Abstract
In this paper we will study the periodic solutions of an inhomogeneous second order equation with time-dependent damping coefficient:(1)x..+(c+ϵcos2t)x.+(m2+α)x+Acosωt=0 where c, α, ϵ, A are small parameters and m, ω positive integers. Physically, the phenomenon of a time-dependent damping coefficient 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, α and A are considered to be small implying that they are expressed in the characteristic small parameter ϵ of the problem:(2)c=ϵc1+ϵ2c2+ϵ3c3,α=ϵα1+ϵ2α2+ϵ3α3,A=ϵA1+ϵ2A2+ϵ3A3, where ci, αi and Ai, i = 1, 2, 3 are of O(1).
For m, ω ∈ {1, 2, 3}, it will be shown that an O(1)-periodic solution exists if m = ω and if m ≠ ω the periodic solution is of order ϵ. Further, if c = O(ϵ), α = O(ϵ), and A = O(ϵ), 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(ϵ2), α = O(ϵ2), and A = O(ϵ2), for m = ω = 2, 3 only stable periodic solutions are found. But for m = 3 and α = (g/64)ϵ2 + O(ϵ3), c = O(ϵ3), A = O(ϵ3) both stable and unstable periodic solution exist. The stability of the periodic solutions follows from a new stability diagram related to equation (3) with A ≡ 0.
A periodic solution of van der Pol's equation with a damping coefficient $\lambda=20$
