PERIODIC SOLUTIONS OF AN INHOMOGENEOUS SECOND ORDER EQUATION WITH TIME-DEPENDENT DAMPING COEFFICIENT

摘要

In this paper we will study the periodic solutions of an inho-mogeneous second order equation with time-dependent damping coefficient: xe + (c +∈cos 2t )x + (m~2 +α) x + A cosωt = 0 (1) where c,α,∈, A are small parameters and m,ω positive integers. Ph ysically , the phenomenon of a time-dependetndamping coef-ficien t can occur in a special electrical circuit (RLC-circuit), or in a model equation for the study of rain-wind induced vibrations of a special oscillator. Because of the presence of a number of small parameters in equation (3) we will use the averaging method (up to third order) for the construction of approximations for the periodic solutions. The parameters c,a and A are considered to be small implying that they are expressed in the characteristic small parameter e of the problem: c = ∈c_1+∈~2C_2 +∈~3c_3, α = ∈α_1+∈~2α_2 +∈~3α_3, (2) A = ∈A_1+∈~2A_2+∈~3A_3, where c_i, α_i and A_i, i = 1, 2, 3 are of O(1). For m,∈{1, 2, 3}, it will be shown that an O(l)-periodic solution exists if m =ωand if m≠ωthe periodic solution is of order∈. Further, if c = O(∈),α= O(∈), and A = O(∈), for m =ω = 1 both stable and unstable periodic solutions exist but for m =ω= 2, 3 only stable periodic solutions are found. For the case that c = O(∈~2),α= O(∈~2), and A = O(∈~2), for m =ω= 2, 3 only stable periodic solutions are found. But for m = 3 andα= 9/64∈~2 + O(∈~3), c = O(∈~3), A = O(∈~3) both stable and unstable periodic solution exist. The stability of the periodic solutions follows from a new stabilit y diagram related to equation (3) with A ≡ 0.