On the classical and quantum dynamics of a class of nonpolynomial oscillators

We consider two one dimensional nonlinear oscillators, namely (i) Higgs oscillator and (ii) a $k$-dependent nonpolynomial rational potential, where $k$ is the constant curvature of a Riemannian manifold. Both the systems are of position dependent mass form, ${displaystyle m(x) = frac{1}{(1 + k x^2)^2}}$, belonging to the quadratic Li$acute{e}$nard type nonlinear oscillators. They admit different kinds of motions at the classical level. While solving the quant