Abstract
We study the problem of existence and uniqueness of invariant toroidal manifolds of countable systems of linear equations given on an infinite-dimensional torus.
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Yu. A. Mitropol’skii, A. M. Samoilenko, and D. I. Martynyuk, Systems of Evolution Equations with Periodic and Quasiperiodic Coefficients [in Russian], Naukova Dumka, Kiev (1984).
D. I. Martynyuk and N. A. Perestyuk, “On the reductibility of difference equations on a torus,” Vych. Prikl. Mat., 26, 42–48 (1975).
D. I. Martynyuk, V. Ya. Danilov, and V. G. Pan’kov, “The second Bogolyubov theorem for systems of difference equations,” Ukr. Mat. Zh., 48, No. 4, 464–475 (1996).
D. I. Martynyuk, “Investigation of the neighborhood of a smooth invariant toroidal manifold of a system of difference equations,” Differents. Uravn., 11, No. 8, 1474–1484 (1975).
A. M. Samoilenko, D. I. Martynyuk, and N. A. Perestyuk, “On the existence of invariant tori of systems of difference equations,” Differents. Uravn., 9, No. 10, 1904–1910 (1973).
A. M. Samoilenko, “Investigation of a discrete dynamical system in the neighborhood of a quasiperiodic trajectory,” Ukr. Mat. Zh., 44, No. 12, 1702–1711 (1992).
A. M. Samoilenko, D. I. Martynyuk, and V. G. Pan’kov, Properties of Invariant Toroidal Sets of Nonlinear Systems of Difference Equations [in Russian], Preprint No. 90.7, Institute of Mathematics, Ukrainian Academy of Sciences, Kiev (1990).
A. M. Samoilenko and Yu. V. Teplinskii, “On the reducibility of countable systems of linear difference equations,” Ukr. Mat. Zh., 47, No. 11, 1533–1541 (1995).
Yu. V. Teplinskii and A. M. Samoilenko, “On periodic solutions of countable systems of linear and quasilinear difference equations with periodic coefficients,” Ukr. Mat. Zh., 48, No. 8, 1144–1152 (1996).
Yu. V. Teplinskii and A. Yu. Teplinskii, “On periodic solutions of countable systems of quasilinear difference equations with periodic coefficients in the resonance case,” Dokl. Ukr. Akad. Nauk, 1, 13–15 (1996).
A. M. Samoilenko, D. I. Martynyuk, and N. A. Perestyuk, “Reducibility of nonlinear almost periodic systems of difference equations on an infinite-dimensional torus,” Ukr. Mat. Zh., 46, No. 9, 1216–1223 (1994).
A. M. Samoilenko and Yu. V. Teplinskii, Countable Systems of Differential Equations [in Russian], Institute of Mathematics, Ukrainian Academy of Sciences, Kiev (1993).
A. M. Samoilenko and Yu. V. Teplinskii, “Truncation method for the construction of invariant tori of countable systems of differential equations,” Differents. Uravn., 30, No. 2, 204–212 (1994).
A. M. Samoilenko and Yu. V. Teplinskii, “On invariant tori of differential systems with pulses in spaces of bounded number sequences,” Differents. Uravn., 21, No. 8, 1353–1361 (1985).
A. M. Samoilenko and Yu. V. Teplinskii, “On the smoothness of an invariant torus of a countable linear extension of a dynamical system on an m-dimensional torus,” Differents. Uravn., 30, No. 5, 781–790 (1994).
A. M. Samoilenko, Elements of the Mathematical Theory of Multifrequency Oscillations [in Russian], Nauka, Moscow (1987).
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Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 2, pp. 244–251, February, 1998.
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Samoilenko, A.M., Teplinskii, Y.V. Invariant tori of linear countable systems of discrete equations given on an infinite-dimensional torus. Ukr Math J 50, 278–286 (1998). https://doi.org/10.1007/BF02513451
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DOI: https://doi.org/10.1007/BF02513451