Nonlocal transformations of the generalized Liénard type equations and dissipative Ermakov-Milne-Pinney systems

Abstract

We employ the method of nonlocal generalized Sundman transformations to formulate the linearization problem for equations of the generalized Liénard type and show that they may be mapped to equations of the dissipative Ermakov-Milne-Pinney type. We obtain the corresponding new first integrals of these derived equations, this method yields a natural generalization of the construction of Ermakov–Lewis invariant for a time-dependent oscillator to (coupled) Liénard and Liénard type equations. We also study the linearization problem for the coupled Liénard equation using nonlocal transformations and derive coupled dissipative Ermakov-Milne-Pinney equation. As an offshoot of this nonlocal transformation method when the standard Liénard equation, [Formula: see text], is mapped to that of the linear harmonic oscillator equation, we obtain a relation between the functions [Formula: see text] and [Formula: see text] which is exactly similar to the condition derived in the context of isochronicity of the Liénard equation.