Generalized quantum theory for accessing nonlinear systems: the case of Levinson-Smith equations

Motivated by a recently developed generalized scheme of quantum mechanics, we touch upon connections with Levinson-Smith classes of nonlinear systems that contain as a particular case the Liénard family of differential equations. The latter, which has coefficients of odd and odd symmetry, admits a closed form solution when converted to the Abel form. Analysis of the governing condition shows that one of the nontrivial equilibrium points is stable in character. Other classes of differential equations that we encounter speak of solutions involving Jacobi elliptic functions for a certain combination of underlying parameters, while, for a different set, relevance to position-dependent mass systems is shown. In addition, an interesting off-shoot of our results is the emergence of solitonic-like solutions from the condition of the level surface in the system.