  The incidence of rare events in fast-slow systems is investigated via
analysis of the large deviation principle (LDP) that characterizes the
likelihood and pathway of large fluctuations of the slow variables away from
their mean behavior -- such fluctuations are rare on short timescales but
become ubiquitous eventually. This LDP involves an Hamilton-Jacobi equation
whose Hamiltonian is related to the leading eigenvalue of the generator of the
fast process, and is typically non-quadratic in the momenta -- in other words,
the LDP for the slow variables in fast-slow systems is different in general
from that of any stochastic differential equation (SDE) one would write for the
slow variables alone. It is shown here that the eigenvalue problem for the
Hamiltonian can be reduced to a simpler algebraic equation for this Hamiltonian
for a specific class of systems in which the fast variables satisfy a linear
equation whose coefficients depend nonlinearly on the slow variables, and the
fast variables enter quadratically the equation for the slow variables. These
results are illustrated via examples, inspired by kinetic theories of turbulent
flows and plasma, in which the quasipotential characterizing the long time
behavior of the system is calculated and shown again to be different from that
of an SDE.
