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The Second Part of Hilbert’s Sixteenth Problem

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Dynamical Systems with Applications using MATLAB®

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Abstract

Aims and Objectives

• To describe the second part of Hilbert’s sixteenth problem.

• To review the main results on the number of limit cycles of planar polynomial systems.

• To consider the flow at infinity after Poincaré compactification.

• To review the main results on the number of limit cycles of Liénard systems.

• To prove two theorems concerning limit cycles of certain Liénard systems.

On completion of this chapter the reader should be able to

• state the second part of Hilbert’s sixteenth problem;

• describe the main results for this problem;

• compactify the plane and construct a global phase portrait which shows the behavior at infinity for some simple systems;

• compare local and global results;

• prove that certain systems have a unique limit cycle;

• prove that a limit cycle has a certain shape for a large parameter value.

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Authors and Affiliations

  1. Department of Computing and Mathematics, Manchester Metropolitan University School of Computing, Mathematics & Digital Technology, Manchester, UK

    Stephen Lynch

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  1. Stephen Lynch
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Lynch, S. (2014). The Second Part of Hilbert’s Sixteenth Problem. In: Dynamical Systems with Applications using MATLAB®. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-06820-6_17

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