Instability of Dynamical Systems with Several Degrees of Freedom
Vladimir I. Arnold
Abstract
Recent progress in perturbation enables us to find many conditionally periodic motions in every nonlinear dynamical system which is close to an integrable system (see [ 1 , 2 ]. The stability of all the motions of the system follows from these results only when the dimension of the phase space is ≤ 4. The purpose of the present note is to give an example (3) of a system with a 5-dimensional phase space which satisfies all the conditions of [ 1 , 2 ] but is nonstable . * The secular changes I 2 in the system (3) have the velocity exp ( − 1 / ϵ ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003069515/b13987dd-0fcc-46d5-9637-5ef0a04fcee5/content/eq4763.tif"/> and consequently cannot be dealt with by any approxi mation of the classical theory of perturbations.