theory of dynamical systems

Top PDF theory of dynamical systems:

Dynamical systems theory sheds new light on compound climate extremes in Europe and Eastern North America

Dynamical systems theory sheds new light on compound climate extremes in Europe and Eastern North America

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).
En savoir plus

23 En savoir plus

Evaluation of Crisis, Reversibility, Alert Management for Constrained Dynamical Systems Using Impulse Dynamical Systems

Evaluation of Crisis, Reversibility, Alert Management for Constrained Dynamical Systems Using Impulse Dynamical Systems

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.
En savoir plus

8 En savoir plus

T HE THEORY OF DYNAMICAL SYSTEMS , AND ITS APPLICATIONS TO ECONOMICS

T HE THEORY OF DYNAMICAL SYSTEMS , AND ITS APPLICATIONS TO ECONOMICS

 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.
En savoir plus

19 En savoir plus

Control theory and dynamical systems

Control theory and dynamical systems

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 :
En savoir plus

119 En savoir plus

Extremes and Recurrence in Dynamical Systems

Extremes and Recurrence in Dynamical Systems

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.
En savoir plus

306 En savoir plus

Reachability in Dynamical Systems with Rounding

Reachability in Dynamical Systems with Rounding

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
En savoir plus

18 En savoir plus

A Core Theory of Delay Systems

A Core Theory of Delay Systems

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.
En savoir plus

21 En savoir plus

Lorentzian 3-manifolds and dynamical systems.

Lorentzian 3-manifolds and dynamical systems.

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
En savoir plus

32 En savoir plus

An evolutionary theory of urban systems

An evolutionary theory of urban systems

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).
En savoir plus

15 En savoir plus

Chaotic dynamical systems associated with tilings of ${R}^{N}$

Chaotic dynamical systems associated with tilings of ${R}^{N}$

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.

20 En savoir plus

Scaling Invariants and Symmetry Reduction of Dynamical Systems

Scaling Invariants and Symmetry Reduction of Dynamical Systems

• 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.
En savoir plus

31 En savoir plus

Dynamic Data-Driven Parametric Dynamical Systems

Dynamic Data-Driven Parametric Dynamical Systems

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.
En savoir plus

5 En savoir plus

Case Studies in Data-Driven Verification of Dynamical Systems

Case Studies in Data-Driven Verification of Dynamical Systems

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.
En savoir plus

7 En savoir plus

Detection of Changes in Nonlinear Dynamical Systems using Multiresolution Entropy

Detection of Changes in Nonlinear Dynamical Systems using Multiresolution Entropy

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]

48 En savoir plus

Dynamical reduction of metabolic networks by singular perturbation theory : application to microalgae

Dynamical reduction of metabolic networks by singular perturbation theory : application to microalgae

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.
En savoir plus

206 En savoir plus

Self-Organization: Complex Dynamical Systems in the Evolution of Speech

Self-Organization: Complex Dynamical Systems in the Evolution of Speech

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).
En savoir plus

22 En savoir plus

Complexity of ten decision problems in continuous time dynamical systems

Complexity of ten decision problems in continuous time dynamical systems

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.
En savoir plus

7 En savoir plus

Spectral theory of damped quantum chaotic systems

Spectral theory of damped quantum chaotic systems

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]
En savoir plus

25 En savoir plus

Variational Inference and Learning of Piecewise-linear Dynamical Systems

Variational Inference and Learning of Piecewise-linear Dynamical Systems

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.
En savoir plus

13 En savoir plus

Galois extensions of height-one commuting dynamical systems

Galois extensions of height-one commuting dynamical systems

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
En savoir plus

17 En savoir plus

Show all 10000 documents...