Mathematical Analysis of a Nonlinear Oscillator with Cubic Velocity Damping: Equilibrium, Stability, and Phase Curves

This paper examines the free response of a nonlinear oscillator that combines linear and cubic velocity damping with hardening cubic stiffness. The analysis identifies the equilibrium and its local behavior, then uses a mechanical-energy Lyapunov function and LaSalle’s invariance principle to establish global asymptotic stability when the mass and linear stiffness are positive, the nonlinear stiffness and damping coefficients are nonnegative, and at least one damping mechanism is active. Numerical simulations with the DOP853 solver compare three initial displacement levels with a reference model that omits cubic damping. Energy decreased throughout all simulated trajectories, consistent with the analytical energy balance. At the final simulation time, adding cubic damping reduced the remaining energy by approximately 9%, 32%, and 58% for initial displacements of 0.5, 1.0, and 1.5, respectively. A parameter sweep also showed lower response peaks and a modest delay in successive peaks as cubic damping increased. These effects were strongest during the large-amplitude part of the motion; near equilibrium, both cases followed the same linearized dynamics. The results are numerical and specific to the selected dimensionless parameters. The paper presents an application of established stability methods rather than a new stability theorem