A family of non-autonomous hybrid Lienard oscillators based on a parametrically extended commutative factorization

Abstract

We introduce a class of nonautonomous nonlinear oscillator equations of mixed Liénard type that arises from a parametric deformation of the commutative factorization procedure applied to second-order ordinary differential equations with periodic solutions. Their solutions, in particular the isochronous waveforms, are obtained in closed form through a Riccati reduction scheme for the power-law choice of the factorization function, $\varphi (x)=k{x}^q$, $k\in \mathbb{R}$, and $q\in \mathbb{N}$, corresponding to the so-called modified Emden oscillators, and do not depend on the arbitrary deformation parameter $A_1$. The variational structure of the equation is characterised by a Lagrangian supplemented with a generalised Rayleigh dissipation function that contains a non-standard cubic term in $\dot{x}$. We also show that multiplying the equation of motion by the Jacobi multiplier $M(x)=x^{-2A_1}$ a position-dependent-mass (PDM) form is obtained with related friction and restoring force.

9 pages, 3 figures, 22 references