THIRD-ORDER NORMAL FORMS OF NONLINEAR OSCILLATIONS.

The case is considered in which for a real autonomous system of differential equations analytic in the neighborhood of zero, the matrix for the linear part has two purely imaginary eigenvalues plus or minus i omega and one negative eigenvalue. On the basis of theorems due to A. D. Bryuno computational formulas for the coefficients of the normalizing transformation and the normal form are derived and the latter is integrated. The article concludes with an electromechanical example. The results can also be applied to electromagnetic oscillations of two coupled oscillators where the natural oscillations of one are described by a first-order nonlinear equation.