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The poincare method for an oscillator with quadratic nonlinearity

In the present paper, we deal with the case of quadratic non-linearity. An additional modification is introduced consisting in the elimination of constant derivation and second harmonic terms. The results obtained show that the domain of application of the Poincare method can be enlarged. | Vietnam Journal of Mechanics, VAST, Vol. 26 , 2004, No. 1 (23 - 30) THE POINCARE MET HOD FOR AN OSCILLAT OR WITH QUAD RATIC NONLINEARITY NGUYEN VAN DINH AND TRAN DUONG TRI Institute of Mechanics Vietnamese Academy of Science and Technology In [2], to evaluate free oscillation period of an undamped oscillator with large cubic restoring nonlinearity, a modified Poincare method has been proposed. There, in the neighbourhood of the free oscillation of interest , the strongly nonlinear system under consideration is assumed to be near certain linear one with unknown (to be evaluated) frequency. In the present paper, we deal with the case of quadratic non-linearity. An additional modification is introduced consisting in the elimination of constant derivation and second harmonic terms . The results obtained show that the domain of application of the Poincare method can be enlarged. 1 Systems under consideration and exact free oscillation period Consider an oscillator described by the differential equation x + x + (3x 2 = (1.1) 0, where (3 > 0 is coefficient of quadratic nonlinearity; other notations retain their significations explained in [2] . In the phase plane Oxx, the origin 0(0, 0) and the point I ( ~l , 0) represent two equilibrium states: the first one is a stable center, the second one is a saddle. T he domain where oscillations occur contains the center 0 and is bounded by an homoclinic orbit IJI starting from and ending at the same point I, intersecting the axis ~ , 0). If (3 is very small, the mentioned oscillatory domain is very 2 large and the standard Poincare method is applicable. Contrarily, if (3 is very large, the Ox at the point J ( oscillatory domain enclosed by the homoclinic orbit IJI is very small and the problem of oscillation becomes insignificant. So , below, we are interested only in the case in which the quadratic nonlinearity coefficient (3 is of "medium magnitude" - not too small and also not too large. The system under .