Linkage between Adiabatic Invariance and Symmetry of Solutions

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, y_s and y_a, so that one is adiabatically symmetric, y_s(?t)=y_s(t)+o(ε^k) for all k≥0, and the other adiabatically antisymmetric, y_a(?t)=?y_a(t)+o(ε^k) for all k≥0. By using the techniques of exponential asymptotics to calculate y′_s(0) and y_a(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 y_s 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.