EGU General Assembly 2021
© Author(s) 2022. This work is distributed under
the Creative Commons Attribution 4.0 License.

Unstable Periodic Orbits Sampling and Its Applications to Climate Models

Chiara Cecilia Maiocchi1,2 and Valerio Lucarini1,2
Chiara Cecilia Maiocchi and Valerio Lucarini
  • 1University of Reading, Mathematics and Statistics, United Kingdom of Great Britain – England, Scotland, Wales (
  • 2Centre for the Mathematics of Planet Earth, University of Reading, Reading, UK

Climate can be interpreted as a complex, high dimensional non-equilibrium stationary system characterised by multiple time and space scales spanning various orders of magnitude. Statistical mechanics and dynamical system theory have been key mathematical frameworks in the study of the climate system. In particular, unstable periodic orbits (UPOs) have been proven to provide relevant insight in the understanding of its statistical properties. In a recent paper Lucarini and Gritsun [1] provided an alternative approach for understanding the properties of the atmosphere.

In general, UPOs decomposition plays a relevant role in the study of chaotic dynamical systems. In fact, UPOs densely populate the attractor of a chaotic system, and can therefore be thought as building blocks to construct the dynamic of the system itself. Since they are dense in the attractor, it is always possible to find a UPO arbitrarily near to a chaotic trajectory: the trajectory will remain close to the UPO, but it will never follow it indefinitely, because of its instability. Loosely speaking, a chaotic trajectory is repelled between neighbourhoods of different UPOs and can thus be approximated in terms of these periodic orbits. The statistical properties of the system can then be reconstructed from the full set of periodic orbits in this fashion.

The numerical study of UPOs thus represents a relevant problem and an interesting research topic for Climate Science and chaotic dynamical systems in general. In this presentation we address the problem of sampling UPOs for the paradigmatic Lorenz-63 model. First, we present results regarding the measure of the system, thus its statistical properties, using UPOs theory, namely with the trace formulas. Second, we introduce a more innovative approach, considering UPOs as global states of the system. We approximate the exact dynamics by a suitable Markov chain process, describing how the system hops on different UPOs, and we compare the two different approaches.  

[1] V. Lucarini and A. Gritsun, “A new mathematical framework for atmospheric blocking events,” Climate Dynamics, vol. 54, pp. 575–598, Jan 2020.

How to cite: Maiocchi, C. C. and Lucarini, V.: Unstable Periodic Orbits Sampling and Its Applications to Climate Models, EGU General Assembly 2021, online, 19–30 Apr 2021, EGU21-5457,, 2021.


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