I am unsure of what (any) "DAE" might encompass and am not highly familiar with them in general, but perhaps you should substitute a function that mm knows is periodic such as Sin[t] and unsubstitute the result when done. Mathematica knows how to do certain solving over modulus but I can't say if your equations fall within that evolving capability. f[z] might be one substitution if used wisely. If you bounds are within one period exactly then periodicity may not be required it can be assumed (I cannot say if many period will be a theorem used to solve your problem set of course).