Non-existence and uniqueness of limit cycles in a class of generalized Liénard equations (Q2141751)

Generalized Liénard systems of the form \[ \dot{x} = y+ax^p+by^k, \dot{y} = cx^m,\hspace{9cm} (1) \] where \(n, k, m\) are positive integers, \(1<n<k,\) and \(a, b, c\in\mathbb{R}\) with \(bc\neq0\) are considered. It is proved that: 1) with a change of variables and a rescaling of the time, system (1) is equivalent to the following two systems \[ \dot{x} = y+ax^p+by^k, \dot{y} = x^m,\hspace{9cm} (i) \] \[ \dot{x} = y+ax^p+by^k, \dot{y} = -x^m,\hspace{8.5cm} (ii) \] 2) for \(m\) even, neither systems (i) nor systems (ii) have limit cycles, 3) for \(m\) odd, systems (i) do not have limit cycles, 4) for \(m,\) odd systems (ii) have limit cycles if and only if \(k\) and \(n\) are odd and \(a<0.\) In this case, the maximum number of limit cycles that system (ii) can have is one and this upper bound is reached. Moreover, whenever the limit cycle exists, it is hyperbolic.