  It is widely believed that resonant orbits play an important role in
formation and evolution of bars and large-scale spirals in galaxy discs. These
resonant orbits have been studied in a number of specific potentials, often
with an imposed bar component. In this paper I show that families of resonant
(e.g., two-dimensional $x_1$) orbits of differing eccentricities can be excited
at a common pattern speed, in a variety of axisymmetric potentials. These
families only exist over finite ranges of frequency in most of these
potentials. Populations of such resonant eccentric orbits (REOs) can provide
the backbone of both bars and spirals. At each frequency in the allowed range
there is a maximum eccentricity, beyond which the REOs generically become
quasi-stable (or `sticky'), then unstable (or chaotic), as the eccentricity
increases, at values that depend on the potential and the orbit frequency.
Sticky and chaotic orbits have been extensively studied recently with
invariant/unstable manifolds in a variety of phase planes, but it is found that
studying them as a function of eccentricity and pattern speed provides a
particularly useful framework for classifying them and their stability
transitions. The characteristics of these orbit families depend on the galaxy
potential and the pattern speed, and as backbones of bars and spirals can help
understand a number of observed or predicted regularities. These include: the
size and speed of bars in different potentials, the range of pattern speeds and
windup rates in spirals within galaxy discs, and constraints wave growth.
