NSFD representations for polynomial terms appearing in the potential functions of 1-dimensional conservative systems

We provide a methodology for the construction of nonstandard finite difference (NSFD) schemes for 1-dimensional conservative dynamical systems. Such systems are used to model a broad range of nonlinear oscillators. To obtain the desired representations, we imposed conditions following from the constraints of parity invariance and the conservation of energy. A further, practical computational requirement is that these schemes be explicit and linear in the dependent variable evaluated at the upper discrete-time level. The current work extends previous studies by Mickens et al.