ABSTRACT
The investigations made in Chapter 1 are closely connected with the study of the theory of preserving invariant tori of dynamical systems under perturbations. When the closure of a set of values in the phase space R k of the quasiperiodic solution y = y(t), t G R, y € R k, of the autonomous system
is considered, in many cases we get a surface homeomorphic to the m-dimensional torus Tm • The investigation of changes of the toroidal surface Tm for small changes of the right-hand of the autonomous system is quite difficult. However it is an important problem in the theory of multifrequence oscillations. The introduction of cyclic coordinates = (<pi, . . . , ifm) and the normal ones x € R n on the torus Tm , m + n = Jfc, allows the perturbed autonomous system to be written as
the linearization of which in the torus neighbourhood yields
Chapter 2 deals with the investigation of the system (2.0.1). Here the existence conditions for both the unique and non-unique Green function Go(r, ip) of the prob lem on invariant tori are established by means of Lyapunov functions of variable sign. The problems of the smoothness of the Green function with respect to phase variables <p are also treated as is its dependence on the parameters. In the author’s opinion, it is a result of interest that if the system
has many Green functions, then the extended system
has a unique 2n x 2n-dimensional Green function. This makes it possible to inves tigate the smoothness and dependence of the family of invariant tori of the system (2.0.1) on parameters. It is noted that similar results can also be obtained when the right-hand side of the system (2.0.1) is not periodic in ip. Essential differences are noticed in this case in the problems of smoothness of bounded invariant manifolds.
