  Since exoplanets were detected using the radial velocity method, they have
revealed a diverse distribution of orbital configurations. Amongst these are
planets in highly eccentric orbits (e > 0.5). Most of these systems consist of
a single planet but several have been found to also contain a longer period
planet in a near-circular orbit. Here we use the latest Keplerian orbital
solutions to investigate four known systems which exhibit this extreme
eccentricity diversity; HD 37605, HD 74156, HD 163607, and HD 168443. We place
limits on the presence of additional planets in these systems based on the
radial velocity residuals. We show that the two known planets in each system
exchange angular momentum through secular oscillations of their eccentricities.
We calculate the amplitude and timescale for these eccentricity oscillations
and associated periastron precession. We further demonstrate the effect of
mutual orbital inclinations on the amplitude of high-frequency eccentricity
oscillations. Finally, we discuss the implications of these oscillations in the
context of possible origin scenarios for unequal eccentricities.
