Normal modes of coupled vortex gyration in two spatially separated   magnetic nanodisks
  We found from analytical derivations and micromagnetic numerical simulations that there exist two distinct normal modes in apparently complex vortex gyrotropic motions in two dipolar-coupled magnetic nanodisks. The normal modes have characteristic higher and lower single angular eigenfrequencies with their own elliptical orbits elongated along the x (bonding axis) and y axes, respectively. The superposition of the two normal modes results in coupled vortex gyrations, which depend on the relative vortex-state configuration in a pair of dipolar-coupled disks. This normal-mode representation is a simple means of understanding the observed complex vortex gyrations in two or more dipolar-interacting disks of various vortex-state configurations. 
Ki-Suk Lee, Hyunsung Jung, Dong-Soo Han, and Sang-Koog Kim
cond-mat.mtrl-sci