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.

Based on the theory of "snapshot/pullback attractors", we show that important features of the climate change that we are observing can be understood by imagining many replicas of Earth that are not interacting with each other. Their climate systems evolve in parallel, but not in the same way, although they all obey the same physical laws, in harmony with the chaotic-like nature of the climate dynamics. These parallel climate realizations evolving in time can be considered as members of an ensemble. We argue that the contingency of our Earth's climate system is characterized by the multiplicity of parallel climate realizations rather than by the variability that we experience in a time series of our observed past. The natural measure of the snapshot attractor enables one to determine averages and other statistical quantifiers of the climate at any instant of time. In this paper, we review the basic idea for climate changes associated with monotonic drifts, and illustrate the large number of possible applications. Examples are given in a low-dimensional model and in numerical climate models of different complexity. We recall that systems undergoing climate change are not ergodic, hence temporal averages are generically not appropriate for the instantaneous characterization of the climate. In particular, teleconnections, i.e. correlated phenomena of remote geographical locations are properly characterized only by correlation coefficients evaluated with respect to the natural measure of a given time instant, and may also change in time. Physics experiments dealing with turbulent-like phenomena in a changing environment are also worth being interpreted in view of the attractor-based ensemble approach. The possibility of the splitting of the snapshot attractor to two branches, near points where the corresponding time-independent system undergoes bifurcation as a function of the changing parameter, is briefly mentioned. This can lead in certain climate-change scenarios to the coexistence of two distinct sub-ensembles representing dramatically different climatic options. The problem of pollutant spreading during climate change is also discussed in the framework of parallel climate realizations. Keywords Climate dynamics • Nonautonomous systems • Ensembles • Snapshot attractors • Natural measures Communicated by Valerio Lucarini.

We investigate the death and revival of chaos under the impact of a monotonous time-dependent forcing that changes its strength with a non-negligible rate. Starting on a chaotic attractor it is found that the complexity of the dynamics remains very pronounced even when the driving amplitude has decayed to rather small values. When after the death of chaos the strength of the forcing is increased again with the same rate of change, chaos is found to revive but with a different history. This leads to the appearance of a hysteresis in the complexity of the dynamics. To characterize these dynamics, the concept of snapshot attractors is used, and the corresponding ensemble approach proves to be superior to a single trajectory description, that turns out to be nonrepresentative. The death (revival) of chaos is manifested in a drop (jump) of the standard deviation of one of the phase-space coordinates of the ensemble; the details of this chaos-nonchaos transition depend on the ratio of the characteristic times of the amplitude change and of the internal dynamics. It is demonstrated that chaos cannot die out as long as underlying transient chaos is present in the parameter space. As a condition for a "quasistatically slow" switch-off, we derive an inequality which cannot be fulfilled in practice over extended parameter ranges where transient chaos is present. These observations need to be taken into account when discussing the implications of "climate change scenarios" in any nonlinear dynamical system.

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Using an intermediate complexity climate model (Planet Simulator), we investigate the so-called Snowball Earth transition. For certain values of the solar constant, the climate system allows two different stable states: one of them is the Snowball Earth, covered by ice and snow, and the other one is today's climate. In our setup, we consider the case when the climate system starts from its warm attractor (the stable climate we experience today), and the solar constant is decreased continuously in finite time, according to a parameter drift scenario, to a state, where only the Snowball Earth's attractor remains stable. This induces an inevitable transition, or climate tipping from the warm climate. The reverse transition is also discussed. Increasing the solar constant back to its original value on individual simulations, we find that the system stays stuck in the Snowball state. However, using ensemble methods i.e., using an ensemble of climate realizations differing only slightly in their initial conditions we show that the transition from the Snowball Earth to the warm climate is also possible with a certain probability. From the point of view of dynamical systems theory, we can say that the system's snapshot attractor splits between the warm climate's and the Snowball Earth's attractor. 1 arXiv:1906.00952v1 [physics.ao-ph] 31 May 2019 Ever since its discovery, the Snowball Earth, i.e. when the Earth's surface is nearly entirely frozen, received much attention within the climate science community. Much of the details of the transition to the planet's frozen state are still unexplored. Here, instead of focusing on the true Snowball events of Earth's history, we investigate the transition in an intermediate complexity climate model (PlaSim), with a continuously drifting solar constant (a hypothetical climate change scenario), in which a full return to the original value occurs. Using an ensemble based method we obtain both of the possible stable states as possible outcomes. We also show that the process is probabilistic and the probabilities of the corresponding outcomes are given by the ensemble's distribution. In addition, the third, unstable state (referred to as the edge state) is also recovered. I. INTRODUCTIONSnowball Earth refers to the planet's coldest possible global climate. In this state, the whole Earth, from the poles to the Equator, is covered in ice and snow. Since the thick ice covering the surface reflects much of the energy radiated by the Sun, the global average temperature is very low, around 220 K 1 .Modern findings suggest that during the Earth's history, there were periods, when such Snowball events occurred. For example, several traces of glacial activity point to the presence of glaciers along the so-called Paleoequator 2 .This suggests that also the current configuration of the Earth system may be bistable, the two stable states being the Snowball state and our current climate. To better understand the phenomenon, there are simple models available that only take the global energy balance...

Our aim is to unfold phase space structures underlying systems with a drift in their parameters. Such systems are non-autonomous and belong to the class of non-periodically driven systems where the traditional theory of chaos (based e.g., on periodic orbits) does not hold. We demonstrate that even such systems possess an underlying topological horseshoe-like structure at least for a finite period of time. This result is based on a specifically developed method which allows to compute the corresponding time-dependent stable and unstable foliations. These structures can be made visible by prescribing a certain type of history for an ensemble of trajectories in phase space and by analyzing the trajectories fulfilling this constraint. The process can be considered as a leaking in history space-a generalization of traditional leaking, a method that has become widespread in traditional chaotic systems, to leaks depending on time.

Pancreatic carcinoma is one of the most malignant diseases and is associated with a poor survival rate. Pituitary adenylate cyclase activating polypeptide (PACAP) is a neuropeptide that acts on three different G protein-coupled receptors: the specific PAC1 and the VPAC1/2 that also bind vasoactive intestinal peptide. PACAP is widely distributed in the body and has diverse physiological effects. Among other things, it acts as a trophic factor and influences proliferation and differentiation of several different cells both under normal circumstances and tumourous transformation. Changes of PACAP and its receptors have been shown in various tumour types. However, it is not known whether PACAP and its specific receptor are altered in pancreatic cancer. Perioperative data of patients with pancreas carcinoma was investigated over a five-year period. Histological results showed Grade 2 or Grade 3 adenocarcinoma in most cases. PACAP and PAC1 receptor expression were investigated by immunohistochemistry. Staining intensity of PAC1 receptor was strong in normal tissues both in the exocrine and endocrine parts of the pancreas, the receptor staining was markedly weaker in the adenocarcinoma. PACAP immunostaining was weak in the exocrine part and very strong in the islets and nerve elements in non-tumourous tissues. The PACAP immunostaining almost disappeared in the adenocarcinoma samples. Based on these findings a decrease or lack of the PAC1 receptor/PACAP signalling might have an influence on tumour growth and/or differentiation.

A primary spectral submanifold (SSM) is the unique smoothest nonlinear continuation of a nonresonant spectral subspace E of a dynamical system linearized at a fixed point. Passing from the full nonlinear dynamics to the flow on an attracting primary SSM provides a mathematically precise reduction of the full system dynamics to a very low-dimensional, smooth model in polynomial form. A limitation of this model reduction approach has been, however, that the spectral subspace yielding the SSM must be spanned by eigenvectors of the same stability type. A further limitation has been that in some problems, the nonlinear behavior of interest may be far away from the smoothest nonlinear continuation of the invariant subspace E. Here, we remove both of these limitations by constructing a significantly extended class of SSMs that also contains invariant manifolds with mixed internal stability types and of lower smoothness class arising from fractional powers in their parametrization. We show on examples how fractional and mixed-mode SSMs extend the power of data-driven SSM reduction to transitions in shear flows, dynamic buckling of beams, and periodically forced nonlinear oscillatory systems. More generally, our results reveal the general function library that should be used beyond integer-powered polynomials in fitting nonlinear reduced-order models to data.