Contact Points And Their Bifurcations In Several Singularly Perturbed Systems

Since Mandelbrot[75]created fractal geometry,this new branch of mathematics has gradually developed into an important research field in modern mathematics.Fractal geometry provides new ideas,methods,and techniques for describing and studying irregular geometric figures.It has important theoretical significance and application value for research in the fields of mathematics,physics,biology,medicine,and engineering.In 1981,Hutchinson[56]systematically studied the iterated function system(IFS for short),laying the theoretical foundation of IFS.Since then,mathematicians have used the IFS theory to construct and analyze many typical fractal sets and fractal measures.The fractal set generated by IFS plays an important role in the study of fractal geometry,and the important means of analyzing fractal sets is the study of fractal measures.In recent years,the issue of the algebraic,analytical,and geometric directions of IFSs and their fractal sets has attracted extensive attention.This thesis will focus on three kinds of problems related to IFSs and their fractal sets.The first kind of problem is to study the finiteness and infiniteness of the groups of isometries of planar IFS fractals.Let K be the attractor of any IFS on the plane,and consider all isometry groups on K.If K is a generalized self-affine(or self-similar)set,we prove that the groups of isometries of K are finite groups.If K is a bi-Lipschitz IFS fractal,we give the necessary and sufficient conditions for the infiniteness of the groups of isometries of K.For the finite case,we also calculate the size of the groups of isometries of self-similar sets.The second kind of problem is to consider the continuous dependence of a class of special self-affine sets and self-affine measures on parameters.Through in-depth analysis of the relation between the convergence of self-affine sets T(An,Dn)and selfaffine measures μAn,Dn,Pn and the convergence of the sequence {An,Dn,Pn}n=1 ∞,we generalize the results of references[32]and further obtain some new results.The third kind of problem is to analyze the geometric properties of a class of deformed Koch curves on the complex plane.According to the definition of the IFS of this class of deformed Koch curve Ka,r,θ,we discuss when Ka,r,θ satisfies the open set condition and the post-critically finite condition,and obtain a necessary and sufficient condition for Ka,r,θ is a simple curve.