Abstract: | In this article, for the symmetric pendulum equation and the symmetric bisuperlinear equation respectively, we show that there are two one-parameter families of solutions, ys and ya, so that one is adiabatically symmetric, ys(?t)=ys(t)+o(εk) for all k≥0, and the other adiabatically antisymmetric, ya(?t)=?ya(t)+o(εk) for all k≥0. By using the techniques of exponential asymptotics to calculate y′s(0) and ya(0), we demonstrate that, in general, they are not genuinely symmetric or antisymmetric, because these quantities are in fact exponentially small. Finally, after establishing a relationship between the total change in the leading-order adiabatic invariant and the quantity y′s(0) for the family of solutions ys of the bisuperlinear equation, we are able to reveal explicitly how the behavior of the adiabatic invariant depends on the complex singularities of the equation. |