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On quasiperiodic oscillations of a nonlinear dynamic system of liapunov type with time lag

The paper is concerned with the investigation of the quasiperiodic oscillations of a nonlinear dynamic system of Liapunov type with time lag. The following results are obtained: The necessary and sufficient conditions for the existence of the quasiperiodic solution describing the oscillating processes. The approximate quasiperiodic solution in the power series. | Vietnam Journal of Mechanics, NCNST of Vietnam T. XX, 1998, No 1 (1- 8) I ON QUASIPERIODIC OSCILLATIONS OF A NONLINEAR DYNAMIC SYSTEM OF LIAPUNOV TYPE WITH TIME LAG LE XUAN CAN Department of Mechanics - VNU Hanoi ABSTRACT. The paper is concerned with the investigation of the quasiperiodic oscillations of a nonlinear dynamic system of Liapunov type with time lag. The following results are obtained: - The necessary and sufficient conditions for the existence of the quasiperiodic solution describing the oscillating processes. - The approximate quasiperiodic solution in the power series. - The quasiperiodic oscillations of a nonlinear dynamic system of Dulling type with the quasiperiodic perturbations. L. Let us consider a nonlinear dynamic systerr. described by the differential equa,ion of the form (1.1) vhere e is a small parameter, X(x) is a power series in x of the form (1.2) IC is a continuous in t and quasiperiodic function with fre1uency basis VJ, v 2 , •.• ,vn d;t md analytical function in x, XL>., ~;, in the domair D. ixLl. d dt = dtx(t- C:.), t. is a positive number. ~ere XL>. = x(t- t.), Together with, given differential equation we consider the following differential ~quation called deg~:d~rate ~quatibn, which cari be~btained from (1.1) by putting : = 0: (1.3) 1 It is easy to see that equation (1.3) is differential equation of Liapunov type because we can find its initial integral in the form (~~) 2 2 2 +w x + 2 J X(x)dx = const. (1.4) It is known that the Liapunov type equation (1.3) has a continuous in t and periodic solution depending on two .parameters with period T in the form [1] T ( 1 + a2c 2 + a3c 3 + . ) , = -271' w (1.5) where the first coefficient ai in expression (1.5) is different from zero and has the even index (denoted by a£), c is value of x in initial moment t = 0. We prove first the following theorem: Theorem 1.1. If the developmer:tt of function X(x) in power series is such that the first coefficient "Yi .