Statistically significant α anomaly means associated with cold extremes are observed over the north-eastern part **of** the North American domain (Figure 4d), reflecting the larger temperature anomalies shown in Figure 4a. α anomaly means linked with wet extremes in Europe are instead significant over Scandinavia, north-western Russia and in the Middle East (Figure 4e). The difference in geographical distribution relative to the composite precipitation anomalies shown in Figure 4b likely reflects differences in the local precipitation distributions. That is, the fact that the large anomalies show in Figure 4b are locally less extreme than the weaker anomalies found over Scandinavia. Note that Figure 4e differs from Figure 3e because **of** the different variables and domains over which α is computed here. Lastly, α anomaly means observed during concurrent cold and wet extremes show statistical significance in the north-eastern North American domain and Iberia/Western France (Figure 4f). These reflect very closely the mean anomaly patterns in Figures 4a-b. Similar to what we found for concurrent wet and windy extremes in Europe (Figure 3f), these α anomaly means for the concurrent extremes display larger values than for the univariate extremes. The monovariate α anomaly means mostly fail the sign test, while the anomalies for the concurrent extremes display extensive sign agreement mainly over Europe (stippling in Figure 4d-f). We interpret these differences as indicating that the α computed on the North American temperature and European precipitation reflect more closely the concurrent extremes in these variables than the monovariate extremes in the individual domains. As for Figure 3c-f, similar results are retrieved for a different definition **of** extremes (Figure S6).

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Abstract. Considering constrained **dynamical** **systems** character- ized by a differential inclusion x 0 ∈ F (x), we are interested in study- ing the situation where, for various reasons, the state leaves the con- strain domain K either because the initial position does not belong to the Viability Kernel **of** K for F or it belongs to a “sustainable or tolerable” but not “comfortable” domain. This question appears in numerous models in Social Sciences or in Genetics as well as for controlling security in Automatics and Robotics, like Aircraft landing, rolling and taking off. After recalling basic concepts in Vi- ability **Theory** and using hybrid calculus, we show how to evaluate and manage crisis in general cases.

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t
t
n , since the marginal product **of** land is a -bN t > 0
and A = a/w where a is the output per unit land (or worker) on the best land , and w is the subsistence wage. Since this is just the logistic equation , values **of** A near 4 will produce chaotic trajectories **of** the model . Rather than setting down at the stationary state, as Ricardo probably envisioned and surely many later economists working on this type **of** model assumed without much question, the population and capital stock being a complex chaos-like oscillation around the stationary state , with periods **of** over accumulation **of** capital alternative with periods **of** under accumulation.

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proof in the literature we would like to give a sketch **of** proof for the sake **of** comple- teness. We shall not repeat all the steps **of** the proof **of** Lemma II.3 in [46] because the arguments extend quite forwardly, we shall just point out where the symplectic assumption matters. The proof **of** Lemma II.3 in [46] has two main parts. The first part is based on the generic linear algebra **of** what Ma˜ n´e calls uniformly contracting families **of** periodic sequences **of** linear isomorphisms, namely, uniformly hyperbolic families **of** periodic sequences where the unstable part **of** each sequence is trivial (see [46] from pages 527 to 532). Since the restriction **of** the dynamics **of** a uni- formly hyperbolic periodic sequence to the stable subspace gives rise to a uniformly contracting periodic sequence the argument consists in proving separatedly uniform contraction properties for the stable part **of** the dynamics and then uniform ex- pansion properties for the unstable part **of** the dynamics. In the case **of** hyperbolic symplectic matrices, the invariant subspaces **of** the dynamics are always Lagrangian, so we have the following elementary result **of** symplectic linear algebra :

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Let’s briefly consider, to take a relevant and instructive example, the case **of** geo- physical extremes, which do not only cost many human lives each year, but also cause significant economic damages [4, 8, 9, 10]; see also the discussion and histor- ical perspective given in [11]. For instance, freak ocean waves are extremely hard to predict and can have devastating impacts on vessels and coastal areas [12, 13, 14]. Windstorms are well-known to dominate the list **of** the costliest natural disasters, with many occurrences **of** individual events causing insured losses topping 1 Billion $ [15, 16]. Temperature extremes, like heat waves and cold spells, have severe im- pacts on society and ecosystems [17, 18, 19]. Notable temperature-related extreme events are the 2010 Russian heat wave, which caused 500 wild fires around Moscow, reduced grain harvest by 30% and was the hottest summer in at least 500 years [20], and the 2003 heat wave in Europe, which constituted the second hottest summer in this period [21]. The 2003 heat wave had significant societal consequences; e.g. it caused additional deaths exceeding 70000 [17]. On the other hand, recent European winters were very cold, with widespread cold spell hitting Europe during January 2008, December 2009 and January 2010. The increasing number **of** weather and cli- mate extremes over the last few decades [22, 23, 24] has led to intense debates, not only amongst scientists but also policy makers and the general public, whether this increase is triggered by global warming.

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We ran experiments on the behaviour **of** rounded orbits induced by rotations in the plane. Four prototypical results are depicted in Figure 4. We note that in every one **of** our examples the orbits eventually become periodic. Moreover, all experiments fall into the four categories **of** Figure 4, i.e., where the resulting set consists **of** (a) a square with cut-off corners, (b) this same square, but with a central square cut out, and (c) all points within the circle with some seemingly randomly added points outside (in the case **of** an irrational multiple **of** π), (d) the initial circle with added ‘tentacles’ occuring in intervals corresponding to the rotational angle (in the case **of** a rational multiple **of** π). We have been unable construct a rotation with an

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However, competing theories **of** delay **systems** have been developped and used over time (Hale 1977; M. Delfour and Karrakchou 1987; Salamon 1984); they may use different definitions **of** delay operators, **of** the solutions to the initial- value problem and may be restricted to **systems** **of** a certain type. For the newcomer in the field, this is an unintended source **of** complexity. To make the subject more widely accessible, we lay out in this paper a simple but general framework for the description **of** delay **systems**, based on a combination **of** linear algebra and measure **theory**. Then we develop on this foundation a core **theory**: we characterize the well-posedness **of** the initial-value problem, then we perform a graph-theoretic analysis **of** this issue and finally, we provide a general stability criterion.

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signature. By local homogeneity **of** Σ, the Gaussian curvature **of** Σ is con- stant. The Lie algebra z X (M) should then be isomorphic to the Lie algebra
sol (case **of** curvature 0) or o(1, 2) (case **of** nonzero constant curvature). This yields a contradiction since the center **of** both sol and o(1, 2) is trivial (we have implicitely used the fact that a local Killing field on M which vanishes on Σ is trivial, what is easily proved using Proposition 3.1 ). Lemma 6.3. — Let (M 3 , g) be a smooth, closed, 3-dimensional Lorentz manifold. Let X be a Killing field on M generating a flow ϕ t X which is not

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At the macro-level **of** the system **of** cities, the process **of** specialization is part **of** the “division **of** labor” between cities. While the classical diffusion **of** innovation waves and their associated growth impulses create and increase the quantitative inequalities **of** sizes within a system **of** cities, the specialization process structures the major qualitative differences that are observed between cities in their employment profiles, professional composition, and average income levels. A classical pattern that was observed in comparing French or European cities as well as North American, Chinese and Indian ones by means **of** multivariate analysis establishes that the major dimension differentiating their societal characteristic today is the trace **of** the location **of** manufacturing activities **of** the first industrial revolution (i.e., including mainly mining, textile and steel industries), while the second dimension reflects the differential adaptation **of** cities to the “revolution **of** services” since the second half **of** 20 th century (i.e., opposing cities where finance, insurance and real estate activities and business services have had intense developments to cities where traditional small trade and craft remained relatively more present). Over time, during each large innovation wave, the specialization process creates cases **of** anomalous rapid growth for single cities or regional pockets **of** cities, which surge in higher ranks and may partly reshape the urban hierarchy. At the meso-level **of** single cities, the specialization by definition represents a large part **of** the economic base and, for a few decades at least, makes the city residents wealthy and increases the attractiveness **of** the city. Therefore, it generates peculiar non-smooth growth trajectories that remain ascending as long as the innovation wave is productive enough, but very often exhibit tipping points and reversal when the highly specialized cities fail to adapt to further innovation (Fig.4). Comparative analysis **of** the **systems** **of** cities **of** the BRICS, Europe and the United States **of** America provides many examples **of** these few but important “anomalous” trajectories that contrast with the generally smooth trend **of** growth in urban hierarchies (Pumain et al., 2015).

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For ease **of** implementation and duplication, a cryptographic scheme must involve a map for which the parameters identification is expected to be a difficult task, while computational require- ments for masking and unmasking information are not too heavy. The second aim **of** this chapter is to show that all these requirements are fulfilled for the class **of** **dynamical** **systems** considered here. The way **of** extracting the masked information is provided through an observer-based syn- chronization mechanism with a finite-time stabilization property.

• n − r generating rational invariants that are algebraically (and functionally) independent • a simple rewriting **of** any (rational) invariant in terms **of** this generating set,
• a rational section to the orbits **of** the scaling.
We thus go much further than the group action transcription **of** the Buckingham π-theorem **of** dimensional analysis [4, 26]. This latter takes any basis **of** the nullspace **of** the matrix A and provides a set **of** functionally generating invariants, some **of** which could involve fractional powers. In the present approach, only integer powers are involved. This spares us the determination **of** proper domains **of** definition. Furthermore, the Buckingham π-theorem gives no indication on how to rewrite an invariant in terms **of** the generators produced. The rewriting we propose is a simple substitution. This is reminiscent **of** the normalized invariants appearing in [8, 14, 23] (or replacement invariants in [13]). Using the terminology **of** those articles, we are in a position to exhibit a global cross-section (or cross-section **of** degree one) to the orbits **of** the scaling. Note though that the substitution is again rational: we do not introduce any algebraic functions as would generally be the case when choosing a local cross-section arbitrarily.

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María de Luna, 7, E-50018 Zaragoza, Spain
ecueto@unizar.es ; gonzal@unizar.es
SUMMARY
Dynamic Data-Driven Application **Systems** – DDDAS – appear as a new paradigm in the field **of** applied sciences and engineering, and in particular in simulation-based engineering sciences. By DDDAS we mean a set **of** techniques that allow the linkage **of** simulation tools with measurement devices for real-time control **of** **systems** and processes. DDDAS entails the ability to dynamically incorporate additional data into an executing application, and in reverse, the ability **of** an application to dynamically steer the measurement process. DDDAS needs for accurate and fast simulation tools making use if possible **of** off-line computations for limiting as much as possible the on-line computations. We could define efficient solvers by introducing all the sources **of** variability as extra-coordinates in order to solve the model off-line only once to obtain its most general solution to be then considered in on-line purposes. In this work we are evaluating one such strategy in both linear and non-linear **dynamical** **systems** as the ones usually encountered in control, epidemiology models, physiological **systems** (diagnostic **of** diabetes …). Such models result defined in highly multidimensional spaces suffering the so-called curse **of** dimensionality that could be efficiently circumvented by applying a Proper Generalized Decomposition that constitutes a work in progress.

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ABSTRACT
We interpret several **dynamical** system verification questions, e.g., region **of** attraction and reachability analyses, as data classification problems. We discuss some **of** the tradeoffs be- tween conventional optimization-based certificate construc- tions with certainty in the outcomes and this new date- driven approach with quantified confidence in the outcomes. The new methodology is aligned with emerging computing paradigms and has the potential to extend systematic verifi- cation to **systems** that do not necessarily admit closed-form models from certain specialized families. We demonstrate its effectiveness on a collection **of** both conventional and un- conventional case studies including model reference adaptive control **systems**, nonlinear aircraft models, and reinforce- ment learning problems.

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L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignemen[r]

Nonlinear Dynamic Metabolic Networks
Abstract: Here we propose a rigorous approach to reduce metabolic nonlinear models. We assume that a metabolic network can be represented with Michaelis-Menten enzymatic reactions, and that it contains at least two different time-scales. We also consider a continuous (slowly) varying input in the model, such as light for microalgae, so that the system is never at steady state. Using a Quasi Steady State Reduction based on Tikhonov’s Theorem, a reduced system with a characterized error is obtained. Furthermore, our analysis proves that the metabolites with slow dynamics reach higher con- centrations (by one order **of** magnitude) than the fast metabolites. A simple example illustrates our approach and the resulting accuracy **of** the reduction method, also showing that it is adequate for **systems** with more than two time-scales.

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3.2 Computer Science and the Origins **of** Language and Languages
It is also possible to use computers and agent-based simulations not only to help us understand the phenomena that characterize self-organization **of** matter, simple biological structures, or insect societies, but also to help us understand phenomena that characterize humans and its societies. The time has come to use computers and robots as scientific tools in human sciences. Thus, building artificial **systems** in the context **of** research into language origins and the evolution **of** languages is enjoying a growing popularity in the scientific community, exactly because it is a crucial tool for studying the phenomena **of** language in relation to the complex interactions **of** its components (Steels, 1997; Oudeyer and Kaplan, 2007; Kaplan and Oudeyer, 2008). These **systems** are put to two main types **of** use: 1) they serve to evaluate the internal coherence **of** verbally expressed theories already proposed by clarifying all their hypotheses and verifying that they do indeed lead to the proposed conclusions (and quite often one discovers errors in the assumptions as well as in the conclusions, which need to be revised); 2) they serve to explore and generate new theories, which themselves often appear when one simply tries to build an artificial system reproducing the verbal behaviour **of** humans. A number **of** decisive results have already been obtained and have opened the way for resolution **of** previously unanswered questions: the decentralized generation **of** lexical and semantic conventions in populations **of** agents (e.g. Kaplan, 2001), the formation **of** shared inventories **of** vowels or syllables in groups **of** agents (e.g. Berrah et al., 1996; de Boer, 2001; Oudeyer, 2001; Oudeyer, 2005a; Oudeyer, 2005b; Oudeyer, 2006), with features **of** structural regularities greatly resembling those **of** human languages (e.g. Pierrehumbert, 2001; Wedel, 2006), the formation **of** conventionalized syntactic (e.g. Batali, 1998, ) and grammatical structures (e.g. Steels, 2005), the conditions under which combinatoriality, the property **of** systematic re-use, can be selected (Kirby, 2001).

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These notions are all formally defined in Section IV. The input to these problems is an ordered list **of** coefficients (expressed as rational numbers) defining the polynomial or trigonometric vector field. Establishing NP-hardness **of** these problems implies that unless P=NP, it is not possible to provide an algorithm that can have a running time bounded by a polynomial in the number **of** bits required to represent the input. Further, all the NP-hardness results in this paper are in the strong sense (as opposed to weakly NP-hard problems like KNAPSACK or SUBSET SUM). This implies that the problems remain NP-hard even when the bit length **of** the coefficients (i.e. the input) is O(log(n)) (here, n is the dimension **of** the state space). Unless P=NP, even pseudo- polynomial time algorithms cannot exist for strongly NP- hard problems; see [15] for more details and definitions. In particular, our results suggest that none **of** the numerous recent techniques for **systems** analysis based on convex optimization (e.g. in terms **of** linear programs, linear matrix inequalities, or sum **of** squares programs) can be exact, unless the size **of** the formulated optimization problems are exponential in the input.

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have a limit when ~ → 0? Is this limit distribution related with the value distributions **of** the averages hqi t ?
3. Spectral estimates on Anosov manifolds
The above questions are open in general. In order to get more precise informations on the spectrum **of** P (~), one needs to make specific assumptions on the geodesic flow on X. For instance, the case **of** a completely integrable dynamics has been con- sidered by Hitrik-Sjöstrand in a sequence **of** papers (see e.g. [HitSjo08]and reference therein). The case **of** nearly-integrable dynamics including KAM invariant tori has been studied by Hitrik-Sjöstrand-V˜ u Ngo.c [HSVN07]. In these cases, one can trans- form the Hamiltonian flow into a normal form near each invariant torus, which leads to a precise description **of** the spectrum “generated” by this torus. A Weyl law for the quantum decay rates was recently obtained in [HitSjo11] (for skew-adjoint per- turbations iθ(~) Op ~ (q), with θ(~) ~). On the other hand, Asch-Lebeau [AschLeb]

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More recently, learning and inference **of** **dynamical** **systems** have been addressed in the framework **of** deep generative models (DGMs), where the linear-Gaussian transition and emission distributions **of** LDS are replaced with non-linear Gaussian models. In detail, the mean and covariance **of** a Gaussian distribution are modeled with neural networks. Because **of** this non-linear dependencies, direct optimization **of** the corresponding data log-likelihood function is intractable. This issue is solved by maximization **of** a variational lower bound **of** the log-likelihood. For example, [57] uses a recurrent neural network (RNN) to model the mean and diagonal covariance matrix. The proposed structured inference network corresponds to a deep Kalman smoother, that needs both past and future observations. This formulation belongs to a wider class **of** non-linear Gaussian state-space models that were recently reviewed in [58]. Deep neural networks can also be used within structured variational inference for pixel-level prediction tasks [59], [60], but we are not aware **of** any works addressing switching LDS with this methodology.

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1. Introduction
The study **of** p-adic **dynamical** **systems** has seen increased interest over the past two decades, reflected most recently in a new MSC category: Arith- metic and non-Archimedean **dynamical** **systems**. This note is concerned with three overlapping ways **of** looking at such **systems**—formal power series that commute under composition, iterated morphisms **of** the open p-adic unit disc, and galoisness **of** extensions that are obtained by adjoining zeros **of** **dynamical** **systems**. Indeed, the proof **of** the main result in this note can be viewed as relating commuting power series to formal groups, analytic

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