  Cycloidal spin orders are common in multiferroics. One of the prototypical
examples is BiFeO3 (BFO) which shows a large polarization and a cycloidal
antiferromagnetic order at room temperature. Here we employ Landau theory and
phase-field simulations to analyze the coupled switching dynamics of
polarization and cycloidal antiferromagnetic orders in BFO. We are able to
identify 14 types of transitional spin structures between two cycloids and 9
electric-field-induced spin switching paths. We demonstrate the
electric-field-induced rotation of wave vectors of the cycloidal spins and
discover 2 types of cycloidal spin switching dynamics: fast local spin flips
and slow rotation of wave vectors. Also, we construct road maps to achieve the
switching between any two spin cycloids through multi-step applications of
electric fields. The work provides a theoretical framework for the
phenomenological description of spin cycloids and a fundamental understanding
of the switching mechanisms to achieve electrical control of magnetic orders.
