Exponential Growth Rates of Periodic Asymmetric Oscillators

In this paper we will study the dynamics of the periodic asymmetric oscillator x" + q(+)(t)x(+) + q(-)(t)x(-) = 0, where q(+), q(-) epsilon L-1(R/2 pi Z) and x(+) = max(x, 0), x(-) = min(x, 0) for x epsilon R. It will be proved that the exponential growth rate X(x) := lim t-+infinity 1/t log root(x(t))(2) + (x'(t))(2) does exist for each non-zero solution x(t) of the oscillator. The properties of these rates, or the Lyapunov exponents, will be given using the induced circle diffeomorphism of the oscillator. The proof is extensively based on the Denjoy theorem in topological dynamics and the unique ergodicity theorem in ergodic theory.