Proceedings of the Royal Society of Edinburgh: Section A Mathematics

Table of Contents - 1988 - Volume 108, Issue 1-2  

Research Article

A system of order 2 in which small sub-harmonics and large sub-harmonics coexist

Chike Obia1

a1 Mathematics Department, University of Lagos, Lagos, Nigeria

Synopsis

This paper establishes the coexistence of small and large subharmonics in a special case of the ordinary non-linear differential equation

S0308210500026494_eqn1

of order 2, where κ, ε are small parameters, λ>0 is a parameter independent of κ, ε, h(t) has the least period 2π and

S0308210500026494_eqn2

It is divided into three sections. In Section 1 a general analysis of the periodic solutions of (.), classified into small, medium or large, is given. In Section 2 the general theory of Section 1 is applied to the special form of (.) where k = vε1+s, v>0, s>0 constants, to obtain results from which we extract in Section 2 a theorem (Section 3) on the coexistence of a small periodic solution of order 1, several small sub-harmonics and several large sub-harmonics of the special case

S0308210500026494_eqnU3

of (.), where Q≧1 is an integer.

(Received August 26 1985)

(Revised October 27 1986)