Effortless estimation of basins of attraction

We present a fully automated method that identifies attractors and their basins of attraction without approximations of the dynamics. The method works by defining a finite state machine on top of the dynamical system flow. The input to the method is a dynamical system evolution rule and a grid that partitions the state space. No prior knowledge of the number, location, or nature of the attractors is required. The method works for arbitrarily high-dimensional dynamical systems, both discrete and continuous. It also works for stroboscopic maps, Poincaré maps, and projections of high-dimensional dynamics to a lower-dimensional space. The method is accompanied by a performant open-source implementation in the DynamicalSystems.jl library. The performance of the method outclasses the naïve approach of evolving initial conditions until convergence to an attractor, even when excluding the task of first identifying the attractors from the comparison. We showcase the power of our implementation on several scenarios, including interlaced chaotic attractors, high-dimensional state spaces, fractal basin boundaries, and interlaced attracting periodic orbits, among others. The output of our method can be straightforwardly used to calculate concepts, such as basin stability and final state sensitivity. © 2022 Author(s)

Earth’s albedo and its symmetry

Abstract The properties of Earth's albedo and its symmetries are analyzed using twenty years of space-based Energy Balanced And Filled product of Clouds and the Earth's Radiant Energy System measurements. Despite surface asymmetries, top of the atmosphere temporally & hemispherically averaged reflected solar irradiance R appears symmetric over Northern/Southern hemispheres. This is confirmed with the use of surrogate time-series, which provides margins of 0.1±0.28Wm?2 for possible hemispheric differences supported by Clouds and Earth's Radiant System data. R time-series are further analyzed by decomposition into a seasonal (yearly and half yearly) cycle and residuals. Variability in the reflected solar irradiance is almost entirely (99%) due to the seasonal variations, mostly due to seasonal variations in insolation. The residuals of hemispherically averaged R are not only small, but also indistinguishable from noise, and thus not correlated across hemispheres. This makes yearly and sub-yearly timescales unlikely as the basis for a symmetry-establishing mechanism. The residuals however contain a global trend that is large, as compared to expected albedo feedbacks. It is also hemispherically symmetric, and thus indicates the possibility of a symmetry enforcing mechanism at longer timescales. To pinpoint precisely which parts of the Earth system establish the hemispheric symmetry, we create an energetically consistent cloud-albedo field from the data. We show that the surface albedo asymmetry is compensated by asymmetries between clouds over extra-tropical oceans, with southern hemispheric storm-tracks being 11% cloudier than their northern hemisphere counterparts. This again indicates that, assuming the albedo symmetry is not a result of chance, its mechanism likely operates on large temporal and spatial scales

Husimi function for electrons moving in magnetic fields

Husimi functions allow one to obtain sensible and useful phase space probability distributions from quantumechanical wavefunctions or classical wave fields, linking them to (semi-)classical methods and intuition. They have been used in several fields of physics, including electronic transport. We show that applying Husimi functions to ballistic electron dynamics in magnetic fields needs special consideration in order for them to obey gauge invariance and energy conservation. We therefore extend the Husimi function formalism to allow for magnetic fields making use of magnetic translation operators. We demonstrate the application in tight-binding magneto-transport calculations in graphene nanodevices, highlighting connections with Klein tunneling. In continuation of recent work, with this paper we further pave the way for using the Husimi function to unravel quantum transport phenomena in nanodevices

Agents.jl: A performant and feature-full agent based modelling software of minimal code complexity

Agent based modelling is a simulation method in which autonomous agents interact with their environment and one another, given a predefined set of rules. It is an integral method for modelling and simulating complex systems, such as socio-economic problems. Since agent based models are not described by simple and concise mathematical equations, code that generates them is typically complicated, large, and slow. Here we present Agents.jl, a Julia-based software that provides an ABM analysis platform with minimal code complexity. We compare our software with some of the most popular ABM software in other programming languages. We find that Agents.jl is not only the most performant, but also the least complicated software, providing the same (and sometimes more) features as the competitors with less input required from the user. Agents.jl also integrates excellently with the entire Julia ecosystem, including interactive applications, differential equations, parameter optimization, and more. This removes any ``extensions library'' requirement from Agents.jl, which is paramount in many other tools

Minimal recipes for global cloudiness

Clouds are primary modulators of Earth’s energy balance. It is thus important to understand the links connecting variabilities in cloudiness to variabilities in other state variables of the climate system, and also describe how these links would change in a changing climate. A conceptual model of global cloudiness can help elucidate these points. In this work we derive simple representations of cloudiness, that can be useful in creating a theory of global cloudiness. These representations illustrate how both spatial and temporal variability of cloudiness can be expressed in terms of basic state variables. Specifically, cloud albedo is captured by a nonlinear combination of pressure velocity and a measure of the low-level stability, and cloud longwave effect is captured by surface temperature, pressure velocity, and standard deviation of pressure velocity. We conclude with a short discussion on the usefulness of this work in the context of global warming response studies

A unified and automated approach to attractor reconstruction

We present a fully automated method for the optimal state space reconstruction from univariate and multivariate time series. The proposed methodology generalizes the time delay embedding procedure by unifying two promising ideas in a symbiotic fashion. Using non-uniform delays allows the successful reconstruction of systems inheriting different time scales. In contrast to the established methods, the minimization of an appropriate cost function determines the embedding dimension without using a threshold parameter. Moreover, the method is capable of detecting stochastic time series and, thus, can handle noise contaminated input without adjusting parameters. The superiority of the proposed method is shown on some paradigmatic models and experimental data from chaotic chemical oscillators

DynamicalSystems.jl - Nonlinear dynamics software for everyone

DynamicalSystems.jl is an award-winning software library for nonlinear dynamics and nonlinear timeseries analysis. It was born out of frustration when facing two problems: (1) the lack of a general-purpose accessible software for nonlinear dynamics for using in the lecture hall, and (2) the complete and utter lack of reproducibility of the entire field. DynamicalSystems.jl was designed to address both of these problems, and also offer much more. It is structured in a way of an encyclopaedia on nonlinear dynamics, and was written with clarity of source code as the highest priority. By now, it has seen contributions by dozens of individuals initially unaffiliated with the library. Besides making nonlinear dynamics accessible, this has also enabled brand new kind of research, born on, and done entirely on, GitHub, an open source software collaboration platform. In this presentation I will discuss how DynamicalSystems.jl came to be, why it is the first software to succeed in what it does, and ultimately, how it can make the field of nonlinear dynamics reproducible

Nonlinear dynamics: A concise introduction interlaced with Code

This concise and up-to-date textbook provides an accessible introduction to the core concepts of nonlinear dynamics as well as its existing and potential applications. The book is aimed at students and researchers in all the diverse fields in which nonlinear phenomena are important. Since most tasks in nonlinear dynamics cannot be treated analytically, skills in using numerical simulations are crucial for analyzing these phenomena. The text therefore addresses in detail appropriate computational methods as well as identifying the pitfalls of numerical simulations. It includes numerous executable code snippets referring to open source Julia software packages. Each chapter includes a selection of exercises with which students can test and deepen their skills

Robustness of ballistic transport in antidot superlattices

The magneto-resistance of antidot lattices shows pronounced peaks, which became a hallmark of ballistic electron transport. While most studies agree that they reflect the interplay of regular and chaotic motion in the quasi-classical dynamics, the exact mechanism has been surprisingly controversial. Inspired by recent experiments on graphene antidot lattices showing that the effect survives strong impurity scattering, we give a new explanation of the peaks linked to a fundamental relation between collision times and accessible phase space volumes, accounting for their robustness. Due to the fundamental nature of the mechanism described it will be relevant in many mesoscopic transport phenomena