Unstable Periodic Orbits: a language to interpret the complexity of chaotic systems

Unstable periodic orbits (UPOs), exact periodic solutions of the evolution equation, offer a very powerful framework for studying chaotic dynamical systems, as they allow one to dissect their dynamical structure. UPOs can be considered the skeleton of chaotic dynamics, its essential building blocks. In fact, it is possible to prove that in a chaotic system, UPOs are dense in the attractor, meaning that it is always possible to find a UPO arbitrarily near any chaotic trajectory. We can thus think of the chaotic trajectory as being approximated by different UPOs as it evolves in time, jumping from one UPO to another as a result of their instability. In this thesis we provide a contribution towards the use of UPOs as a tool to understand and distill the dynamical structure of chaotic dynamical systems. We will focus on two models, characterised by different properties, the Lorenz-63 and Lorenz-96 model. The process of approximation of a chaotic trajectory in terms of UPOs will play a central role in our investigation. In fact, we will use this tool to explore the properties of the attractor of the system under the lens of its UPOs. In the first part of the thesis we consider the Lorenz-63 model with the classic parameters’ value. We investigate how a chaotic trajectory can be approximated using a complete set of UPOs up to symbolic dynamics’ period 14. At each instant in time, we rank the UPOs according to their proximity to the position of the orbit in the phase space. We study this process from two different perspectives. First, we find that longer period UPOs overwhelmingly provide the best local approximation to the trajectory. Second, we construct a finite-state Markov chain by studying the scattering of the trajectory between the neighbourhood of the various UPOs. Each UPO and its neighbourhood are taken as a possible state of the system. Through the analysis of the subdominant eigenvectors of the corresponding stochastic matrix we provide a different interpretation of the mixing processes occurring in the system by taking advantage of the concept of quasi-invariant sets. In the second part of the thesis we provide an extensive numerical investigation of the variability of the dynamical properties across the attractor of the much studied Lorenz ’96 dynamical system. 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 shows 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

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.Comment: 28 pages, 11 figures, final accepted versio

Decomposing the dynamics of the Lorenz 1963 model using unstable periodic orbits: averages, transitions, and quasi-invariant sets

Unstable periodic orbits (UPOs) are a valuable tool for studying chaotic dynamical systems, as they allow one to distill their dynamical structure. We consider here the Lorenz 1963 model with the classic parameters’ value. We investigate how a chaotic trajectory can be approximated using a complete set of UPOs up to symbolic dynamics’ period 14. At each instant, we rank the UPOs according to their proximity to the position of the orbit in the phase space. We study this process from two different perspectives. First, we find that longer period UPOs overwhelmingly provide the best local approximation to the trajectory. Second, we construct a finite-state Markov chain by studying the scattering of the orbit between the neighborhood of the various UPOs. Each UPO and its neighborhood are taken as a possible state of the system. Through the analysis of the subdominant eigenvectors of the corresponding stochastic matrix, we provide a different interpretation of the mixing processes occurring in the system by taking advantage of the concept of quasi-invariant sets. The attractor of a chaotic system is densely populated by an infinite number of unstable periodic orbits (UPOs), which are exact periodic solutions of the evolution equations. UPOs can be used to decompose the complex phenomenology of a chaotic flow into elementary components and have shown great potential for the understanding of macroscopic features in turbulent fluid flows. Here, we investigate how a long forward trajectory of the celebrated Lorenz 1963 model featuring the classical parameters’ value can be seen as a scattering process where the scatterers are the UPOs. This process helps elucidate how a generic ensemble of initial conditions converges to the invariant measure through diffusion and provides a new interpretation of quasi-invariant sets of the system in terms of UPOs

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

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

Tocilizumab for treatment of severe covid-19 patients: Preliminary results from smatteo covid19 registry (smacore)

Objective: This study aimed to assess the role of Tocilizumab therapy (TCZ) in terms of ICU admission and mortality rate of critically ill patients with severe COVID-19 pneumonia. Design: Patients with COVID-19 pneumonia were prospectively enrolled in SMAtteo COvid19 REgistry (SMACORE). A retrospective analysis of patients treated with TCZ matched using propensity score to patients treated with Standard Of Care (SOC) was conducted. Setting: The study was conducted at IRCCS Policlinico San Matteo Hospital, Pavia, Italy, from March 14, 2020 to March 27, 2020. Participants: Patients with a confirmed diagnosis of COVID-19 hospitalized in our institution at the time of TCZ availability. Interventions: TCZ was administered to 21 patients. The first administration was 8 mg/kg (up to a maximum 800 mg per dose) of Tocilizumab intravenously, repeated after 12 h if no side effects were reported after the first dose. Main Outcomes and Measures: ICU admission and 7-day mortality rate. Secondary outcomes included clinical and laboratory data. Results: There were 112 patients evaluated (82 were male and 30 were female, with a median age of 63.55 years). Using propensity scores, the 21 patients who received TCZ were matched to 21 patients who received SOC (a combination of hydroxychloroquine, azithromycin and prophylactic dose of low weight heparin). No adverse event was detected following TCZ administration. This study found that treatment with TCZ did not significantly affect ICU admission (OR 0.11; 95% CI between 0.00 and 3.38; p = 0.22) or 7-day mortality rate (OR 0.78; 95% CI between 0.06 and 9.34; p = 0.84) when compared with SOC. Analysis of laboratory measures showed significant interactions between time and treatment regarding C-Reactive Protein (CRP), alanine aminotransferase (ALT), platelets and international normalized ratio (INR) levels. Variation in lymphocytes count was observed over time, irrespective of treatment. Conclusions: TCZ administration did not reduce ICU admission or mortality rate in a cohort of 21 patients. Additional data are needed to understand the effect(s) of TCZ in treating patients diagnosed with COVID-19