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The accuracy of solutions obtained by numerical analysis of self-oscillating microwave systems in the deterministic chaos mode

Keywords:

I.V. Semernick – Post-graduate, Department of Antennas and Radio Transmitting Equipments, Southern Federal University, Taganrog, Russia. E-mail: ione7nick@yandex.ru
Yu.I. Alekseev – Dr.Sci. (Eng.), Professor, Department of Antennas and Radio Transmitting Equipments, Southern Federal University, Taganrog, Russia
A.V. Demyanenko – Ph.D. (Eng.), Associate Professor, Department of Antennas and Radio Transmitting Equipments, Southern Federal University, Taganrog, Russia


Fractal analysis is becoming more common in the theoretical study of self-oscillating microwave systems. Particularly relevant application of the fractal analysis elements becomes recently in the areas of radio physics and radiolocation, when in radio systems mentioned areas get increasingly application the devices with chaotic generation. In this context, as well as results of quantitative analysis are available mainly on the basis of numerical methods, it is lawful to raise the question about the accuracy of such solutions and the validity of variations of certain parameters, which are inevitable in the numerical solutions of differential equations that describe the various states of oscillatory systems. It is always necessary to know whether effects and phenomena, obtained as a result of investigations of the dynamic system, are the consequences of its irregular behavior, or the reason lies in the accumulation of errors in the calculations related to the inaccuracy problem to be solved. One of the methods to assess fidelity of the solution is to compare the solutions obtained at different given values for the local error. In the paper we define influence of the preset maximum allowable local error (tolerance) on the accuracy of the results obtained from the numerical solution of nonlinear differential equations describing the dynamic system in different modes: harmonic, mode with a small amplitude instability, multi-frequency and chaotic mode. As a dynamic system adopted the microwave oscillator based on the IMPATT diode. It is shown that the type of motion is not changed at a variation the integration step, that says that the effects obtained in the present work as a result of the numerical solution, seen on the time realization, phase portraits and spectral characteristics are the consequences of dynamics development of the studied nonlinear dynamic system instead consequence of integration errors. In solving real tasks is always necessary to find a compromise between the accuracy of solutions and the time spent on the investigation of the system. Therefore it is possible to say that for the determination the parameters in the single-frequency oscillator must be set the value of tolerance, not more than 10–6. However, to establish the parameters of motion in a chaotic mode – bandwidth noise generation is acceptable to carry out calculations at the maximum allowable local error the order of 10–4. At the same time, to identify the type of motion of the dynamic system and determine the statistical characteristics of the attractor in the chaotic mode it is sufficient to establish the value of tolerance, not more than 10–3. At values less than said tolerance values in the each case, the time required to obtain the numerical solution increases manifold, but the gain in accuracy is negligible.
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