the nonlinear resonator
nonlinear-parametrical zone system
a parametrical resonance
zones of instability of fluctuations
dissipation of system
attenuation of fluctuations
rigid and soft nonlinear systems
the nonlinear differential equation
lines of equal amplitudes
borders of zones of the instability
the truncated equations
a method consecutive approximations
the higher zones of parametrical excitation
the higher harmonics
At research and designing of the electronic devices containing nonlinear resonators (oscillatory resonant contours, or nonlinear-parametrical zone (so-called pazone) systems, NPS), the great value has presence of the information about zones of parametrical instability of system in which the work of external forces spent for periodic change of parameter of system, creates increasing of supply of its oscillatory energy. To borders of zones of instability of oscillations there correspond the periodic decisions of the differential equation describing NPS.
In article the question, as behaviour of lines of the equal amplitudes corresponding to borders of areas of instability of fluctuations resonant of strongly nonlinear parametrical zone system, with the account of attenuation of fluctuations owing to losses, i.e. dissipation in system, is considered such important for the theory and practice of radio electronics.
Zones of instability strongly nonlinear NPS with the attenuation account (dissipation) oscillations are defined. It is found out that dissipative rigid nonlinear parametrical system (with other things being equal) to excite much easier, than dissipative linear parametrical system, and dissipative soft nonlinear system – it is more difficult, than linear. It means that for excitation of oscillations of dissipative NPS, is not required the more intensity of modulation of parameter, than for linear parametrical system.
The interrelation of zones of parametrical excitation of NPS with phase trajectories and system portraits is found. Phase portraits of NPS are received; modes and excitation zones are compared. On the equation of isoclinals lines there are constructed and analyzed the phase portraits of NPS for cases of soft and rigid nonlinearity. The results of construction of phase portraits in case of soft and rigid modes of parametrical excitation of oscillations are generalized. The diagramme of stability of the decision of dissipative equations of NPS near to the basic parametrical resonance is constructed. It is established that to steady stationary oscillations of system there correspond the points of type of «steady focus», and unstable stationary fluctuations are answered with points of type of a «saddle». Received on the equation of isoclinals lines the phase portraits of NPS have considerable graphic conformity with the phase portraits, received experimentally on the screen of an oscillograph.
From the analysis of the received correlation follows, that the common account of strong nonlinearity and attenuation in differential equation of NPS with periodic factors leads to displacement of lines of equal amplitudes (zones of parametrical instability) on both coordinates in a plane and to their turn concerning the bottom points. Thus displacement and turn by that more than is more on the module degree of nonlinearity and amplitude of stationary oscillations. For rigid nonlinear system the displacement of the bottom points of lines of equal amplitudes occurs towards smaller frequencies and downwards in comparison to the case k=0, and for soft nonlinear system – towards the big frequencies and upwards.
Thus, the dissipative rigid nonlinear-parametrical system (with other things being equal) to excite much easier, than dissipative linear parametrical system, and dissipative soft nonlinear system – it is more difficult, than linear. It means that for excitation of oscillations of dissipative NPS, is not required the more intensity of modulation of parameter, than for linear parametrical system.