Existence and uniqueness of limit cycles for generalized \(\phi\)-Laplacian Liénard equations (Q268567)

The paper deals with a generalized \(\varphi\)-Laplacian Liénard system NEWLINE\[NEWLINE \dot{x}=y\varphi'(y), \quad \dot{y}=-g(x)-f(x)\psi(y),NEWLINE\]NEWLINE which cannot to be transformed to the classical one. The aim of this work is to provide sufficient conditions on the functions \(f\), \(g\), \(\varphi\) and \(\psi\) for the existence of periodic orbits of such system and, in this case, conditions for uniqueness. Some known corresponding results for the classical Liénard system are extended to the case under the consideration. In particular, the derived results ensure the existence and uniqueness of a periodic orbit for the relativistic van der Pol equation NEWLINE\[NEWLINE(x'/\sqrt{1-(x'/c)^2})'+\mu(x^2-1)x'+x=0NEWLINE\]NEWLINE for \(\mu \neq 0\).