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