A survival kit in phase plane analysis: some basic models and problems

The aim of this note is to show how phase plane analysis is a strong tool for the study of a mathematical model, in view of its application in water waves theory. This because only in recent work such method was actually used in water waves theory and people working in this field area might be interested in a discussion of the basic ideas of phase plane analysis, which we call ”a survival kit”. In this light, at first we review some classical results in Dynamics of Population and Epidemiology, and then we investigate more carefully the phase portrait of the classical Liènard equation. In particular, starting from the Van Der Pol equation, the problem of existence and uniqueness of limit cycles will be treated and the methods used to attack this problem will be presented.Finally we come back to water waves theory and present in details the resultsof a joint paper with A. Constantin in which, as far as we know, for the first time phase plane analysis was used in this kind of problems