# Complete classification on center of cubic planar systems with a line of symmetry

kߣlrg2022-06-17g[Δ10

RʴW

rg20226181430

cvӍh 241 673 194

ekλWԺ

v˽BhRʴWWcyӋWԺڣTʿWuՓuՓTnˮԭWʿ2012ϴWWcӋƌWWԺ@WʿWλ2014꣬܇WYôԴWLWһꡣFևȻƌWĿ1헡Journal of Differential Equations, ЇƌWNonlinear AnalysissIlՓ20ƪ2016@ɽ|ʡȻƌWWgª

ݽBIn this talk, bi-center and bi-isochronous center problems in cubic planar systems which are symmetric with respect to a straight line will be discussed. These systems can be transformed to ones which are symmetric with respect to the $y$-axis and have two symmetric singular points at $(\pm 1,0)$, which can be classified as elementary and nilpotent singular points. A complete classification is given on the centers, including nine conditions for elementary singular points and four conditions for nilpotent singular points. Moreover, six bi-isochronous center conditions are obtained for the elementary singular points.