The methodology of the studying of dynamic processes in two-dimensional systems by mathematical models containing nonlinear equation of Klein-Gordon was developed. The methodology contains such underlying: the concept of the motion wave theory; the single - frequency fluctuations principle in nonlinear systems; the asymptotic methods of nonlinear mechanics. The aggregate content allowed describing the dynamic process for the undisturbed (linear) analogue of the mathematical model of movement. The value determining the impact of nonlinear forces on the basic parameters of the waves for the disturbed analogue is defined.

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[8] Ye. V. Kharchenko, and M. B. Sokil, “Neliniyni protsesy u seredovyshchakh, yaki kharakteryzuyutʹsya pozdovzhnim rukhom, i vplyv sposobu zakriplennya na yikh kolyvannya” [“Nonlinear processes in media characterized by longitudinal motion and the influence of the method of fixation on their oscillations”], *Avtomatizacìâ virobničih procesìv u mašinobuduvannì ta priladobuduvannì [Industrial Process Automation in Engineering and Instrumentation]*, vol. 41, pp. 156–159, 2007. [in Ukrainian].

[9] Martyntsiv M. P., and Sokil M. B., “Odne uzahalʹnennya metodu D’Alambera dlya system, yaki kharakteryzuyutʹsya pozdovzhnim rukhom” [“One generalization of the D’Alembert method for systems characterized by longitudinal motion”], *Naukovyy visnyk Derzhavnoho lisotekhnichnoho universytetu Ukrayiny [Scientific Bulletin of Ukrainian State Forestry University]*, vol. 13.4, pp. 64–67, 2003. [in Ukrainian].

[10] Y. A. Mitropol'skii, and B. I. Sokil, “On the application of Ateb-functions to the construction of an asymptotic solution of the perturbed nonlinear Klein-Gordon equation”, *Ukrainian Mathematical Journal*, no. 5 (50), pp. 754–760, 1998.

[11] B. I. Sokil, “On a method for constructing one-frequency solutions of a nonlinear wave equation”, *Ukrainian Mathematical Journal*, no. 6 (46), pp. 853–856, 1994.