Periodic solutions of periodic differential equations

Abstract

In this paper we extend the work of Bello [4] where he considered the periodic solutions of certain dynamical systems inside a cylindrical phase space with differential equations of the form y^n-1α₁y^{n-1+...+α_n-1}y⁽¹⁾ + f(y¹..,y^n-1,y) = 0 (\'=ddt (+) with the property that there is a ω>0 and a natural number K such that y (t+w) = y(t) + k ∀t (**)
with necessary and sufficient condition that the fundamental matrix φ (λ) of the characteristic equation
φ (λ) λ^n-1 + α₁ λ^n-2+...+ λ_n-1 = 0 (***)
of (*) have negative real parts (See [1], [7]), φ (λ) is stable asymptotically. The extension considered the periodic solutions of the differential equations of the type
y\'+λ(y\')y\"+b(y)y\'+f(y) = 0 (****)
with the property (**). The periodic solutions and the asymptotic behaviour of the solutions were investigated and analysed. Some theorems were proved and examples given to illustrate certain properties of the solutions.


Journal of the Nigerian Association of Mathematical Physics Vol. 10 2006: pp. 391-398