Autonomous Systems in the Plane

Abstract

There is an old joke well known, probably, in most countries of the world. In the middle of the night the policeman sees a drunkard on all fours at the foot of a lamppost on the main street and asks him: ‘What are you doing here, friend’? The man replies: ‘I am looking for my lost purse, officer’. ‘Have you lost it here at this lamp?’ ‘No, I have lost it in that side-street’. ‘Then why aren’t you looking for it there? ’ ‘I can’t, officer, it is too dark over there’. ... No doubt, this is one of the reasons why two dimensional systems have been treated so extensively: there is some clarity in the two dimensional plane which disappears as the dimension of the system is increased. The clarity is mainly due to “Jordan’s Theorem” according to which a simple closed Jordan curve divides the plane into two disconnected components. Jordan curves do not generate a similar division of three or higher dimensional spaces, and that is the main reason why the existence problem of closed trajectories, i.e. periodic solutions, is much more difficult in dimensions higher than two than on the plane. Therefore, one can say more about periodic solutions of two dimensional systems than about those of higher dimensional ones, and one may illustrate general situations relatively easily on the former. One of the purposes of this chapter is the introduction of those classical two dimensional autonomous systems (Van der Pol, Liénard, Duffing, Volterra) that will serve as standard references in the general theory.