Introduction

Abstract

We observe periodic motions, periodic variations in every field of science and everywhere in real life. We observe the revolution of the Moon around the Earth, the rotation of the Earth around its axis, the swinging movement of the pendulum of a clock, the wheel of a moving car, an engine in working order, the effect of alternating electric current; we observe the periodic ups and downs of economy, heart beat, respiration, etc. To be sure, some of these phenomena can better be described as “almost periodic motions”; still, even these can be approximated in a satisfactory way by periodic variations. If the phenomenon is characterised by some parameters, the so-called, phase coordinates, then the periodicity of the phenomenon is represented by the periodic variation of these phase coordinates, i.e. by a periodic motion in the phase space. A periodic motion in the phase space is a path that after some fixed time returns to its starting point and is continuing to do this in the whole future. This means that a periodic motion is represented by a closed path in the phase space. Behind such a periodic motion, usually there stands a natural (mechanical, physical, biological, etc.) or an economic law. This natural or economic, etc., law is, usually, represented by a differential equation which determines how the rate of change of the phase coordinates depends on their actual or past values, on the actual time and on other parameters. In this book we shall consider and call the “independent variable ” or one of the independent variables, time. We shall be concerned with variations periodic in time, though naturally there are important phenomena that show spatial periodicity or periodicity with respect to some other variable. We shall denote time by t if possible, and assume that either t ∈ R or t ∈ R ₊ = [0, ∞] or t ∈ I where I is some interval on the real line.