الفهرس 
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Abstract
Mathematics fields are the philosophy of sciences. There are several branches of mathematics some of them are ifferential equations, geometry, geometric topology and numerical analysis. Many mathematical models which attempt to interpret physical problems can be formulated in terms of the rate of change of one or more variables and as such naturally lead to differential equations. Furthermore, most differential equations cannot be integrated to give a solution in terms of elementary functions and it may be necessary to integrate them numerically. Therefore, there are an important connection between differential equations, numerical analysis, geometry and geometric topology. In this thesis, we deduce a relation between three fields, differential equations, geometry, and geometric topology. Some properties of a manifold will be discussed from viewpoint of differential equations and their solutions. The properties of some, geometric maps will be discussed using some restrictions on the differential equations and their solutions. The thesis consists of four chapters: Chapter 1 is an introductory chapter in which we introduce some preliminary concepts in differential equations, numerical analysis, some geometric transformations which affect the covering of chaotic 1 -manifolds will be discussed. . * V In chapter 2, we introduce a new connection between differential equations and some geometric transformations on 1-manifolds. The concepts of folding and unfolding on the 1 -manifold will be characterized .by using the differential equations. Some geometric transformations.