  Recent studies have established that, in addition to the well-known kicked
Harper model (KHM), an on-resonance double kicked rotor model (ORDKR) also has
Hofstadter's butterfly Floquet spectrum, with strong resemblance to the
standard Hofstadter's spectrum that is a paradigm in studies of the integer
quantum Hall effect. Earlier it was shown that the quasi-energy spectra of
these two dynamical models (i) can exactly overlap with each other if an
effective Planck constant takes irrational multiples of 2*pi and (ii) will be
different if the same parameter takes rational multiples of 2*pi. This work
makes some detailed comparisons between these two models, with an effective
Planck constant given by 2*pi M/N, where M and N are coprime and odd integers.
It is found that the ORDKR spectrum (with two periodic kicking sequences having
the same kick strength) has one flat band and $N-1$ non-flat bands whose
largest width decays in power law as K to the power of N+2, where K is a
kicking strength parameter. The existence of a flat band is strictly proven and
the power law scaling, numerically checked for a number of cases, is also
analytically proven for a three-band case. By contrast, the KHM does not have
any flat band and its band width scales linearly with K. This is shown to
result in dramatic differences in dynamical behavior, such as transient (but
extremely long) dynamical localization in ORDKR, which is absent in KHM.
Finally, we show that despite these differences, there exist simple extensions
of KHM and ORDKR (upon introducing an additional periodic phase parameter) such
that the resulting extended KHM and ORDKR are actually topologically
equivalent, i.e., they yield exactly the same Floquet-band Chern numbers and
display topological phase transitions at the same kick strengths. A theoretical
derivation of this topological equivalence is provided.
