On a measure of the closeness of neutral systems to internal resonance (Q1094899)

For certain classes of parametrically perturbed resonance systems that are neutral in a linear approximation, a quantitative characteristic is introduced for the closeness of the system of resonance: the magnitude of the critical detuning value for resonance \(\delta^*\) at which the change in stability occurs as the system with draws from resonance. The problem of finding this critical value is made complicated by the nonlinear nature of the change in stability in neutral systems. It is solved below for third-order resonances in a situation that guarantees the passage of instability into asymptotic stability as the system withdraws from resonance. Knowledge of the quantity \(\delta^*\) enables the strong instability domain in parameter space to be estimated, enables the danger of resonance to be characterized, and enables the structual parameter in the system, the shift of the resonance phases, to be clarified, whose variation would enable the danger of resonance to be increased or reduced.