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Recent studies have established that, in addition to the well-known kickedHarper model (KHM), an on-resonance double kicked rotor model (ORDKR) also hasHofstadter's butterfly Floquet spectrum, with strong resemblance to thestandard Hofstadter's spectrum that is a paradigm in studies of the integerquantum Hall effect. Earlier it was shown that the quasi-energy spectra ofthese two dynamical models (i) can exactly overlap with each other if aneffective Planck constant takes irrational multiples of 2*pi and (ii) will bedifferent if the same parameter takes rational multiples of 2*pi. This workmakes some detailed comparisons between these two models, with an effectivePlanck 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 havingthe same kick strength) has one flat band and $N-1$ non-flat bands whoselargest width decays in power law as K to the power of N+2, where K is akicking strength parameter. The existence of a flat band is strictly proven andthe power law scaling, numerically checked for a number of cases, is alsoanalytically proven for a three-band case. By contrast, the KHM does not haveany flat band and its band width scales linearly with K. This is shown toresult in dramatic differences in dynamical behavior, such as transient (butextremely long) dynamical localization in ORDKR, which is absent in KHM.Finally, we show that despite these differences, there exist simple extensionsof KHM and ORDKR (upon introducing an additional periodic phase parameter) suchthat the resulting extended KHM and ORDKR are actually topologicallyequivalent, i.e., they yield exactly the same Floquet-band Chern numbers anddisplay topological phase transitions at the same kick strengths. A theoreticalderivation of this topological equivalence is provided.

Kicked-Harper model vs On-Resonance Double Kicked Rotor Model: From Spectral Difference to Topological Equivalence