Weak Stability Boundary And Its Fractal Research

Weak stability boundary(WSB)is a method to distinguish the weakly stable orbit and unstable orbit,which initially proposed for finding the low energy transfer orbit.As a way of low energy transfer,the weak stability boundary has been applied effectly.According to weak stability boundary method,the regions of the initial position of weakly stable orbit continuously,and defined by the weak stability boundary.The research for weak stability boundary are of important meaming for both the design of low energy transfer orbit and the associated dynamics.The structure of weak stability boundary and corresponding weakly stable set(WSS)has been verified complex and difficult understood.Most of existing studies for weak stability boundary are about the calculation of structure and the relationship with other dynamical method such as Poincare section and invariant manifold,which barely involve the mathmetical explanation or a physical understanding.However,this does not affect the contribution of the weak stability boundary method to the practical application of low energy transfer orbit design.From the geometric point of view,the structure of weak stability boundary has the characteristics of fine structure which exists in fractal geometry.Analysed and interpreted by fractal theory,it may be reached a more clear understanding of the theory of weak stability boundary.The main purpose of this thesis is to analyze the complex structure related to weak stability boundary,to find out the variation of structure and its characteristic with respect to various parameters,and to attempt to understand the related dynamics.Start-ing from the definition of weakly stable orbit,this thesis attempts to find the trajectory patterns of related structures in weak stability boundary,and thus to provide reference for the practical application of low energy transfer orbit.The main achievements and innovations of this thesis are as follows:1.Considering the uncertainty of the structure of weak stability boundary,we think the weakly stable set is more accurate and adaptive than weak stability boundary to describe and analyse the structure.In the framework of the planar circular restricted three-body problem,we describe and analyse the structures of the weakly stable set with different conditions,and optimize the algorithm of the weakly stable orbit.With the proposed method of filtered weak stable set for different order,we summarise the trajectory patterns of weakly stable orbits;2.Considering the fine structure of weakly stable set,the fractal geometry is introduced to understand the weakly stable set.From the view of fractal,the intrinsic structure of weakly stable set representing the accurate weak stability boundary is obtained and deeply interpreted from global aspect;3.Based on the fractal idea,the method(named moving box)for describing the distri-bution of characteristics of local structures is proposed and the fractal image of weak stability boundary is constructed.In this image,we get not only the structure informa-tion of weakly stable set,but also the information of weakly stable orbits.In the sight of structure,the structure feature and the fine level of all the local regions in weakly stable set are described.In the sight of weakly stable orbit,it's obtained the distribution of aggregation degree and the level of sensitivity with the initial position changing;4.By the fractal analysis of weakly stable set,we obtain the method of obtaining all the complete trajectory patterns in the weakly stable set.Taking the case of Sun-Jupiter system as an example,the main local structures of weakly stable sets and their corresponding trajectory patterns of weak stable orbits are given.By the fractal analysis of weak stability boundary,the fractal method was con-firmed effectly description for the dynamics under the geometrical structure.In nonlin-ear celestial mechanics,there are many methods which map the dynamcs to geometry structure.Thus we hope that the fractal analysis of the weak stability boundary on this thesis can provide a reference for other similar studies which attempt to analyze the dynamics by the geometric structure.