Abstract

One of the most studied planar polynomial differential systems are the so--called generalized Liénard polynomial differential equations $x\prime \prime \; +\; f$_n(x)x′+ g_m(x) = 0,     (1) which were studied by many researchers, where $g(0)=0$ and $f$ and $g$ are polynomials of degree $n$ and $m$ respectively. Such dynamical systems appear very often in several branches of the sciences, such as biology, chemistry, mechanics, electronics, etc. The differential equation (1) of second order can be written as the equivalent $2$--dimensional Liénard polynomial differential system of first order $x\prime \prime \; +\; f$_n(x)x′+ g_m(x) = 0,     (2) When $g(x)=\; x$ the Liénard differential systems (2) are called the classical Liénard systems.
The main objective of this talk is to present a survey on the old and new results on the limit cycles of the Liénard polynomial differential systems (2) algebraic or not in function of $m$ and $n$.

We shall put special emphasis on the number of the so--called small, medium and large limit cycles in function of $m$ and $n$. These are the limit cycles of the polynomial Liénard differential system (2) which bifurcate from the origin, or which bifurcate from the linear center $x\prime =\; y$, $y\prime $ =\; -x$perturbed\; inside\; the\; class\; of\; all\; polynomial\; Liénard\; differential\; systems\; (2),\; or\; which\; bifurcate\; from\; the\; graphics\; that\; the\; Liénard\; polynomial\; Liénard\; differential\; system\; (2)\; can\; have.$