L.V. Cherkesova

In the article are considered important for the theory and practice of radio electronics questions: the construction and analysis of zones of instability of the equation lost-free resonant nonlinear-parametrical zone (pazone) systems (NPS), without the account of losses - conservative system, a finding of invariants of movement NPS, and also finding-out of interrelation of phase portraits with zones of parametrical excitation. As a result of the spent researches it has been found out that nonlinearity introduction in equation NPS leads to turn and deformation of lines of equal amplitudes, i.e. borders of areas of excitation concerning points of resonant frequencies, and an angle of rotation to those than more than the coefficient, characterizing the degree of nonlinearity and type of system, and also amplitude of stationary oscillations is more on the module. The inclination of lines of equal amplitudes towards the bigger or smaller frequencies depends on a sign of coefficient, characterizing degree and type of nonlinearity of system. For soft nonlinear system (k>0) lines of equal amplitudes, corresponding to borders of area of excitation, turn relatively points of resonant frequency clockwise (to the right) in comparison with borders at k=0 (in classical case of the equation of Matier), and for rigid system (k<0) ? counter-clockwise (to the left). It is shown that in rigid NPS the right line of equal amplitudes is corresponding to stable stationary oscillations with phase , and left - to unstable stationary oscillations with phase . For soft NPS, on the contrary, the right line corresponds to unstable stationary oscillations with phase , and left - steady with phase . The presence of nonlinearity of system leads to restriction of amplitude of parametrical excited oscillations of a resonant contour. The interrelation of zones of excitation NPS with phase portrait of the investigated systems in case of soft and rigid modes of exciting oscillations is considered. It is established that the zone of soft mode coincides with Matier-s zone, and the zone of rigid mode is to the left of it at values of coefficient k<0, and to the right of it at k>0. For a zone of a soft mode presence of a phase portrait with three special points and unstable near zero oscillations is characteristically, and for a zone of a rigid mode phase portraits with five special points and a stable state of rest take place. For rigid nonlinear system to borders of Matier-s zone of in case of cubic nonlinearity there is corresponds rest state, thus to the right border - stable, and left - unstable. For soft nonlinear system on the contrary, the right border corresponds - an unstable state of rest, and the left border - steady. In case of soft nonlinear system on the right border of Matier-s zone (or the left border in case of rigid nonlinear system) there are stable nonzero stationary oscillations. For the first zone of parametrical excitation of oscillations dependence through elliptic functions of Jacobi, and the period of change of dependence through full elliptic integral of the first kind were found in an explicit form. The researches that were realized in this article represent theoretical and practical interest for the experts who are engaged in designing of radio-electronic equipment on the basis of nonlinear resonators, in base of functioning of which there is functioning oscillatory nonlinear-parametrical zone system in the higher zones of instability of oscillation on the higher harmonics.