ABSTRACT
Throughout this chapter we consider a map with one parameter. These results also apply to differential equations near a periodic orbit by considering the Poincare´ map. We write fµ(x) = f(x, µ), where µ ∈ R and x ∈ Rn. We proved in Theorem V.6.4 that if fµ0(x0) = x0 is a fixed point and 1 is not an eigenvalue of D(fµ0)x0 , then the fixed point can be continued for values of the parameter µ near µ0. This is a non-bifurcation result. Notice that the tool we used to show that the fixed point could be continued in this case is the Implicit Function Theorem. We repeatedly use this theorem to study the bifurcations considered in this chapter.
