On the Tk→Tk+1 Bifurcation Problem

Abstract

Publisher Summary This chapter analyzes the T k → T k+1 bifurcation problem. The chapter discusses about G. R. Sell, A. Chenciner, and G. Iooss who presented results describing a Hopf-type bifurcation of a k ‑dimensional into a ( k +1)-dimensional invariant torus within a one-parameter family of differential equations or maps. The chapter gives an extension of these results because it is not assumed that (i) the normal spectrum of the k -dimensional torus crosses from R – to R + in a transversal fashion, and (ii) the quadratic and cubic terms are the only significant nonlinearities.. An outline of the main ideas is presented in the chapter, and emphasis is given on the derivation of the setup used for this kind of bifurcation problem. The chapter also investigates what the corresponding hypotheses for the T k → T k+1 bifurcation are that allow to answer questions involving the number of branches of ( k +l)-dimensional tori bifurcating from a given k -dimensional torus, their direction of bifurcation, and their stability properties.