Exploring branched Hamiltonians for a class of nonlinear systems

One of the less well-understood ambiguities of quantization is emphasized to result from the presence of higher-order time derivatives in the Lagrangians resulting in multiple-valued Hamiltonians. We explore certain classes of branched Hamiltonians in the context of nonlinear autonomous differential equation of Lienard type. Two eligible elementary nonlinear models that emerge are shown to admit a feasible quantization along these lines.