  We present our study on the emergent states of two interacting nonlinear
systems with differing dynamical time scales. We find that the inability of the
interacting systems to fall in step leads to difference in phase as well as
change in amplitude. If the mismatch is small, the systems settle to a
frequency synchronized state with constant phase difference. But as mismatch in
time scale increases, the systems have to compromise to a state of no
oscillations. We illustrate this for standard nonlinear systems and identify
the regions of quenched dynamics in the parameter plane. The transition curves
to this state are studied analytically and confirmed by direct numerical
simulations. As an important special case, we revisit the well-known model of
coupled ocean atmosphere system used in climate studies for the interactive
dynamics of a fast oscillating atmosphere and slowly changing ocean. Our study
in this context indicates occurrence of multi stable periodic states and steady
states of convection coexisting in the system, with a complex basin structure.
