We construct some explicit formulas of rational maps
and transcendental meromorphic functions having Herman rings
of period strictly larger than one.
This gives an answer to a question raised by Shishikura in the 1980s.
Moreover, the formulas of some rational maps with nested Herman rings
are also found.
To obtain the formulas of transcendental meromorphic functions having
periodic Herman rings,
a crucial step is to find an explicit family of transcendental entire
functions
having bounded Siegel disks of all possible periods and rotation numbers.
This is based on proving the existence of a Mandelbrot-like set of
period one
in the corresponding parameter space.