Heterogeneity of the attractor of the Lorenz '96 model: Lyapunov analysis, unstable periodic orbits, and shadowing properties

Maiocchi, C. C., Lucarini, V. ORCID: https://orcid.org/0000-0001-9392-1471, Gritsun, A. and Sato, Y. (2024) Heterogeneity of the attractor of the Lorenz '96 model: Lyapunov analysis, unstable periodic orbits, and shadowing properties. Physica D: Nonlinear Phenomena, 457. 133970. ISSN 1872-8022 doi: 10.1016/j.physd.2023.133970

Abstract/Summary

It is well known that the predictability of weather and climate is strongly state-dependent. Special, easily recognisable, and extremely relevant atmospheric states like blockings are associated with anomalous instability. This reflects the general property that the attractors of chaotic dynamical systems can feature considerable heterogeneity in terms of dynamical properties, and specifically, of their instability. The attractor of a chaotic dynamical systems is 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 - has considerable implications in terms of the structural stability of the system and of the possibility to model accurately its behaviour through numerical models. As a step in the direction of better understanding the properties of high-dimensional chaotic systems, here we provide an extensive numerical investigation of the variability of the dynamical properties across the attractor of the much studied Lorenz '96 model. By combining the Lyapunov analysis of the tangent space with the study of the shadowing of the chaotic trajectory performed by a very large set of unstable periodic orbits, we show that the observed variability in the number of unstable dimensions, which is a serious breakdown of hyperbolicity, is associated with the presence of a substantial number of 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 is associated with a significant drop of the quality of the shadowing. By performing a coarse graining based on the shadowing unstable periodic orbits, we are able to characterise 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.

Item Type	Article
URI	https://reading-clone.eprints-hosting.org/id/eprint/113738
Item Type	Article
Refereed	Yes
Divisions	Interdisciplinary Research Centres (IDRCs) > Centre for the Mathematics of Planet Earth (CMPE) Science > School of Mathematical, Physical and Computational Sciences > Department of Mathematics and Statistics
Publisher	Elsevier
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