  Highly eccentric binary systems appear in many astrophysical contexts,
ranging from tidal capture in dense star clusters, precursors of stellar
disruption by massive black holes, to high-eccentricity migration of giant
planets. In a highly eccentric binary, the tidal potential of one body can
excite oscillatory modes in the other during a pericenter passage, resulting in
energy exchange between the modes and the binary orbit. These modes exhibit one
of three behaviors over multiple passages: low-amplitude oscillations, large
amplitude oscillations corresponding to a resonance between the orbital
frequency and the mode frequency, and chaotic growth. We study these phenomena
with an iterative map, fully exploring how the mode evolution depends on the
pericenter distance and other parameters. In addition, we show that the
dissipation of mode energy results in a quasi-steady state, with gradual
orbital decay punctuated by resonances, even in systems where the mode
amplitude would initially grow stochastically. A newly captured star around a
black hole can experience significant orbital decay and heating due to the
chaotic growth of the mode amplitude and dissipation. A giant planet pushed
into a high-eccentricity orbit may experience a similar effect and become a hot
or warm Jupiter.
