Nonlinear Analysis 73 (2010) 2071–2077 Contents lists available at ScienceDirect Nonlinear Analysis journal homepage: www.elsevier.com/locate/na The stability of the equilibrium of the damped oscillator with damping changing sign Qihuai Liu ∗ , Dingbian Qian, Zhiguo Wang School of Mathematical Sciences, Suzhou University, Suzhou 215006, People’s Republic of China article info Article history: Received 9 November 2009 Accepted 12 May 2010 MSC: 34C99 58F13 34D20 Keywords: Stability Damped oscillator Canonical transformation Moser’s twist theorem abstract In this paper, we prove a sufficient and necessary condition for the stability of the equilib- rium x = x ′ = 0 of the damped oscillator with damping changing sign x ′′ + h(t )x ′ + a(t )x 2n+1 + e(t , x) = 0, n ≥ 1 where a(t ), h(t ) are continuous and 1-periodic with h =  1 0 h(t )dt = 0, e(t , x) is contin- uous, 1-periodic in t and dominated by the power x 2n+2 in a neighborhood of x = 0. Crown Copyright © 2010 Published by Elsevier Ltd. All rights reserved. 1. Introduction The damped oscillator of one and a half degrees of freedom is described by the second order differential equation x ′′ + h(t )x ′ + g (t , x) = 0, (1.1) where the damping coefficient h(t ) is continuous, 1-periodic and the function g : R 2 → R is continuous and 1-periodic in the first variable t . The stability of the equilibrium x = x ′ = 0 of Eq. (1.1) is an important problem in the theory of dynamical systems. If damping coefficient h(t ) is positive, there are many interesting results concerning the asymptotic stability and stability by using the classical analysis methods. We refer to [1–5] for the discussions for damped linear oscillator x ′′ + h(t )x ′ + k 2 x = 0. (1.2) If h(t ) changes sign, the problem becomes more delicate. Generally speaking, the equilibrium is not asymptotic stable, but under some conditions the equilibrium still preserves the stability. In this paper, we are concerned with the stability of the equilibrium x = x ′ = 0 of the following damped oscillator with damping changing sign x ′′ + h(t )x ′ + a(t )x 2n+1 + e(t , x) = 0, n ≥ 1, (1.3) where a(t ), h(t ) is 1-periodic in the time t and h(t ) is continuous a.e. t ∈[0, 1] with h =  1 0 h(t )dt = 0, e(t , x) is also 1-periodic in the time t and dominated by the power x 2n+2 in a neighborhood of x = 0, both a(t ) and e(t , x) are continuous. ∗ Corresponding author. E-mail addresses: lqh520@tom.com (Q. Liu), dbqian@suda.edu.cn (D. Qian), zgwang@suda.edu.cn (Z. Wang). 0362-546X/$ – see front matter Crown Copyright © 2010 Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2010.05.035