ABSTRACT
This chapter discusses the development of the blowing-up techniques to show that under certain conditions on a differential equation there is a bifurcation to an attracting invariant sphere. As it turns out, this result is extremely useful in analyzing the formation of heteroclinic cycles. Appropriate hyperbolicity conditions hold in the normal directions, it is again possible to prove persistence of the invariant sphere when we allow in higher order terms. The chapter presents the invariant sphere theorem in the context of real vector spaces. It is also possible to repeat the constructions in the framework of complex vector spaces and Hermitian inner products. The chapter concludes with some exercises that illustrate this approach.
