A second-order nonlinear system subjected to parametric excitation is investigated. The nonlinear factors included are nonlinear damping and a cubic term in displacement. The primary purpose of the paper is to study the limiting effects of these nonlinear factors on the growth of motion for those systems which are otherwise unstable and have an exponential growth. Through an asymptotic analysis formulas are found for evaluating the limit cycle response amplitude in the first and second instability regions of the Ince-Strutt chart. Some results calculated from these formulas for the important case of velocity square damping are compared against those obtained by direct numerical integration in order to assess their accuracy.

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