Generalized van der Pol equation and Hilbert's 16th problem (Q2877603)

The problem under consideration is an upper bound for the number of limit cycles bifurcating from the linear center in the generalized Van der Pol system NEWLINE\[NEWLINE \dot{x}=y, \quad \dot{y}=-x +\varepsilon f(y)(1-x^{2}),NEWLINE\]NEWLINE when the parameter \(\varepsilon\) crosses zero value. The author proves that \(n+1\) is an upper bound for the number of limit cycles for the case that \(f\) is an odd polynomial \(f\) of degree \(2n+1\). For the proof the Poincaré method and some results of Iliev are applied. It is shown how to construct a polynomial \(f\) such that the upper bound is attained. Some examples illustrating the presented theory are included too.