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We performed micromagnetic numerical calculations to explore the novel dynamics of two vortex cores magnetically coupled via long-range dipolar interactions and short-range exchange interactions in a single element of spherical nano-shells. One vortex core is located in the upper half-shell and the other in the lower half-shell. Under a given static magnetic field, they show unique coupled dynamic motions that have yet to be reported so far. For the parallel core orientations, they show slow and large-amplitude orbital motions around the static field direction, and both cores’ positions on the surface are out-of-phase with each other. This motion resembles the precession motion of a single vortex core around the static magnetic field direction in a nanosphere. In addition to such a precession-like motion, each core simultaneously shows relatively fast and small-amplitude gyration-like motions that move along with the precession-like motion. The superposition of the two distinct dynamic modes results in an overall nutation-like motion as observed in other non-linear systems such as gyroscopes and geodynamics. For the anti-parallel configuration of both cores, further complex fast decaying, irregular motions are shown, because their opposite core orientations show an almost null surface-normal component of the average magnetization of both cores as well as the opposite rotational senses of their gyration motions. This work provides not only a further understanding of dynamic-coupled motions of magnetic vortices in 3D nano-structures but also an efficient way to utilize nano-spherical shells in other possible applications related to novel dynamic-modes excitation. [ABSTRACT FROM AUTHOR]