Periodic solutions and their stability of some higher-order positively homogenous differential equations

In the present paper we study periodic solutions and their stability of the m-order differential equations of the form x((m)) + f(n)(x) = mu h(t), where the integers m, n >= 2, f(n)(x) = delta x(n) or delta vertical bar x vertical bar(n) with delta = +/- 1, h(t) is a continuous T-periodic function of non-zero average, and mu is a positive small parameter. By using the averaging theory, we will give the existence of T-periodic solutions. Moreover, the instability and the linear stability of these periodic solutions will be obtained. (c) 2017 Elsevier Ltd. All rights reserved.