  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
quantum versions of the systems, we consider a generalized position dependent
mass Hamiltonian in which the ordering parameters of the mass term are treated
as arbitrary. We observe that the quantum version of the Higgs oscillator is
exactly solvable under appropriate restrictions of the ordering parameters,
while the second nonlinear system is shown to be quasi exactly solvable using
the Bethe ansatz method in which the arbitrariness of ordering parameters also
plays an important role to obtain quasi-polynomial solutions. We extend the
study to three dimensional generalizations of these nonlinear oscillators and
obtain the exact solutions for the classical and quantum versions of the three
dimensional Higgs oscillator. The three dimensional generalization of the
quantum counterpart of the $k$-dependent nonpolynomial potential is found out
to be quasi exactly solvable.
