Study of steady modes of disturbed autonomous systems in critical cases: PMM vol. 39, no. 5, 1975, pp. 817–826

Abstract

A method due to Poincaré is used to study the critical cases in essentially nonlinear autonomous systems with one degree of freedom, and the situations leading to the splitting of the trajectories. The first Liapunov method is used to study the problems of stability of the steady modes. A selfrotating, almost conservative system is considered as an example. Previous papers concerned with the analysis of the motions near the generating family of periodic or rotational motions of an unperturbed system dealt, as a rule, with relatively simple cases in which the equations of the parameters of the family defining the steady mode admit, in the first approximation, simple real roots /1 – 6/. Subtler and more complex cases in which the roots are multiple, or when some of the equations of the defining system are satisfied identically, were given much less attention /1, 7 – 11/.