Abstract: In this talk we will follow the paper "Introduction to bifurcation theory" by John David Crawford. We will focus on the basic concepts, including some definitions, basic questions, and the Hartman-Grobman Theorem. This theorem tells us the relation between the linear dynamics and the nonlinear dynamics under the "Hyperbolicity" hypothesis. We will discuss the case about the loss of the "Hyperbolicity". Finally, we will give the definition of invariant manifolds and some examples.