Cyclicity of some Liénard systems (Q746494)

The paper is devoted to the cyclicity problem of two families of Liénard systems \[ \dot{x}=y-\frac{q_n(x)}{(1-x)^m}, \quad \dot{y}=-g(x) \] and \[ \dot{x}=y-\frac{q_n(x)}{(1-\alpha x)(1-\beta x)}, \quad \dot{y}=-g(x), \quad \alpha\beta\neq 0, \] where \(q_n(x)=\sum_{i=0}^n a_ix^i\) and \(g(x)\) satisfies the following conditions \[ g(0)=0, \quad g'(0)>0, \quad g(-x)=-g(x). \] The authors study small-amplitude limit cycles and find the exact number of such limit cycles bifurcating from a center or focus at the origin for these families.