  Orbital resonances play an important role in the dynamics of planetary
systems. Classical theoretical analyses found in textbooks report that
libration widths of first order mean motion resonances diverge for nearly
circular orbits. Here we examine the nature of this divergence with a
non-perturbative analysis of a few first order resonances interior to a
Jupiter-mass planet. We show that a first order resonance has two branches, the
pericentric and the apocentric resonance zone. As the eccentricity approaches
zero, the centers of these zones diverge away from the nominal resonance
location but their widths shrink. We also report a novel finding of "bridges"
between adjacent first order resonances: at low eccentricities, the apocentric
libration zone of a first order resonance smoothly connects with the
pericentric libration zone of the neighboring first order resonance. These
bridges may facilitate resonant migration across large radial distances in
planetary systems, entirely in the low eccentricity regime.
