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The Second Part of Hilbert’s Sixteenth Problem

In: Dynamical Systems with Applications using MATLAB®

Author

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  • Stephen Lynch

    (Manchester Metropolitan University School of Computing, Mathematics & Digital Technology, Department of Computing and Mathematics)

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.

Suggested Citation

  • Stephen Lynch, 2014. "The Second Part of Hilbert’s Sixteenth Problem," Springer Books, in: Dynamical Systems with Applications using MATLAB®, edition 2, chapter 0, pages 355-376, Springer.
    • Handle: RePEc:spr:sprchp:978-3-319-06820-6_17
    DOI: 10.1007/978-3-319-06820-6_17
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