Non-linear argumental oscillators: Stability criterion and approximate implicit analytic solution

•A multiple-bifurcation system is brought out.•A simple analytic expression of the system's stability criterion is given.•Attractors are identified, which duplicate in a rotational-symmetry figure.•An approximate implicit analytic solution to the averaged system is given.•Cases exist, where motion starting near the origin evolves towards large amplitude.

The behaviour of a space-modulated, so-called “argumental” oscillator, is studied. The oscillator is submitted to an external harmonic force, which is amplitude-modulated by the oscillator's position in space. An analytic expression of a stability criterion is given. Using the averaging method, an integrating factor and a Van der Pol representation in the (amplitude, phase)-space, an exact implicit analytic solution is given when there is no damping, and an approximate implicit analytic solution is given when there is damping, allowing the plotting of the separatrix curve. An attractor is identified.