# American Institute of Mathematical Sciences

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

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

Received  December 2010 Revised  June 2011 Published  January 2012

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.
Citation: Freddy Dumortier. Sharp upperbounds for the number of large amplitude limit cycles in polynomial Lienard systems. Discrete & Continuous Dynamical Systems - A, 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.  doi: 10.1016/j.jde.2007.11.011.  Google Scholar [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.   Google Scholar [3] F. Dumortier, Compactification and desingularisation of spaces of polynomial Liénard equations,, Journal of Differential Equations, 224 (2006), 296.  doi: 10.1016/j.jde.2005.08.011.  Google Scholar [4] F. Dumortier and C. Herssens, Polynomial Liénard equations near infinity,, Journal of Differential Equations, 153 (1999), 1.  doi: 10.1006/jdeq.1998.3543.  Google Scholar

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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.  doi: 10.1016/j.jde.2007.11.011.  Google Scholar [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.   Google Scholar [3] F. Dumortier, Compactification and desingularisation of spaces of polynomial Liénard equations,, Journal of Differential Equations, 224 (2006), 296.  doi: 10.1016/j.jde.2005.08.011.  Google Scholar [4] F. Dumortier and C. Herssens, Polynomial Liénard equations near infinity,, Journal of Differential Equations, 153 (1999), 1.  doi: 10.1006/jdeq.1998.3543.  Google Scholar
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