ABSTRACT
In this chapter, the author examines the structure of the fixed point subspaces and the dynamics near the points (local dynamics). A central task for the physicist is to identify the ‘relevant’ dynamical variables of a natural system and to describe its time evolution. This identification process is a delicate matter that depends of a number of factors including the system, the observer and the models she/he has in mind. In the new coordinate system, the readers immediately realize that the motions in the different eigenspaces are independent but the motion inside each subspace may not be further decomposed into independent components by means of linear transformations. Readers familiar with quantum mechanics can notice that the oscillation theorem is directly related to the Bloch theorem describing the possible energies of an electron in a periodic potential. Finally, the authors discuss additively forced systems and parametrically forced systems; the latter already share some ‘non-trivial’ features with the nonlinear flows.
