The stability of the parametric oscillations of second-order non-linear systems

The parametric oscillations of strongly non-linear systems with one degree of freedom are considered using a more general definition of these oscillations than the generally accepted definition. Stability criteria, that are verifiable using the signs of the derivatives of the amplitude-frequency characteristics, are found for the two families of periodic solutions corresponding to the fundamental parametric resonance. A condition is indicated under which the latter are monotonic and, as a result, one of the families is stable and the other is unstable. It is shown that, in a system with a concave non-monotonic elastic characteristic, the stable family loses stability for fairly large amplitudes and this effect is not revealed by the well-known analytical methods of non-linear mechanics.