Branching of periodic solutions

Abstract

where g(x) is a sufficiently regular odd function with xg(x) > 0 for x # 0, and f(t) is a continuous (or perhaps piecewise continuous) X~-periodic function. We also assume that f(t) is an even function which is odd-harmonic (i.e. f(t +~r) =f(t)). Such equations as (i. i) occur in nonlinear mechanics, and it is of interest to learn about their periodic solutions. The very existence of such solutions is a substantial problem, since there are simple linear equations of the form (i. i) which have no periodic solutions. Other questions which arise after the establishment of existence are those of stability and constructibility. When equation(l, i) is linear, where g(x) = kZx, there is a unique 21r periodic solution provided that k 2 is not the square of an integer. What is more, this unique periodic solution is an even function which is odd-harmonic. This leads to the expectation that periodic solutions of (i. i) in general will be even and odd-harmonic, and in certain cases, this proves to be the case. However, other phenomena are known to occur with nonlinear equations. For some values of E, (i.i) may have more than one 2~ periodic solution, with the number of such solutions changing abruptly at some values of E.