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Albrecht, Felix Yao, Leummim 2011-05-07T13:54:13Z 2011-05-07T13:54:13Z 10000-01-01 1995 We consider the Lienard equation of the form$$\ddot x + \epsilon f(x)\dot x + x\sp{2m+1} = 0,\eqno(1)\cr$$which is equivalent to the planar system$$\eqalignno{\dot x &= y&\enspace\cr \dot{y} &= -x\sp{2m+1} - \epsilon f(x)y,&(2)\cr}$$where m is a nonnegative integer, f is a real polynomial and $\epsilon$ is a small parameter. We establish a sharp upper bound for the number of limit cycles (nontrivial isolated periodic orbits of system (2) depending on m and the degree of f provided $\epsilon$ is sufficiently small. This is done by investigating the fixed points of the Poincare return map associated with system (2). This result is an important step in the study of planar polynomial vector fields in connection with the second part of Hilbert's Sixteenth Problem, which is still an open question. Made available in DSpace on 2011-05-07T13:54:13Z (GMT). No. of bitstreams: 2 license.txt: 4922 bytes, checksum: 910b249b4beec47e7ab768910c8f966f (MD5) 9624545.pdf: 1576418 bytes, checksum: aa4db2535ed9c62557c8df23e86e8928 (MD5) Previous issue date: 1995 Item marked as restricted to the 'UIUC Users [automated]' Group (id=2) by Howard Ding (hding2@illinois.edu) on 2011-05-07T15:00:34Z Item is restricted indefinitely. Restriction data tranferred 2014-07-01T11:28:40-05:00 Original Data Group with Access UIUC Users [automated] Release Date: none Reason: ETDs are only available to UIUC Users without author permission ETDs are only available to UIUC Users without author permission U of I Only AAI9624545 (UMI)AAI9624545 http://hdl.handle.net/2142/22869 eng Copyright 1995 Yao, Leummim Mathematics On the maximum number of limit cycles of certain polynomial Lienard equations text Mathematics Mathematics University of Illinois at Urbana-Champaign Dissertation Ph.D.