On the Theory of Quasiperiodic Motions

Abstract

Introduction. The object of this paper is to discuss some recent developments in the theory of almost periodic solutions of ordinary differential equations. This concerns particularly the theory of nonlinear oscillations which to a large extent has been restricted to the study of periodic solutions. These can be viewed as special cases of almost periodic solutions having merely one basic frequency. As soon as one attempts to find almost periodic solutions with more frequencies for nonlinear differential equations one runs into subtle problems which are very familiar to workers in astronomical dynamics. We mention one of the typical difficulties occurring in perturbation theory: If for a system containing a small parameter, say E, an almost periodic solution is known for e = 0 one would like to find an almost periodic solution for small values of e. This may involve certain nondegeneracy restrictions. However, for almost periodic solutions (i.e., at least 2 independent basic frequencies) such almost periodic solutions need not exist-no matter how small e-as one sees even for the flow on a torus (see ?5). However, the availability of such a perturbation theory is an absolute necessity if the concept of an almost periodic solution is to have any physical significance. While under general small perturbation of the differential equation almost periodic solutions may change their character or may disappear, they may persist for more restricted classes of differential equations. We point to the fundamental result by Kolmogorov [13], [14] and Arnol'd [3] who show that for Hamiltonian systems of N degrees of freedom which are close to integrable ones such a perturbation theory of almost periodic solutions can be established. The special feature of the Hamiltonian character of the differential equation is responsible for the of almost periodic solutions under small perturbations. This result has important implications and, in fact, Arnol'd [2] established a large class of almost periodic solutions for the n-body problem. Our object is to discuss a perturbation theory for quasiperiodic solutions' for other classes of differential equations so as to reveal the properties of the differential equations which make the permanence of quasiperiodic solutions possible. Even for Hamiltonian systems there remains an interesting problem: So far only quasiperiodic solutions of N basic frequencies have been established, where N is the number of degrees of freedom. We will show the number of basic frequencies n of a quasiperiodic solution satisfies n _ N. The extreme case n = 1 corresponds to periodic solutions for which the perturbation theory is well known, and was