Existence, uniqueness, and stability of periodic solutions of an equation of duffing type

Abstract

Abstract. We consider a second-order equation of Duffing type. Bounds for the derivative of the restoring force are given which ensure the existence and uniqueness of a periodic solution. Furthermore, the unique periodic solution is asymptotically stable with sharp rate of exponential decay. In particular, for a restoring term independent of the variable t, a necessary and sufficient condition is obtained which guarantees the existence and uniqueness of a periodic solution that is stable. §1. Introduction. This paper is devoted to the existence, uniqueness and stability of periodic solutions of the Duffing-type equationwhere g(t, x) is a T -periodic function in t and h(t) is a T -periodic function. The existence and multiplicity of periodic solutions of (1.1) or more general types of nonlinear second-order differential equations have been investigated extensively by many authors since C. Fabry, J. Mawhin and M.N. Nkashama initiated the study of the Ambrosetti-Prodi problem with periodic boundary condition [7]. However, the stability of periodic solutions is less extensively studied. In [18] R. Ortega studied (1.1) from a stability point of view and obtained an Ambrosetti-Proditype theorem under an assumption of convex nonlinearity. A.C. Lazer and P.J. McKenna established stability results by converting the equation (1.1) to a fixedpoint problem [14]. Recently, more complete results concerning the stability of periodic solutions of (1.1) were obtained by J.M. Alonso and R. Ortega [1,2]. Under the condition that the derivative of the restoring force is independent of t and positive, they found sharp bounds that guarantee global asymptotic stability. In [1], optimal bounds for stability are obtained. But the above results do not cover our Theorem 1, since in the theorem the derivative of the restoring force may be negative 2000 Mathematics Subject Classification. 34C25, 34C10.

The rate of decay of stable periodic solutions for Duffing equation with L p -conditions

“…To analyze the rate of exponential decay of x 0 (t), Chen and Li [5,6] suppose (1.3) and (1.4). They are equivalent to the conditions that there exist functions…”

“…Assuming (1.3) and T = 2π, the rate of exponential decay of 2πperiodic solution of (1.1) is c/2 (see also Chen and Li [5,6]). However, when G(t, x) is beyond π 2 /(2π) 2 = 1/4, the rate of decay of 2π-periodic solution of (1.1) could be less than c/2 (see the examples of [5,6,28]). Therefore, if (1.1) has a unique and stable 2π-periodic solution, then the rate of exponential decay c/2 is sharp.…”

“…Njoku and Omari [24] present the existence and stability of T -periodic solutions of (1.1) with the method of upper and lower solutions. Recently, Chen and Li [5,6] prove that (1.1) has a unique and stable T -periodic solution x 0 (t), and every solution other than the unique T -periodic solution x 0 (t) exponentially decays to x 0 (t) at the sharp rate c/2 (see [5,6] and Remark 1.3), under the classical conditions…”

“…two constant functions of t). And in (1.3), the upper bound of π 2 /T 2 is optimal in the sense of the L ∞ -norm, by the examples in [5,6,28]. The question arises whether we have some conclusions or not if G(t, x) is beyond the optimal bounds of the L ∞ -norm.…”

Bifurcation and stability of periodic solutions of Duffing equations

“…The stability of periodic solutions follows by computing the local index given by Ortega in [31]. More recent results concerning the stability and the sharpness of the rate of decay of periodic solutions can be found in [1,2,6,19,20].…”