Error of small parameter methods in solving shortened equations of a synchronized oscillator
DOI:
https://doi.org/10.30837/rt.2021.2.205.12Keywords:
synchronized oscillator, shortened equations, small parameter methods, discrepancy, errorAbstract
The paper considers the use of recently appeared analytical methods for solving shortened equations of a synchronized oscillator. These are a quasi-small parameter method and a combined small parameter method. Both methods use the classic small parameter method. A peculiarity of their application is that in this case they are used for solving nonlinear differential equations that do not contain a small parameter. The difference between the above methods is in obtaining the equations of the first approximation. In the quasi-small parameter method, they are linear differential equations obtained by linearizing the original nonlinear differential equations in the area of the zero frequency detuning. In the combined small parameter method, the equations of the first approximation are obtained by approximating the original nonlinear differential equations. Of course, a number of transformations of these equations were made for this. The approximation made it possible to obtain better representation of the original nonlinear differential equations by means of linear differential equations. This representation provided a smaller error, which in both cases was presented as a discrepancy. The discrepancy does not allow obtaining a relative error and investigating its peculiarity.
A study of the relative error of the quasi-small parameter method shows that this error is a continuous function of the frequency detuning with a zero value for a zero frequency detuning.
A function representing relative error has a gap at zero frequency detuning for the combined small parameter method. However, this kind of gap can be eliminated by additional function definition.
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