Perturbation theory of an invariant torus of a non-linear system

Abstract

The problems of the linear theory considered in the previous chapter are also important for the general theory of invariant tori of non-linear systems. This deals with the invariant surface

$$h = u\left( \phi \right),{\text{ }}\phi \in {{\mathcal{T}}_{m}},{\text{ }}u \in C\left( {{{\mathcal{T}}_{m}}} \right)$$

((1.1))

of the system of equations

$$\frac{{d\phi }}{{dt}} = a\left( {\phi ,h} \right),{\text{ }}\frac{{dh}}{{dt}} = F\left( {\phi ,h} \right)$$

((1.2))

the right hand side of which is defined, continuous in ø, h in the domain

$$ \parallel h\parallel \leqslant d, \phi \in {\mathcal{T}_m} $$

and periodic in ø _v (v = 1,…, m) with period 2π. The study of system (1.2) is complicated by the fact that the functions a and F are non-linear with respect to the variable h and are accessible by modern methods of investigation only within the framework of perturbation theory, where it is assumed that the quantities $\parallel a\left( {\phi ,h} \right) - a\left( {\phi ,0} \right)\parallel$ and $\parallel F\left( {\phi ,h} \right) - F\left( {\phi ,0} \right)\parallel$ are small in domain (1.3). Here progress is achieved by using iteration procedures that linearize the problem at each stage of the iteration.