This Focus Issue is, along with the one on Nonautonomous dynamical systems: Theory, methods, and applications, part of the Double Focus Issue on Nonautonomous dynamical systems in the sciences. We refer to the two twin Focus Issues as NDS-G and NDS-C, for short, where "G" stands for General and "C" for Climate. A key area of inquiry for understanding climate behavior in this century is the impact of anthropogenic and natural forcing on a system that is highly nonlinear and exhibits both chaotic and random aspects. The theory of NDSs is the perfect framework for exploring this impact. The 16 papers in this issue address several questions within this broad area: (i) various types of tipping that have arisen in the system's history or may occur in its future; (ii) the effects that one component of the climate system might have on another one; and (iii) how we may better learn from observations and model simulations about all that is going on within the system.

This paper offers a review while also studying yet unexplored features of the area of chaotic systems subjected to parameter drift of non-negligible rate, an area where the methods of traditional chaos theory are not applicable. Notably, periodic orbit expansion cannot be applied since no periodic orbits exist, nor do long-time limits, since for drifting physical processes the observational time can only be finite. This means that traditional Lyapunov-exponents are also ill-defined. Furthermore, such systems are non-ergodic, time and ensemble averages are different, the ensemble approach being superior to the single-trajectory view. In general, attractors and phase portraits are time-dependent in a non-periodic fashion. We describe the use of general methods which remain nevertheless applicable in such systems. In the phase space, the analysis is based on stable and unstable foliations, their intersections defining a Smale horseshoe, and the intersection points can be identified with the chaotic set governing the core of the drifting chaotic dynamics. Because of the drift, foliations and chaotic sets are also time-dependent, snapshot objects. We give a formal description for the time-dependent natural measure, illustrated by numerical examples. As a quantitative indicator for the strength of chaos, the so-called ensemble-averaged pairwise distance (EAPD) can be evaluated at any time instant. The derivative of this function can be considered the instantaneous (largest) Lyapunov exponent. We show that snapshot chaotic saddles, the central concept of transient chaos, can be identified in drifting systems as the intersections of the foliations, possessing a time-dependent escape rate in general. In dissipative systems, we find that the snapshot attractor coincides with the unstable foliation, and can consist of more than one component. These are a chaotic one, an extended snapshot chaotic saddle, and multiple regular time-dependent attractor points. When constructing the time-dependent basins of attraction of the attractor points, we find that the basin boundaries are time-dependent and fractal-like, containing the stable foliation, and that they can even exhibit Wada properties. In the Hamiltonian case, we study the phenomenon of the break-up of tori due to the drift in terms of both foliations and EAPD functions. We find that time-dependent versions of chaotic seas are not always fully chaotic, they can contain non-chaotic regions. Within such regions we identify time-dependent non-hyperbolic regions, the analogs of sticky zones of classical Hamiltonian phase spaces. We provide approximate formulas for the information dimension of snapshot objects, based on time-dependent Lyapunov exponents and escape rates. Besides these results, we also give possible applications of our methods e.g. in climate science and in the area of Lagrangian Coherent Structures.

We review the recent results of large ensemble climate projections considering them to be the simulations of chaotic systems. The quick spread of an initially localized ensemble in the first weeks after initialization is an appearance of the butterfly effect, illustrating the unpredictability of the dynamics. We show that the growth rate of uncertainty (an analog of the Lyapunov exponent) can be determined right after initialization. The next phase corresponds to a convergence of the no longer localized ensemble to the time-dependent climate attractor and requires a much longer time. After convergence takes place, the ensemble faithfully represents the climate dynamics. Concerning a credible simulation, the observed signal should then wander within the spread of the converged ensemble all the time, i.e., to behave just as any of the ensemble members. As a manifestation of the chaotic-like climate dynamics, one can imagine that beyond the single, observed time-dependent climate, a plethora of parallel climate realizations exists. Converged climate ensembles also define the probability distribution by which the physical quantities of the different climate realizations occur. Large ensemble simulations were shown earlier to be credible in the sense formulated. Here, in addition, an extended credibility condition is given, which requires the ensemble to be a converged ensemble, valid also for low-dimensional models. Interestingly, to the best of our knowledge, no low-order physical or engineering systems subjected to time-dependent forcings are known for which a comparison between simulation and experiment would be available. As illustrative examples, the CESM1-LE climate model and a chaotic pendulum are taken.

In view of the growing importance of climate ensemble simulations, we propose an ensemble approach for following the dynamics of extremes in the presence of climate change. A strict analog of extreme events, a concept based on single time series and local observations, cannot be found. To study nevertheless typical properties over an ensemble, in particular if global variables are of interest, a novel, statistical approach is used, based on a zooming in into the ensemble. To this end, additional, small sub-ensembles are generated, small in the sense that the initial separation between the members is very small in the investigated variables. Plume diagrams initiated on the same day of a year are generated from these sub-ensembles. The trajectories within the plume diagram strongly deviate on the time scale of a few weeks. By defining the extreme deviation as the difference between the maximum and minimum values of a quantity in a plume diagram, i.e., in a sub-ensemble, a growth rate for the extreme deviation can be extracted. An average of these taken over the original ensemble (i.e., over all sub-ensembles) characterizes the typical, exponential growth rate of extremes, and the reciprocal of this can be considered the characteristic time of the emergence of extremes. Using a climate model of intermediate complexity, these are found to be on the order of a few days, with some difference between the global mean surface temperature and pressure. Measuring the extreme emergence time in several years along the last century, results for the temperature turn out to be roughly constant, while a pronounced decaying trend is found in the last decades for the pressure.

We present a new secondary school teaching method of quantum uncertainties of two-state systems. Intending to be a material teachable in schools, only two-state systems described by real numbers can be considered. An elementary argumentation based on school statistics leads to the identification of the uncertainty of a physical quantity in such systems with the standard deviation of two random variables. We provide a qualitative picture on the state-dependence of the uncertainty, leading to a pictorial representation in the form of four petals of a flower. When considering the product of uncertainty of two essentially different physical quantities we conclude that the general feature: “if the measurement of one of the quantities is certain, the other remains uncertain”, cannot be faithfully expressed by means of an inequality, the product has no lower bound different from zero. The application of techniques used by school materials for teaching quantum physics leads to an exact formula for the state-dependence of the uncertainty valid in any two-state system described by real numbers, in full harmony with the qualitative picture. We compare the two-state case with the celebrated Heisenberg position-momentum uncertainty relation and show that these are both specific facets, but only the Heisenberg relation can be expressed by an inequality. The latter hardly provides any hint on the uncertainties of physical quantities in two-state systems. We conclude that the two-state approach is worth teaching in schools also in relation to the uncertainty relation, even if the Heisenberg relation is not part of the curriculum.

Recent advances in the field of complex, transiently chaotic dynamics are reviewed, based on the results published in the focus issue of J. Phys. Complex. on this topic. One group of achievements concerns network dynamics where transient features are intimately related to the degree and stability of synchronization, as well as to the network topology. A plethora of various applications of transient chaos are described, ranging from the collective motion of active particles, through the operation of power grids, cardiac arrhythmias, and magnetohydrodynamical dynamos, to the use of machine learning to predict time evolutions. Nontraditional forms of transient chaos are also explored, such as the temporal change of the chaoticity in the transients (called doubly transient chaos), as well as transients in systems subjected to parameter drift, the paradigm of which is climate change.

To characterize chaos in systems subjected to parameter drift, where a number of traditional methods do not apply, we propose viable alternative approaches, both in the qualitative and quantitative sense. Qualitatively, following stable and unstable foliations is shown to be efficient, which are easy to approximate numerically, without relying on the need for the existence of an analog of hyperbolic periodic orbits. Chaos originates from a Smale horseshoe-like pattern of the foliations, the transverse intersections of which indicate a chaotic set changing in time. In dissipative cases, the unstable foliation is found to be part of the so-called snapshot attractor, but the chaotic set is not dense on it if regular time-dependent attractors also exist. In Hamiltonian cases stable and unstable foliations turn out to be not equivalent due to the lack of time-reversal symmetry. It is the unstable foliation, which is found to correlate with the so-called snapshot chaotic sea. The chaotic set appears to be locally dense in this sea, while tori with originally quasiperiodic character might break up, their motion becoming chaotic as time goes on. A quantity called ensemble-averaged pairwise distance evaluated in relation to unstable foliations is shown to be an appropriate tool to provide the instantaneous strength of time-dependent chaos.

Chaotic phenomena are not part of standard curricula, although this subject offers several interesting aspects which can help students better understand basic features of science. A central observation is that even simple physical systems, if chaotic, are unpredictable, just like the weather. We present the principles applied when developing a freely available short interactive material and outline the material itself. By the means of this material students can become acquainted with the elements of chaos science and also learn that the regular motions taught in physics lessons are the exceptions rather than the rule since a slight modification of the set-up (e.g. letting the suspension point of a pendulum move periodically) converts the motion to chaotic. The material can be used in both classroom and online teaching.

The aim of this study is to investigate the bouncing dynamics of a small elastic ball on a staircase consisting of rounded edge steps, as an example of a dissipative gravitational billiard, and to determine if its dynamics is chaotic. We derive a nonlinear recursion for the coordinates of the collisions, completed with numerical simulations, which indicate that the bouncing dynamics is chaotic, as also follows from elementary considerations regarding the Lyapunov exponent. It is, however, surprising that instead of permanent chaos, only the transient form is present. The main reason behind this is that a collision with the rounded edge of the step enhances the horizontal velocity leading to larger and larger jumps. Not even the introduction of a tangential coefficient of restitution (COR) on the curvature can hinder the flying away of some trajectories. There is also a chance for remaining trapped on a single step in the form of sliding, representing another possibility for escape. Therefore, chaoticity holds for long trajectories before any kind of escape takes place. We also show that an impact-velocity-dependent COR converts the dynamics to permanently chaotic with an underlying fractal attractor. Only elementary mathematics is required for the analytic calculations used, and we offer a set of problems to solve, as well as a user-friendly demo software on our website: https://theorphys.elte.hu/fiztan/stairs to facilitate experimentation and further understanding of this complex phenomenon.

In dissipative systems without any driving or positive feedback all motion stops ultimately since the initial kinetic energy is dissipated away during time evolution. If chaos is present, it can only be of transient type. Traditional transient chaos is, however, supported by an infinity of unstable orbits. In the lack of these, chaos in undriven dissipative systems is of another type: it is termed doubly transient chaos as the strength of transient chaos is diminishing in time, and ceases asymptotically. Here we show that a clear view of such dynamics is provided by identifying KAM tori or chaotic regions of the dissipation-free case, and following their time evolution in the dissipative dynamics. The tori often smoothly deform first, but later they become disintegrated and dissolve in a kind of shrinking chaos. We identify different dynamical measures for the characterization of this process which illustrate that the strength of chaos is first diminishing, and after a while disappears, the motion enters the phase of ultimate stopping.

Based on the example of a paradigmatic area preserving low-dimensional mapping subjected to different scenarios of parameter drifts, we illustrate that the dynamics can best be understood by following ensembles of initial conditions corresponding to the tori of the initial system. When such ensembles are followed, snapshot tori are obtained, which change their location and shape. Within a time-dependent snapshot chaotic sea, we demonstrate the existence of snapshot stable and unstable foliations. Two easily visualizable conditions for torus breakup are found: one in relation to a discontinuity of the map and the other to a specific snapshot stable manifold, indicating that points of the torus are going to become subjected to strong stretching. In a more general setup, the latter can be formulated in terms of the so-called stable pseudo-foliation, which is shown to be able to extend beyond the instantaneous chaotic sea. The average distance of nearby point pairs initiated on an original torus crosses over into an exponential growth when the snapshot torus breaks up according to the second condition. As a consequence of the strongly non-monotonous change of phase portraits in maps, the exponential regime is found to split up into shorter periods characterized by different finite-time Lyapunov exponents. In scenarios with plateau ending, the divided phase space of the plateau might lead to the Lyapunov exponent averaged over the ensemble of a torus being much smaller than that of the stationary map of the plateau.

The increasing public attention attracted by environmental issues requires the understanding of the physics behind large scale motions of the atmosphere and the oceans. It is therefore necessary to extend the scope of secondary education to the physical principles behind these phenomena related to the Earth's rotation. The main message to convey is the fascinating effect that the importance of the Coriolis deflection increases with extension in space. The communication of size making such a difference requires quantitative considerations, whereas high school treatment is scarce and only provides a qualitative explanation of the Coriolis deflection. To bridge this gap, we have developed secondary level teaching material on the physical background of large-scale environmental processes and made it available on the internet for interested students and teachers. Aimed at regular secondary school students, it requires no further knowledge beyond the very basics of mechanics. The material consists of a core study and several extensions. The core study only uses elementary formulae and is easily followed by as independent learner or completed in three teaching periods in class. It begins with simple hands-on experiments from which students can derive the expression of the Coriolis acceleration. Students can also use data from their own experiments or everyday life observations to obtain a numerical measure of the strength of the Coriolis effect. Through the three sections, the material proceeds from small scales, where the Coriolis effect is practically negligible, to planetary scales, where the effect inevitably becomes dominant. As an illustration of the difference between the large and small scales, it also reveals why a video pretending to demonstrate the different rotation of plughole vortices around the Equator must be faked. The material has been developed in permanent contact with students, the paper also shows steps of improvement based on the feedback received from them.

We develop a conceptual coupled atmosphere–phytoplankton model by combining the Lorenz'84 general circulation model and the logistic population growth model under the condition of a climate change due to a linear time dependence of the strength of anthropogenic atmospheric forcing. The following types of couplings are taken into account: (a) the temperature modifies the total biomass of phytoplankton via the carrying capacity; (b) the extraction of carbon dioxide by phytoplankton slows down the speed of climate change; (c) the strength of mixing/turbulence in the oceanic mixing layer is in correlation with phytoplankton productivity. We carry out an ensemble approach (in the spirit of the theory of snapshot attractors) and concentrate on the trends of the average phytoplankton concentration and average temperature contrast between the pole and Equator, forcing the atmospheric dynamics. The effect of turbulence is found to have the strongest influence on these trends. Our results show that when mixing has sufficiently strong coupling to production, mixing is able to force the typical phytoplankton concentration to always decay globally in time and the temperature contrast to decrease faster than what follows from direct anthropogenic influences. Simple relations found for the trends without this coupling do, however, remain valid; just the coefficients become dependent on the strength of coupling with oceanic mixing. In particular, the phytoplankton concentration and its coupling to climate are found to modify the trend of global warming and are able to make it stronger than what it would be without biomass.

is the slope of the time evolution of the temperature contrast. This shows that the temperature contrast, and in particular, the strength of the climate change, becomes influenced by the phytoplankton concentration. In order to specify the unknown constants S and δ, let us rewrite (S1) as S (αDt+1−αβ(δ− 1)) = r(St+ δ) [(αD−S)t+1−αβ(δ− 1)− δ] . (S5) For long times, i.e. t≫ 1/r, the quadratic term dominates on the right hand side which cannot be 20 compensated by anything on the left hand side. The coefficient of the quadratic term should vanish, i.e. αD = S from which, since D =D0− βS, S = D0α 1+ βα , (S6)

Based on the example of a paradigmatic low-dimensional Hamiltonian system subjected to different scenarios of parameter drifts of non-negligible rates, we show that the dynamics of such systems can best be understood by following ensembles of initial conditions corresponding to tori of the initial system. When such ensembles are followed, toruslike objects called snapshot tori are obtained, which change their location and shape. In their center, one finds a time-dependent, snapshot elliptic orbit. After some time, many of the tori break up and spread over large regions of the phase space; however, one may find some smaller tori, which remain as closed curves throughout the whole scenario. We also show that the cause of torus breakup is the collision with a snapshot hyperbolic orbit and the surrounding chaotic sea, which forces the ensemble to adopt chaotic properties. Within this chaotic sea, we demonstrate the existence of a snapshot horseshoe structure and a snapshot saddle. An easily visualizable condition for torus breakup is found in relation to a specific snapshot stable manifold. The average distance of nearby pairs of points initiated on an original torus at first hardly changes in time but crosses over into an exponential growth when the snapshot torus breaks up. This new phase can be characterized by a novel type of a finite-time Lyapunov exponent, which depends both on the torus and on the scenario followed. Tori not broken up are shown to be the analogs of coherent vortices in fluid flows of arbitrary time dependence, and the condition for breakup can also be demonstrated by the so-called polar rotation angle method.

The phenomenonVortices generated by some kind of stirrer device are standard tools for illustrating the structure of tornadoes in schools (see, e.g.: [1]).In the von Kármán Laboratory for Environmental Flows at Eötvös University [2] for this purpose we use a vortex generated by a magnetic stirrer.This experiment is always a hit among all types of audiences, ranging from kindergarten kids, via high school and university students to parents.Commercial magnetic stirrers are commonly used to mix different materials dissolved in fluids.The rotation of the magnet brings a magnetic stirrer bar, whose rotational axis is not fixed, into rotation at the bottom of a container.If the container is filled with a liquid, the bar generates fluid motion, which is believed to cause efficient stirring and mixing.A striking pattern of such flows is a big vortex accompanied with a depression of the free surface, the funnel.With our experiments we demonstrate that there is a strong downwelling in the middle, underneath the funnel and that particles lighter than the ambient fluid are attracted towards the vortex axis.The most surprising effect appears, however, when one drops dye into the central region of the vortex from the surface, marked by the location of the funnel: the injected material stretches in the vertical direction in a compactly arranged manner, forming a cylinder of high dye concentration lasting over minutes (figure 1(a)).All this happens despite the fact that the vortex (and the dyed domain) is time-dependent since the stirrer bar is not fixed to an axis.If a certain amount of dye happens to enter a region outside a critical radius measured from the center, it becomes rapidly

Tipping phenomena, i.e. dramatic changes in the possible long-term performance of deterministic systems subjected to parameter drift, are of current interest but have not yet been explored in cases with chaotic internal dynamics. Based on the example of a paradigmatic low-dimensional dissipative system subjected to different scenarios of parameter drifts of non-negligible rates, we show that a number of novel types of tippings can be observed due to the topological complexity underlying general systems. Tippings from and into several coexisting attractors are possible, and one can find fractality-induced tipping, the consequence of the fractality of the scenario-dependent basins of attractions, as well as tipping into a chaotic attractor. Tipping from or through an extended chaotic attractor might lead to random tipping into coexisting regular attractors, and rate-induced tippings appear not abruptly as phase transitions, rather they show up gradually when the rate of the parameter drift is increased. Since chaotic systems of arbitrary time-dependence call for ensemble methods, we argue for a probabilistic approach and propose the use of tipping probabilities as a measure of tipping. We numerically determine these quantities and their parameter dependence for all tipping forms discussed.

A state-dependent vulnerability of synchronization is shown to exist in a complex network composed of numerically simulated electronic circuits. We demonstrate that disturbances to the local dynamics of network units can produce different outcomes to synchronization depending on the current state of its trajectory. We address such state dependence by systematically perturbing the synchronized system at states equally distributed along its trajectory. We find the states at which the perturbation desynchronizes the network to be complicatedly mixed with the ones that restore synchronization. Additionally, we characterize perturbation sets obtained for consecutive states by defining a safety index between them. Finally, we demonstrate that the observed vulnerability is due to the existence of an unstable chaotic set in the system's state space.

Presentation given at the 21st EGU General Assembly (EGU2019), published on the Proceedings from the conference held 7-12 April, 2019 in Vienna, Austria, id.17414.