The non-stationary delay of establishment nonlinear vibrations in the system of two connected oscillators. Part 4. Circular interaction.
Authors:   Syktyvkar State University, Oktyabrskiy prosp. 55, Syktyvkar 167001, Russia, Ivanov A.P. , Shavrov V.G. ,   Kotelnikov Institute of Radioengineering and Electronics of Russian Academy of Sciences, Mokhovaya 11-7, Moscow 125009, Russia, Shcheglov V.I. ,   Kotelnikov Institute of Radioengineering and Electronics of Russian Academy of Sciences, Mokhovaya 11-7, Moscow 125009, Russia
Published: April 2021

This work is the continuation of investigation of non-stationary delay of establishment nonlinear vibrations in the system of two connected oscillators. The physical foundation of this task is the excitation of power hypersound in ferrite plate having magnetoelastic properties and also excitation of intensive electromagnetic vibrations in ferrite disc placed in electro-dynamic resonator. It is found that under certain conditions the vibrations in these systems take character of non-stationary large-amplitude chaos which excites with large delay after the system is switched. The foundation of this work is received in the first part of this work the enough general system of two connected differential second order equations having nonlinearity as the power row to third order. As a result of elimination of secondary parameters from fundamental system it was selected the reduced system – «initial core» which preserves the properties of non-stationary delay and chaos. It is found the circular character of reduced system. In this character the determined by first equation displacement of first oscillator influences to the displacement of second oscillator through the linear connection of second equation. The received in this case displacement of second oscillator again return to the first equation through the item of nonlinear connection of first equation. In this case the whole connection of equations between ones forms the «ring», which is closed by itself. The first oscillator in this ring has the «mediator» – the second oscillator which plays the subordinate role. In the scheme of break ring the successive influence of oscillators one to other is investigated. It is shown that the influence of sinusoidal signal with constant amplitude to second oscillator leads to the excitation of its vibrations which has the beating character. The influence of these beatings to first oscillator leads to the large delay of excitation chaotic vibrations. The possibility of existence of ring in classic system of Lorents which admit chaotic vibrations. By the replacement of variables it is execute the transformation of Lorents system having three equations of first order to equivalent system having two equations. In this system the first equation has second order and second equation has first order. For this system it is established the circular character. In this case the first order equation for second oscillator is the «mediator» for second order equation for first oscillator. It is shown that the classic Lorents system which is described as a system of two equations has circular character which is the similar to the circular character for two oscillators. In Lorents system the development of large-amplitude chaos is accompanied by time-delay which has form of interval of regularity which is the similar as time-delay in system for two oscillators. The circular character of relax system for two oscillators is investigated. It is proposed the algorithm of step by step calculation hawing two stages character. The first stage is the passage of first half of whole ring which is determined by first equation. The second stage is the passage of second half of whole ring which is determined by second equation. It is established the possibility of presentation of system from two equations in the form when each equation contains selected from corresponding equation the own variable quantity which is equated for the remaining part of initial equation. It is established the circular character of this presentation. It is shown that the substitution the second variable form second equation to the first equation leads to the equation which contain only first variable. The excitation which is found by this substitution is named as «ring excitation» and its solution is named by «ring function». It is established that by the reason of large complexity of ring equation the unique method of its solution is step by step method. In this case the meaning of the variable on present step is determined by the function which contains the same variable which is determined on previous step. It is attended the analogy of this solution with the calculation of discrete single size reflection. On the example of classic logistical reflection it is calculated the development of vibrations by steps for some meanings of parameters which spread the regular dissipative, regular stationary and chaotic regimes. It is established that the delay by steps for logistical reflection is absent. It is established the possibility of ring equation solution as ring function using the same step by step method which is used for the calculation of reflections. It is proposed two variety of ring functions – first and second kind which in the quality of discrete varying variable uses the time. The left part of vibration function of both kinds is the variety – the displacement of first oscillator on the present step. The right part contains the trigonometric function (sine or cosine) which argument is the product of generalized frequency on the time which correspond to the previous step. For the ring function of first kind the generalized function is proportional to the displacement of oscillator plus unit. For the ring function of second kind the generalized frequency is proportional to the displacement of oscillator plus value of frequency which is determined by previous step. It is investigated the development of vibrations on time which is described by ring functions of first and second kinds. It ir proposed the algorithm of calculation which is similar to algorithm which is used for the reflections. It is shown that the vibrations in time are developed in two stages. For the ring function of first kind during the first stage the vibrations are the regular sine and during the second stage on the regular sine is deposited the chaotic vibrations having more lesser period. For the ring function of second kind during the first stage the vibrations are absent which is its delay and during the second stage the large-amplitude chaos is developed. As a result of performed investigation it is shown that the ring functions of second kind provide the convenient imitation of delay and development of the large-amplitude chaos in the system of two oscillators. For the ring function of second kind it is investigated the beginning of development of delay the large-amplitude vibrations and its end and transition to the large-amplitude chaos. It is proposed the model which is consist of the aspiration of function argument during the time to decrease to value π/2 and the increasing of displacement which try to increase of this argument. The exit from delay and beginning of the large-amplitude vibrations is passed through the breach of this balance which is determined by the excitation of vibrations having triangular form. It is proposed some comments about possible development of this work. As an important task it is mentioned the construction of model idea about duration of delay time. This question may be expanded not only the system of two oscillators and also on the ring functions and supplemented the investigation of interval of regularity in Lorents system. It is mentioned the importance of investigation of the role of nonlinear connection having high orders. This investigation may be applied to more wide round of phenomena.