Periodic and Almost Periodic Motions

Abstract

By Def. 38.1 a motion described by p(t, a, t₀) is called periodic with period ω if for all t ≥ t₀ the relation

$$ {\rm{ }}(t{\rm{ }} + {\rm{ }}\omega ,{\rm{ }}a, {t_0}){\rm{ }} = {\rm{ }}p{\rm{ }}(t,{\rm{ }}a, {t_0}) $$

(71.1)

is satisfied. Periodic motions in R _n which are described by differential equations or difference equations are exceedingly important in practice: The motion of planets can be described by differential equations and so can the operating behavior of an electric motor or steam engine. This explains the great importance of the theory of periodic motions and the numerous publications in this area. Strictly speaking, however, most of the “periodic” motions are actually not periodic but almost periodic (cf. also sec. 73) and the purely periodic motion is only approached as a limit. Still the study of this type of motions is indispensible for understanding many phenomena.