Electron storage rings are widely used for high luminosity colliders, damping rings in high-energy linear colliders, and synchrotron light sources. They have become essential facilities to study high-energy physics and material and medical sciences. To further increase the luminosity of colliders or the brightness of synchrotron light sources, the beam emittance is being continually pushed downward, recently to the nanometer region. In the next decade, another order of reduction is expected. This requirement of ultra-low emittance presents many design challenges in beam dynamics, including better analysis of maps and improvement of dynamic apertures. To meet these challenges, we have refined transfer maps of common elements in storage rings and developed a new method to compute the resonance driving terms as they are built up along a beamline. The method is successfully applied to a design of PEP-X as a future light source with 100-pm emittance. As a result, we discovered many unexpected cancelations of the fourth-order resonance terms driven by sextupoles within an achromat. Show Partial Abstract