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The effect of dynamical tide ``kicks" on eccentric binary orbits isconsidered using the orbital mapping method. It is demonstrated that when modedamping is negligible the mode amplitude will generically grow in time for allvalues of orbital eccentricity and semi-major axis, even for small kicksoutside the regime exhibiting diffusive growth. The origin of the small-kickgrowth is the change in kick size from orbit to orbit, an effect quadratic inthe mode amplitude. When damping of the mode is included, the growth is shutoff when the damping time is shorter than the growth time. Hence, in practice,kicks of sufficient size and long mode damping times are required forinteresting levels of growth to occur. Application to the circularization ofhot Jupiters is discussed. Previous investigations found that diffusive growthof the planetary f-mode in the large-kick regime would lead to rapid orbitalshrinkage, but upon exiting the diffusive regime at $e \sim 0.9$ the theorywould predict a large population of highly eccentric orbits. Simulationspresented here show that subsequent orbital evolution relying on the small-kickregime may further decrease the eccentricity to $e \sim 0.2$ on timescales muchless than the Gyrs ages of these systems.

Large Dynamical Tide Amplitudes from Small Kicks at Pericenter