lunar orbit plane, the four-body problem of sun-earth-moon-spacecraft is divided into two three-body problems, the sun-earth-spacecraft in the ecliptic plane and the earth-moon-spacecraft in the lunar orbit plane. Using the orbit maneuver at the place where the two planes and the invariant manifolds intersect, a general method to design low energy lunar landing trajectory is given. It is found that this method can save the energy about 20% compared to the traditional Hohmann transfer trajectory. The mechanism that the method can save energy is investigated in the point of view of energy and the expression of the amount of energy saved is given. In addition, some rules of selecting parameters with respect to orbit design are provided. The method of energy analysis in the paper can be extended to energy analysis in deep space orbit design.

关键词： three-body problem Lagrange point Halo orbit invariant manifold lunarlanding trajectory

THE invariant manifold METHOD AND THE CONTROLLABILITY OF NONLINEAR CONTROL SYSTEM

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Applied Mathematics and Mechanics(English Edition) 2000年第11期21卷 1320-1330页

作者：杨柳 Department of Chemical Processing Machinery Sichuan UniversityChengdu 610065P R China

The problem of controllability of nonlinear control system is a significant field which has an extensive prospect of application. A. M. Kovalev of Ukraine Academy of Science applied the oriented manifold method developed in dynamics:of rigid body to nonlinear control system for the first time,and obtained a series of efficient results. Based on Kovalev's oriented manifold method, firstly, by invariant manifold method the problem of controllability of nonlinear control system was studied and the necessary condition of the controllability of a kind of affine nonlinear system was given out. Then the realization of the necessary condition wars discussed. At last, the motion of a rigid body with two rotors,vas, investigated and the necessary condition which is satisfied by this system,vas proved.

关键词： nonlinear control system controllability oriented manifold invariant manifold basic system dynamics of rigid body?

invariant manifoldS AND THEIR STABILITY IN A THREE－DIMENSIONAL MEASURE－PRESERVING MAPPING SYSTEMS

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Applied Mathematics and Mechanics(English Edition) 1995年第10期16卷 943-950页

作者：刘杰 Center for Nonlinear Studies Institute of Applied Physies and Computational Mathematics Bejing P. R. Clina

In the paper researches on a three-dimensional measure-preserving mappingsystem are made,which is the three-dimensional extension of the Keplerian *** formal series method the expressions of the invariant curves and i... 详细信息

关键词： measure-preserving mapping chaos invariant manifold stability

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Smoothness of invariant manifolds and foliations for in nite dimensional random dynamical systems--Dedicated to Professor Shantao Liao

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Science China Mathematics 2020年第9期63卷 1877-1912页

作者： Jun Shen Kening Lu Weinian Zhang Department of Mathematics Sichuan UniversityChengdu 610064China Department of Mathematics Brigham Young UniversityProvoUT 84602USA

In this paper,we investigate the smoothness of invariant manifolds and foliations for random dynamical systems with nonuniform pseudo-hyperbolicity in Hilbert *** discuss on the effect of temperedness and the spectral gaps in the nonuniform pseudo-hyperbolicity so as to prove the existence of invariant manifolds and invariant foliations,which preserve the CN,τ;(ω)Holder smoothness of the random system in the space variable and the measurability of the random system in the sample ***,we also prove that the stable foliation is CN-1;(ω)in the base point.

关键词： random dynamical system invariant manifold invariant foliation pseudo-hyperbolicity measurability

Comparison of invariant manifolds for Model Reduction in Chemical Kinetics

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Communications in Computational Physics 2007年第5期2卷 964-992页

作者： Eliodoro Chiavazzo Alexander N.Gorban Iliya V.Karlin Aerothermochemistry and Combustion Systems Laboratory LAV ETH Zurich8092 ZurichSwitzerland Department of Mathematics University of LeicesterLeicester LE17RHUK.

A modern approach to model reduction in chemical kinetics is often based on the notion of slow invariant *** goal of this paper is to give a comparison of various methods of construction of slow invariant manifolds using a simple Michaelis-Menten catalytic *** explore a recently introduced Method of invariant Grids(MIG)for iteratively solving the invariance *** initial approximations for the grid are considered such as Quasi Equilibrium manifold,Spectral Quasi Equilibrium manifold,Intrinsic Low Dimensional manifold and Symmetric Entropic Intrinsic Low Dimensional *** invariant manifold was also computed using the Computational Singular Perturbation(CSP)method.A comparison between MIG and CSP is also reported.

关键词： Chemical kinetics model reduction invariant manifold entropy nonlinear dynamics mathematical modeling.

THE invariant manifoldS FOR A PERTURBED QUINTIC-CUBIC SCHRDINGER EQUATION

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Acta Mathematica Scientia 2004年第4期24卷 536-548页

作者：陈翰林郭柏灵 School of Science Southwest University of Science and Technology Mianyang 621002 China Institute of Applied Physics and Computational Mathematics Beijing 100088 Chinan this paper the authors apply the analytical method to establish the persistence of invariant manifolds for certain perturbation of the quintic-cubic Schrodinger equation under even periodic boundary conditions.

In this paper, the authors apply the analytical method to establish the persistence of invariant manifolds for certain perturbation of the quintic-cubic Schrodinger equation under even periodic boundary conditions.

关键词： invariant manifold persistence perturbation

Orbits Heteroclinic to invariant manifolds

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Acta Mathematica Sinica,English Series 1996年第4期12卷 372-373,375-378页

作者： Zhu Deming Department of Mathematics East China Normal University Shanghai, 200062 China Department of Mathematics East China Normal University Shanghai China

Using the theory of invariant manifolds, we give local expressions of the stable and unstable manifolds for normally hyperbolic invariant tori, and study the existence of transverse orbits heteroclinic to hyperbolic i... 详细信息

关键词： invariant manifold Exponential dichotomy Heteroclinic orbit Transversality

invariant SUB-manifoldS AND MODES OF NONLINEAR AUTONOMOUS SYSTEMS

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Applied Mathematics and Mechanics(English Edition) 1998年第7期19卷 687-693页

作者：赵国景魏建国 Peking Graduate School China University of Mining and Technology 100083 Beijing P.R. China College of Urban and Rural Construction Agricultural University of Hebei 071001 Baoding P.R. China

A definition of the modes of a nonlinear autonomous system was developed. The existence conditions and orbits' nature of modes are given by using the geometry theory of invariant manifolds that include stable manifold theorem, center maifold theorm and sub-center manifold theorem. The Taylor series expansion was used in order to approach the sub-manifolds of the modes and obtain the motions of the mods on the manifolds. Two examples were given to demonstrate the applications.

关键词： invariant manifold mode nonlinear system

The Role of Thermodynamics in Model Reduction when Using invariant Grids

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Communications in Computational Physics 2010年第9期8卷 701-734页

作者： Eliodoro Chiavazzo Iliya V.Karlin Alexander N.Gorban Aerothermochemistry and Combustion Systems Lab ETH Zurich8092 ZurichSwitzerland School of Engineering Sciences University of SouthamptonSO171BJ SouthamptonUnited Kingdom Department of Mathematics University of LeicesterLE17RH LeicesterUnited Kingdom

In the present work,we develop in detail the process leading to reduction of models in chemical kinetics when using the Method of invariant Grids(MIG).To this end,reduced models(invariant grids)are obtained by refining initial approximations of slow invariant manifolds,and used for integrating smaller and less stiff systems of equations capable to recover the detailed description with high ***,we clarify the role played by thermodynamics in model reduction,and carry out a comparison between detailed and reduced solutions for a model hydrogen oxidation reaction.

关键词： Chemical kinetics model reduction invariant manifold entropy