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A Survival Kit in Phase Plane Analysis: Some Basic Models and Problems

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Nonlinear Water Waves

Part of the book series: Lecture Notes in Mathematics ((LNMCIME,volume 2158))

Abstract

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 Liénard 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 results of a joint paper with Constantin (Boyce and DiPrima, Elementary Differential Equations and Boundary Value Problems, Wiley, New York, 1977) in which, as far as we know, for the first time phase plane analysis was used in this kind of problems.

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Acknowledgements

I thank Dr. Francesco Mugelli for his friendly help in the writing of this note.

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Authors and Affiliations

  1. Department of Mathematics and Informatics “Ulisse Dini”, University of Florence, Florence, Italy

    Gabriele Villari

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  1. Gabriele Villari
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Correspondence to Gabriele Villari .

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Editors and Affiliations

  1. Faculty of Mathematics, University of Vienna, Vienna, Austria

    Adrian Constantin

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Villari, G. (2016). A Survival Kit in Phase Plane Analysis: Some Basic Models and Problems. In: Constantin, A. (eds) Nonlinear Water Waves. Lecture Notes in Mathematics(), vol 2158. Springer, Cham. https://doi.org/10.1007/978-3-319-31462-4_4

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