Polynomial Liénard Equations near Infinity

We study the behaviour near infinity of the (generalized) Liénard equationsx=y,y=−∑mk=0 akxk−y ∑nk=0 bkxkproviding a complete classification using Poincaré compactification as well as Poincaré–Lyapunov compactification. We show that all the necessary information is contained inamandbn, except for the so called “center-focus” problem occuring in casem 2n+1 withm,nodd, and where the behaviour also depends on the value of the other coefficientsaiandbj. We also indicate how to take care about obtaining uniform information near infinity.