__Keywords:__oscillation systems mathematical model nonlinear processes nonlinear parametrical resonator parametrical resonance

L.V. Cherckesova

Nonlinear oscillations take paramount importance in different areas of science and techniques in XXI century. Nonlinear-parametrical zone (pazone) resonators (or nonlinear pazone system, NPS) presents the variety of nonlinear resonance system that constitute the base of quantum electronics, interest to which was appeared thanks to their own distinctive features of appearance of sub-harmonious resonance.
This mentioned phenomenon as long ago, as 60-th years of XX century, underlay in base of functioning of paramrtronic scheme, sub-harmonious generators and etc.; later they became to use for reception of parametrical generation and amplification of electromagnetic oscillations, phase stability, spasmodic (uneven) excitation of oscillations, multiplication, division, transformation of frequency and etc., in the equipment of radioelectronic, automatics, computing techniques. Nonlinear controlled elements of inductive type have the series of advantages in comparison with capacious elements, in the whole frequency diapason from tens hertz to hundreds gigahertz: high dependence, economy, radiation stableness, possibility of fast and precise tuning of resonance oscillatory circuit on corresponding voltage of feeding – harmonica of spectrum of feeding. The advantage of elements of inductive type also is the using of deep modulation of element, since it is possible to select the according characteristic of element and to use its nonlinear section in the condition of saturation that enables to fluently pass on from the one to another zone of instability without frustration of the oscillations. There are of interest the nonlinear pazone system and with concentrative, and with distributive parameters, that often using when studying the laser pencil of rays (laser beams).
In the capacity of model of resonance system there is considered parametric amplifier-modulator of inductive type that is presented as three-contour circuit electric scheme. Mathematical model of greatly nonlinear processes in asymmetry parametric zone resonator, functioning in the highest zones of oscillations instability, was constructed. The model takes into account the dynamics of changes of jet reaction system by polyharmonious external influence.
From the presented graphs and illustrations, from analysis of the received correlation it follows that asymmetry (nonidentity of magnetic cores of NPS) reduce to non-simplicity of nonlinearity oft he system. In the asymmetrical system with non-symmetrical nonlinear characteristic and presence of displacement is observed the trigger effect: the changing of amplitude of external influence (inflating) provoke the great advance of voltage in the outlet – amplitude of oscillations in the contour divide in two stable states without changing of frequency. The similar effect is observed in the whole areas of oscillation instability and is conditioned by difference of core magnetizing.
By considerable intensity of external influence, nonlinear pazone system passes in the highest zones of excitation of oscillations. The arising oscillations carrying very intricate character; so, the amplitude increase of cosines component of current and voltage (tension) harmonicas involves to appearance even harmonicas of external influence in resulting oscillations of system. The oscillation amplitude is limited by dissipative properties of system and there take place the «smoothing out» this both factors.
One of the most important energetic showing of considering oscillate system is potential energy, that is conditioned by availability of parametric interactions. The value of supplied energy is determine the stability of oscillate process in system by its disorder. The received equations describe energetic correlations in asymmetric dissipative NPS by polyharmonious external influence in arbitrary zone of oscillate instability, allows to state the value of influence on amplitude-and-frequency characteristics of system by constant and variable component of polyharmonious external influence.
There was conducted the analysis of energetic processes of NPS; for all this, was revealed interconnect of energetic supplies of the system with its phase trajectories and portraits.
There were received phase portraits of system oscillations, on which it is easy to determine the character of location of zone of stable and instable oscillations and the type of moving of nonlinear parametrical zone systems. The equation for determinate the dispersion of magnetic induction of was found.
The constructed mathematical models can be used for computation and design of electronic equipment, based on NPS, because system’s constructive parameters are taken into account in the coefficients by according members of nonlinear equation.

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