  A numerical method to construct quasi-invariants (QI) has been explored in
double- and multi-bend achromat (DBA and MBA) lattices. The Fermilab's
Integrable Optics Test Accelerator (IOTA) and the University of Maryland
Electron Ring (UMER) built with exact invariants, pave the way for discovery of
a new type of nonlinear lattice. But it might be impossible to construct exact
invariants in various achromatic lattices for dedicated synchrotron light
source storage rings. Quasi-invariants, however, were found to be constructible
by tuning the sextupoles and octupoles in the DBA and MBA lattices. In our
exploration, QIs were constructed by confining the simulated trajectories to
tori in the phase space at a certain Poincare section, while simultaneously
maintaining monochromatic betatron tunes. The lattices with QIs share some
important features as integrable ones, such as a large dynamic aperture,
confined trajectory tori, robustness to resonances and errors, and a large
amplitude-dependent tune-spread.
