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Most multiferroic materials with coexisting ferroelectric and magnetic order exhibit cycloidal antiferromagnetism with wavelength much larger than lattice spacing. The prototypical example is bismuth ferrite (BiFeO$_3$ or BFO), a room-temperature multiferroic considered for a number of technological applications, including magnetic memories with electric-write capability. While most applications requires small sizes such as nanoparticles, little is known about the state of these materials when their sizes are comparable to the cycloid wavelength. This work describes a theory of cycloid magnetism in nanoparticles. It is argued that magnetic anisotropy close to the surface has a huge impact on the ground state cycloidal wavevector, leading to several observable consequences. For certain sizes the cycloidal wavevector is bistable, an effect that may be exploited in the design of novel memory devices.
