Abstract
The theory developed in [1] is applied to a certain class of ordinary differential equations in a Hubert space. Sufficient conditions are indicated for the appearance of auto-oscillations on the passage of a certain parameter through its critical value (existence of that value is established by the theory of normal operators). A nonlinear parabolic equation is investigated as an example. The auto-oscillating modes are given for the general equation of [1] in the form of series in fractional powers of the supercritical parameter δ, and in the form of series in powers of the amplitude coefficient accompanying the neutral perturbation in the Fourier expansion. The perturbation theory is used to study the stability of the auto-oscillations. The terminology and the basic notation of [1] are all retained.
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