A new approach to nonlinear quartic oscillators

In this study, we proved that damped quadratic nonlinear oscillators similar to Duffing and Helmholtz–Duffing damped equations which emerge in bubble dynamics with time-periodic straining flows and solitary-like wave's dynamics may be derived from a new functional approach based on nonstandard Lagrangians and fractional frictions. The solutions of these equations are given in terms of the Jacobi elliptic functions. It was observed that the dynamical model constructed in this study is comparable to dynamical systems with natural Lagrangians for which the Riemann structure is conformally flat which has important implications in dynamical systems with position-dependent mass.