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Русская версия

**Vladimir V. Basov**

Universitetsky prospekt, 28,

198504, Peterhof, St. Petersburg, Russia,

Saint-Petersburg State University,

The Faculty of Mathematics and Mechanics,

Differential Equations Department

**Artyom Zhukov**

PhD student

The Faculty of Mathematics and Mechanics, Differential Equations Department

Saint-Petersburg State University

Universitetsky prospekt, 28, 198504, Peterhof, St. Petersburg, Russia,

We study two classes of two-dimensional time-periodic systems of ODEs with a small positive parameter --- systems with "slow" and "fast" time. Right-hand sides are assumed to be three times continuously differentiable on the phase variables and the parameter, and the corresponding unperturbed systems are autonomous, conservative and have nine steady points such that both coordinates take values -1,0 or 1. For parameter-independent perturbations of a system explicit conditions guaranteeing the existence of a certain number of invariant two-dimensional surfaces homeomorphic to tori for all small enough values of positive parameter have been obtained. Formulae of these surfaces are given. As an example of practical usage of the obtained result the~class of systems with three invariant surfaces enclosing some of steady points has been found in Refs 10.

- averaging
- bifurcation
- invariant surface

- Bibikov Yu. N. : “Stability and bifurcation under periodic perturbations of the equilibrium position of an oscillator with an infinitely large or infinitely small oscillation frequency”, Mat. Zametki, 65 (3), 323-335 (1999) [in Russian] (!!!! ERROR!!! INVALID URL: ! http://mi. mathnet. ru/eng/mz1056). English version: Mathematical Notes, 1999, 65:3, 269-279
- Bibikov Yu. N. : “Bifurcation of the generation of invariant tori with infinitesimal frequency”, Algebra i Analiz, 10 (2), 81-92 (1998) [in Russian] (http://mi. mathnet. ru/eng/aa987). Eng. version: St. Petersburg Mathematical Journal, 1999, 10:2, 283-292
- Basov V. V. : “Bifurcation of the Equilibrium Point in the Critical Case of Two Pairs of Zero Characteristic Roots”, Differential equations and dynamical systems, Collected papers. Dedicated to the 80th anniversary of academician Evgenii Frolovich Mishchenko, Tr. Mat. Inst. Steklova, 236, Nauka, Moscow, 2002, 45-60 [in Russian] (!!!! ERROR!!! INVALID URL: ! http://mi. mathnet. ru/eng/tm275). English version: Proceedings of the Steklov Institute of Mathematics, 2002, 236, 37-52
- Basov V. V. : “Bifurcation of the Point of Equilibrium in Systems with Zero Roots of the Characteristic Equation”, Mat. Zametki, 75 (3), 323-341 (2004) [in Russian] (http://mi. mathnet. ru/eng/mz35). Eng. version: Mathematical Notes, 2004, 75:3, 297-314
- Basov V. V. : “Invariant surfaces of two-dimensional periodic systems with bifurcating rest points in the first approximation”, Journal of Mathematical Sciencies, 147 (1), 6398-6415 (2007). Translated from Contemporary Mathematics and Its Applications, Vol. 38, Suzdal Conference-2004, Part 3, 2006 (http://link. springer. com/article/10. 1007/s10958-007-0474-x)
- Basov V. V. : “Invariant Surfaces of Standard Two-Dimensional Systems with Conservative First Approximation of the Third Order”, Diff. Equations, 44 (1), 1-18, 2008 (http://link. springer. com/article/10. 1134\%2FS0012266108010011). Original Russian Text © V. V. Basov, 2008, published in Differentsial’nye Uravneniya, 2008, Vol. 44, No. 1, pp. 3-18
- Arnol’d V. I. : Matematicheskie metody klassicheskoi mekhaniki (Mathematical Methods of Classical Mechanics), Moscow: Nauka, 1989
- Hale J. K. : “Integral Manifolds of Perturbed Differential Systems”, Annals of Mathematics Second Series, 73 (3), 496-531 (1961)
- Lyapunov A. M. : Sobr. soch. T. 2 (Collected Papers. Vol. 2), Moscow- Leningrad: Idzat. Akad. Nauk SSSR, 1956, pp. 272-331
- Bibikov Yu. N. : “Mnogochastotnye nelineinye kolebaniya i ikh bifurkatsii” (Multifrequency Nonlinear Oscillations and Their Bifurcations), Leningrad: Leningrad Univ., 1991