Algunos elementos teóricos de la teoría de la bifurcación

Abstract

In this work we state some concepts about the theory of qualitative analysis of the systems of diferential equations or dinamic systems in the plane and bifurcation theory by using the theory from Hirch and Smale's book, and Perko's one to solve some applicativve examples proposed there. In section 2.3. we'll show and examine examples of explicit solutions to X' = F(X) where F is linear, there we'll characterize the diferent kinds of equilibrium points and by using matrix algebra we'll examine some methods for that. Then, in sections 2.4 and 2.5 we'll consider F nonlinear, and we'll show that F can be approximated by a linear function near the equlibrium points state two theorems to characterize the equilibrium points and the periodic solutions in certain regions. In section 2.6 we show a way to sketch the direction field by using nullclines. By last, in the chapter 3 we define the concept of bifurcations and show some examples which allow to increase the usefulness of the previous sections to give some applications of this theory.