  We study the stroboscopic dynamics of a spin-$S$ object subjected to
$\delta$-function kicking in the transverse magnetic field which is generated
following the Fibonacci sequence. The corresponding classical Hamiltonian map
is constructed in the large spin limit, $S \rightarrow \infty$. Upon evolving
such a map for large kicking strength and time period, the phase space appears
to be chaotic; interestingly, however, the geodesic distance increases linearly
with the stroboscopic time implying that the Lyapunov exponent is zero. We
derive the Sutherland invariant for the underlying $SO(3)$ matrix governing the
dynamics of classical spin variables and study the orbits for weak kicking
strength. For the quantum dynamics, we observe that although the phase
coherence of a state is retained throughout the time evolution, the
fluctuations in the mean values of the spin operators exhibit fractality which
is also present in the Floquet eigenstates. Interestingly, the presence of an
interaction with another spin results in an ergodic dynamics leading to
infinite temperature thermalization.
