V.M. Bogachev - Ph.D. (Eng.), professor, National Research University "MPEI". E-mail: BogachevVM@mail.ru
I.N. Leonov - Post-graduate student, National Research University "MPEI". E-mail: leonovc3h7@gmail.com

Peculiarities of the local and global stability/instability of steady-state periodic modes in the multi-circuit oscillators (MCO) with closed natural frequencies are studied by methods of D-fragmentation on the basis of Evtianov-s abbreviated characteristic equations for essentially inertial active elements (AE). It is shown that "isolated singular points" corresponding to the frequency offsets between circuits with the horizontal tangent to the oscillating system-s (OS) locus are globally stable at the "soft" complex oscillation characteristic or they are globally instable at any shape of oscillating characteristics. For the other periodic modes (for unlimited requirements to the oscillating characteristics) there are always areas of local stability in the D-fragmentation diagram. These areas are located near the steady-state point if it is situated in the locally stable hysteresis region (for non-inertia AE) and in the far zone in the opposite case. Outside of hysteresis regions the D-fragmentation areas are discovered in which the periodic oscillations are locally stable at "rigid" ("non-soft") oscillating characteristics and for essentially inertial AE. Two approaches - the Neimark-s frequency approach and the Rauth-Hurvitz determinant criterion are used to create D-fragmenta¬tions. It is shown that significant advantage of the Neimark-s approach is a possibility of parametric problem solution without characteristic equation reduction to the polynomial form. Nevertheless, such a solution is possible in the case only when one of parameters is linearly included into the characteristic equation. The Rauth-Hurvitz method is free from this limitation but the characteristic equation-s reduction to the polynomial of three variables p, A, B is necessary for its realization. If this procedure is fulfilled (for instance, by the method of the multi-dimensional digital Furrier transform interpolation), the final solution becomes simpler and can be reduced to solution of algebraic equations of rather low order according to variables A, B - the plane section parameters of D-fragmentation-s space of the dynamic system.

 

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