  We study 3D chaotic dynamics through an analysis of transport in a granular
flow in a half-full spherical tumbler rotated sequentially about two orthogonal
axes (a bi-axial "blinking" tumbler). The flow is essentially quasi-2D in any
vertical slice of the sphere during rotation about a single axis, and we
provide an explicit exact solution to the model in this case. Hence, the
cross-sectional flow can be represented by a twist map, allowing us to express
the 3D flow as a linked twist map (LTM). We prove that if the rates of rotation
about each axis are equal, then (in the absence of stochasticity) particle
trajectories are restricted to 2D surfaces consisting of a portion of a
hemispherical shell closed by a "cap"; if the rotation rates are unequal, then
particles can leave the surface they start on and traverse a volume of the
tumbler. The period-one structures of the governing LTM are examined in detail:
analytical expressions are provided for the location of period-one curves,
their extent into the bulk of the granular material, and their dependence on
the protocol parameters (rates and durations of rotations). Exploiting the
restriction of trajectories to 2D surfaces in the case of equal rotation rates
about the axes, a method is proposed for identifying and constructing 3D
Kolmogorov-Arnold-Moser (KAM) tubes around the normally elliptic period-one
curves. The invariant manifold structure arising from the normally hyperbolic
period-one curves is also examined. When the motion is restricted to 2D
surfaces, the structure of manifolds of the hyperbolic points in the bulk
differs from that corresponding to hyperbolic points in the flowing layer. Each
is reminiscent of a template provided by a non-integrable perturbation to a
Hamiltonian system, though the governing LTM is not. This highlights the novel
3D chaotic behaviors observed in this model dynamical system.
