  The predictability of weather and climate is strongly state-dependent:
special and extremely relevant atmospheric states like blockings are associated
with anomalous instability. Indeed, typically, the instability of a chaotic
dynamical system can vary considerably across its attractor. Such an attractor
is in general densely populated by unstable periodic orbits that can be used to
approximate any forward trajectory through the so-called shadowing. Dynamical
heterogeneity can lead to the presence of unstable periodic orbits with
different number of unstable dimensions. This phenomenon - unstable dimensions
variability - implies a serious breakdown of hyperbolicity and has considerable
implications in terms of the structural stability of the system and of the
possibility to describe accurately its behaviour through numerical models. As a
step in the direction of better understanding the properties of
high-dimensional chaotic systems, we provide here an extensive numerical study
of the dynamical heterogeneity of the Lorenz '96 model in a parametric
configuration leading to chaotic dynamics. We show that the detected
variability in the number of unstable dimensions is associated with the
presence of many finite-time Lyapunov exponents that fluctuate about zero also
when very long averaging times are considered. The transition between regions
of the attractor with different degrees of instability comes with a significant
drop of the quality of the shadowing. By performing a coarse graining based on
the shadowing unstable periodic orbits, we can characterize the slow
fluctuations of the system between regions featuring, on the average,
anomalously high and anomalously low instability. In turn, such regions are
associated, respectively, with states of anomalously high and low energy, thus
providing a clear link between the microscopic and thermodynamical properties
of the system.
