Boundedness and convergence of solutions of Duffing’s equation

Abstract

In this paper, we shall discuss boundedness of solutions of the equationunder suitable conditions. And we shall discuss asymptotic stability of a periodic solution and convergence of solutions for the equationfor a positive constant cand a periodic function e(t)under some restricted conditions.

The limiting amplitude principle for the nonlinear Lamb system

“…In the case m = 0, any solution of equation 14is either ω/a-periodic or tends to an ω/a-periodic solution y p (t), i. e., |y(t) − y p (t)| → 0 as t → ∞ (see Lemma 4 below). Therefore, using the explicit formula (19) for u(x, t), we obtain the results of Theorem 1.…”

“…where Finally, Corollary 2 and the explicit formula (19) for u(x, t) imply the convergence (17). However, the properties of S do not imply, in general, the results (28) and (12).…”

“…Then u p (x, t) is the solution of problem (2) -(4) with the initial data (ū 0 , u 1 ,ȳ 1 ). Since (ū 0 , u 1 ) ∈ H ω , the functionsf − (±x−at) andḡ + (±x+at) in (31) are ω/a-periodic in t. Hence, the equality (11) holds, and the convergence (12) follows from (19) and (28), sincē…”

“…Furthermore, every solution to equation (14) tends to the set S as t → ∞, i. e., ρ(Y (τ ), S τ ) → 0 as τ → ∞. Hence, the explicit formula (19) for the solutions u(x, t) implies that, for any R > 0,…”

“…If m = 0, then the behavior of solutions to equation (14) is more complicated. In the case F (y) = −ay − by 3 , the equation of a form (14) is called the Duffing equation with damping, see, for example, [11,19]. equation 14is a particular case of the generalized Liénard equations with a forcing term e(t) = 2κp (at), y + f (y)ẏ + g(y) = e(t).…”