  The classical and quantum dynamics of two ultra-strongly coupled and weakly
nonlinear resonators cannot be explained using the Discrete Nonlinear
Schr\"odinger Equation or the Bose-Hubbard model, respectively. Instead, a
model beyond the Rotating Wave Approximation must be studied. In the classical
limit this model is not integrable and becomes chaotic for a finite window of
parameters. For the quantum dimer we find corresponding regions of stability
and chaos. The more striking consequence for both classical and quantum chaos
is that the tunneling time between the sites becomes unpredictable. These
results, including the transition to chaos, can be tested in experiments with
superconducting microwave resonators.
