Sharp upperbounds for the number of large amplitude limit cycles in polynomial Lienard systems

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May 2012, 32(5): 1465-1479. doi: 10.3934/dcds.2012.32.1465

Sharp upperbounds for the number of large amplitude limit cycles in polynomial Lienard systems

Freddy Dumortier ^1,

1.	Hasselt University, Campus Diepenbeek, Agoralaan gebouw D, B-3590 Diepenbeek

Received December 2010 Revised June 2011 Published January 2012

Abstract
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In [1] and [2] upperbounds have been given for the number of large amplitude limit cycles in polynomial Liénard systems of type $(m,n)$ with $m<2n+1$, $m$ and $n$ odd. In the current paper we improve the upperbounds from [1] and [2] by one unity, obtaining sharp results. We therefore introduce the "method of cloning variables" that might be useful in other cyclicity problems.

Keywords: Limit cycle, Liénard equation, Cyclicity., Heteroclinic connection.

Mathematics Subject Classification: 34C07, 34C37, 34D1.

Citation: Freddy Dumortier. Sharp upperbounds for the number of large amplitude limit cycles in polynomial Lienard systems. Discrete and Continuous Dynamical Systems, 2012, 32 (5) : 1465-1479. doi: 10.3934/dcds.2012.32.1465

References:

[1]	M. Caubergh and F. Dumortier, Hilbert's 16th problem for classical Liénard equations of even degree, Journal of Differential Equations, 244 (2008), 1359-1394. doi: 10.1016/j.jde.2007.11.011.
[2]	M. Caubergh, F. Dumortier and S. Luca, Cyclicity of unbounded semi-hyperbolic 2-saddle cycles in polynomial Liénard equations, Discrete and Continuous Dynamical Systems, 27 (2010), 963-980.
[3]	F. Dumortier, Compactification and desingularisation of spaces of polynomial Liénard equations, Journal of Differential Equations, 224 (2006), 296-313. doi: 10.1016/j.jde.2005.08.011.
[4]	F. Dumortier and C. Herssens, Polynomial Liénard equations near infinity, Journal of Differential Equations, 153 (1999), 1-29. doi: 10.1006/jdeq.1998.3543.

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References:

[1]	M. Caubergh and F. Dumortier, Hilbert's 16th problem for classical Liénard equations of even degree, Journal of Differential Equations, 244 (2008), 1359-1394. doi: 10.1016/j.jde.2007.11.011.
[2]	M. Caubergh, F. Dumortier and S. Luca, Cyclicity of unbounded semi-hyperbolic 2-saddle cycles in polynomial Liénard equations, Discrete and Continuous Dynamical Systems, 27 (2010), 963-980.
[3]	F. Dumortier, Compactification and desingularisation of spaces of polynomial Liénard equations, Journal of Differential Equations, 224 (2006), 296-313. doi: 10.1016/j.jde.2005.08.011.
[4]	F. Dumortier and C. Herssens, Polynomial Liénard equations near infinity, Journal of Differential Equations, 153 (1999), 1-29. doi: 10.1006/jdeq.1998.3543.

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