# Article

Full entry | PDF   (0.3 MB)
Keywords:
almost periodic differential systems; almost periodic solutions
Summary:
Let $f(t,x)$ be a vector valued function almost periodic in $t$ uniformly for $x$, and let ${\mathrm mod}(f)=L_1\oplus L_2$ be its frequency module. We say that an almost periodic solution $x(t)$ of the system $\dot{x} = f (t, x), \quad t\in \mathbb{R}, \ \ x\in D \subset \mathbb{R}^n$ is irregular with respect to $L_2$ (or partially irregular) if $({\mathrm mod}(x)+L_1) \cap L_2 = \lbrace 0\rbrace$. Suppose that $f(t,x) = A(t)x + X(t, x),$ where $A(t)$ is an almost periodic $(n\times n)$-matrix and ${\mathrm mod} (A)\cap {\mathrm mod}(X)= \lbrace 0\rbrace .$ We consider the existence problem for almost periodic irregular with respect to ${\mathrm mod} (A)$ solutions of such system. This problem is reduced to a similar problem for a system of smaller dimension, and sufficient conditions for existence of such solutions are obtained.
References:
[1] Samoilenko, A. M.: The Elements of Mathematical Theory of Multifrequency Oscillations. Nauka, Moskva, 1987. (Russian) MR 0928806
[2] Levitan, B. M., Zhikov, V. V.: Almost Periodic Functions and Differential Equations. Izdatelstvo Moskovskogo Universiteta, Moskva, 1978. (Russian) MR 0509035
[3] Hale, J. K.: Oscillations in Nonlinear Systems. Mc Graw-Hill, New York, 1963. MR 0150402 | Zbl 0115.07401
[4] Fink, A. M.: Almost Periodic Differential Equations. Lecture Notes in Mathematics 377, Springer, Berlin, 1974. MR 0460799 | Zbl 0325.34039
[5] Demidovich, B. P.: Lectures on Mathematical Stability Theory. Nauka, Moskva, 1967. (Russian) MR 0226126 | Zbl 0155.41601
[6] Bibikov, Y. N.: Multifrequency Nonlinear Oscillations and Their Bifurcations. Izdatelstvo Leningradskogo Universiteta, Leningrad, 1991. MR 1126680 | Zbl 0791.34032
[7] Zubov, V. I.: Oscillation Theory. Nauka, Moskva, 1979. (Russian)
[8] Kurzweil, J., Vejvoda, O.: On periodic and almost periodic solutions of the ordinary differential systems. Czechoslovak Math. J. 5 (1955), 362–370. MR 0076127
[9] Massera, J. L.: Observationes sobre les soluciones periodicas de ecuaciones differentiales. Boletin de la Facultad de Ingenieria 4 (1950), 37–45.
[10] Erugin, N. P.: On periodic solutions of the linear homogeneous system of differential equations. Doklady Akademii Nauk BSSR 6 (1962), 407–410. (Russian) MR 0137889
[11] Grudo, E. I.: On periodic solutions with incommensurable periods of periodic differential systems. Differentsial’nye uravneniya 22 (1986), 1409–1506. (Russian) MR 0865386
[12] Grudo, E. I., Demenchuk, A. K.: On periodic solutions with incommensurable periods of linear nonhomogeneous periodic differential systems. Differentsial’nye uravneniya 23 (1987), 409–416. (Russian) MR 0886568
[13] Demenchuk, A. K.: On almost periodic solutions of ordinary differential systems. Izvestiya Akademii Nauk BSSR. Ser. fiz.-mat. nauk. 4 (1987), 16–22. (Russian) MR 0913280 | Zbl 0632.34050
[14] Grudo, E. I., Demenchuk, A. K.: On periodic and almost periodic solutions of ordinary differential systems. Issues of qualitative theory of differential equations. Novosibirsk (1987), 23–29. (Russian) MR 0991144
[15] Demenchuk, A. K.: Quasiperiodic solutions of differential systems with frequency bases of solutions and right sides that are linearly independent over $\mathbb{Q}$. Differentsial’nye uravneniya 27 (1991), 1673–1679. (Russian) MR 1157672
[16] Demenchuk, A. K.: On a class of quasiperiodic solutions of differential systems. Doklady Akademii Nauk Belarusi 36 (1992), 14–17. (Russian) MR 1165405 | Zbl 0756.34049

Partner of