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The predictability of weather and climate is strongly state-dependent:special and extremely relevant atmospheric states like blockings are associatedwith anomalous instability. Indeed, typically, the instability of a chaoticdynamical system can vary considerably across its attractor. Such an attractoris in general densely populated by unstable periodic orbits that can be used toapproximate any forward trajectory through the so-called shadowing. Dynamicalheterogeneity can lead to the presence of unstable periodic orbits withdifferent number of unstable dimensions. This phenomenon - unstable dimensionsvariability - implies a serious breakdown of hyperbolicity and has considerableimplications in terms of the structural stability of the system and of thepossibility to describe accurately its behaviour through numerical models. As astep in the direction of better understanding the properties ofhigh-dimensional chaotic systems, we provide here an extensive numerical studyof the dynamical heterogeneity of the Lorenz '96 model in a parametricconfiguration leading to chaotic dynamics. We show that the detectedvariability in the number of unstable dimensions is associated with thepresence of many finite-time Lyapunov exponents that fluctuate about zero alsowhen very long averaging times are considered. The transition between regionsof the attractor with different degrees of instability comes with a significantdrop of the quality of the shadowing. By performing a coarse graining based onthe shadowing unstable periodic orbits, we can characterize the slowfluctuations of the system between regions featuring, on the average,anomalously high and anomalously low instability. In turn, such regions areassociated, respectively, with states of anomalously high and low energy, thusproviding a clear link between the microscopic and thermodynamical propertiesof the system.

Heterogeneity of the Attractor of the Lorenz '96 Model: Lyapunov Analysis, Unstable Periodic Orbits, and Shadowing Properties