This thesis is to study the \extit{adoptive local dynamics of strategy functions.} These functions have application in \extit{game theory} and \extit{adaptive dynamics} of many real world problems. In this direction we first define certain properties of singular points of strategy functions, namely, the \extit{evolutionary stable strategy} (ESS) , \extit{convergence stable strategy} (CvSS) and \extit{singular strategy}. Next an equivalence relation, strategy equivalence, is defined so that it naturally preserves these properties. Singularity theory is a well-known theory for the dynamics study of zeros of singular functions. However, it is not well-suited for our purpose. In fact, due to the differences in equivalence relations between these functions, the algebraic structures are completely different from the ones in the The first and second chapters deal with the literature on singularity and game theories, basic ideas, theorems and definitions. In the third chapter, we will study some local properties of smooth strategy functions near their singular points and will concentrate on some properties including zero sets, CvSS and ESS. Then, we will define the strategy equivalence. The strategy functions will be Chapter four will raise a new notion about strategy functions, that is, payoff functions. Furthermore, we discuss about finite determinacy codimensions sufficiency of strategy functions. Indeed, we answer the question for which perturbations like p, the payoff function g+tp is strategy equivalent to g, for all small t. This question leads us to define the \extit{restricted tangent space} by using an ideal structure. In the rest of this chapter, we will define the concept of \extit{codimension, unfolding and universal unfolding.} We shall use \extit{Nakayama’s Lemma} in chapter 5 to obtain some important The persistent bifurcation diagrams will be ltr"