Analysis of System Signals Based on Cross Recurrence Method for System Dynamics Characterization

Natural systems are inherently non linear. Recurrent behaviours are typical of natural systems. Recurrence is a fundamental property of non linear dynamical systems which can be exploited to characterize the system behaviour effectively. Cross recurrence based analysis of sensor signals from non linear dynamical system is presented in this thesis. The mutual dependency among relatively independent components of a system is referred as coupling. The analysis is done for a mechanically coupled system specifically designed for conducting experiment. Further, cross recurrence method is extended to the actual machining process in a lathe to characterize the chatter during turning. The result is verified by permutation entropy method. Conventional linear methods or models are incapable of capturing the critical and strange behaviours associated with the dynamical process. Hence any effective feature extraction methodologies should invariably gather information thorough nonlinear time series analysis. The sensor signals from the dynamical system normally contain noise and non stationarity. In an effort to get over these two issues to the maximum possible extent, this work adopts the cross recurrence quantification analysis (CRQA) methodology since it is found to be robust against noise and stationarity in the signals. The study reveals that the CRQA is capable of characterizing even weak coupling among system signals. It also divulges the dependence of certain CRQA variables like percent determinism, percent recurrence and entropy to chatter unambiguously. The surrogate data test shows that the results obtained by CRQA are the true properties of the temporal evolution of the dynamics and contain a degree of deterministic structure. The results are verified using permutation entropy (PE) to detect the onset of chatter from the time series. The present study ascertains that this CRP based methodology is capable of recognizing the transition from regular cutting to the chatter cutting irrespective of the machining parameters or work piece material. The results establish this methodology to be feasible for detection of chatter in metal cutting operation in a lathe.

KAM: Automatic Planning and Interpretation of Numerical Experiments Using Geometrical Methods

KAM is a computer program that can automatically plan, monitor, and interpret numerical experiments with Hamiltonian systems with two degrees of freedom. The program has recently helped solve an open problem in hydrodynamics. Unlike other approaches to qualitative reasoning about physical system dynamics, KAM embodies a significant amount of knowledge about nonlinear dynamics. KAM's ability to control numerical experiments arises from the fact that it not only produces pictures for us to see, but also looks at (sic---in its mind's eye) the pictures it draws to guide its own actions. KAM is organized in three semantic levels: orbit recognition, phase space searching, and parameter space searching. Within each level spatial properties and relationships that are not explicitly represented in the initial representation are extracted by applying three operations ---(1) aggregation, (2) partition, and (3) classification--- iteratively.

Automatic Analysis and Synthesis of Controllers for Dynamical Systems Based On P

I present a novel design methodology for the synthesis of automatic controllers, together with a computational environment---the Control Engineer's Workbench---integrating a suite of programs that automatically analyze and design controllers for high-performance, global control of nonlinear systems. This work demonstrates that difficult control synthesis tasks can be automated, using programs that actively exploit and efficiently represent knowledge of nonlinear dynamics and phase space and effectively use the representation to guide and perform the control design. The Control Engineer's Workbench combines powerful numerical and symbolic computations with artificial intelligence reasoning techniques. As a demonstration, the Workbench automatically designed a high-quality maglev controller that outperforms a previous linear design by a factor of 20.

Cambio político y consolidación democrática en México. Los límites del modelo de las transiciones a la democracia

 Este texto plantea una reflexión sobre el estado actual de la democracia en México. Para ello se basa en una lectura crítica de las interpretaciones de la transición mexicana. Estas últimas subrayan el carácter sui generis e incompleto de dicha transición, cuya conclusión no ha significado una discontinuidad clara entre la etapa autoritaria y la de consolidación democrática. Nuestro planteamiento consiste en mostrar que los límites actuales de la democracia mexicana, en parte identificables como problemas de consolidación, dejan dudas sobre la llegada de la democracia a México. Lo ilustramos en torno a la pluralización aparente del sistema de partidos, a la cual los actores y estructuras autoritarias han sabido adaptarse, y en torno a la débil consolidación del Estado de derecho y las instituciones, amenazadas por la ascensión vertiginosa del crimen organizado. La combinación de las recomposiciones autoritarias y de esas nuevas amenazas no se reduce a meras herencias autoritarias, en un contexto supuestamente democrático, ni tampoco admite la hipótesis de una transición inconclusa. Implica más bien una huida hacia delante, la cual oscila entre la democracia y su negación, así como entre las herencias del pasado y un cambio mal asimilado, obligando a revisar el modelo de análisis de las transiciones a la democracia, parcialmente pertinente, pero demasiado lineal y reductor.-----This paper proposes a reflection on the current status of democracy in Mexico. For this purpose, it has been based on an analytical reading of interpretations to the Mexican transition. They highlight the sui generis and unfinished nature of such transition, whose completion has not entailed a clear discontinuity between the authoritarian phase and the democratic consolidation phase. Our approach consists in showing that the current limitations of Mexican democracy, which are in part identifiable as consolidation issues, leave doubts about the arrival of democracy to Mexico. This is illustrated around the supposed plurality of the party system, to which the actors and power structures have adapted, and around the weak consolidation of the constitutional state and the institutions, threatened by the dramatic escalation of organized crime. The combination of authoritarian recomposition and such new threats is not a mere authoritarian inheritance, in what is a supposed democratic context, nor does it agree with the hypothesis of an unconcluded transition. It rather implies fleeing forwards, which in itself oscillates between democracy and its denial as well as between past legacies and a badly assimilated change, thus forcing us to review the analytical model for transitions to democracy which is relevant in part, but far too linear and minimizing.

Annular variability and eddy-zonal flow interactions in a simplified atmospheric GCM. Part 1: Characterization of high and low frequency behaviour

Experiments have been performed using a simplified, Newtonian forced, global circulation model to investigate how variability of the tropospheric jet can be characterized by examining the combined fluctuations of the two leading modes of annular variability. Eddy forcing of this variability is analyzed in the phase space of the leading modes using the vertically integrated momentum budget. The nature of the annular variability and eddy forcing depends on the time scale. At low frequencies the zonal flow and baroclinic eddies are in quasi equilibrium and anomalies propagate poleward. The eddies are shown primarily to reinforce the anomalous state and are closely balanced by the linear damping, leaving slow evolution as a residual. At high frequencies the flow is strongly evolving and anomalies are initiated on the poleward side of the tropospheric jet and propagate equatorward. The eddies are shown to drive this evolution strongly: eddy location and amplitude reflect the past baroclinicity, while eddy feedback on the zonal flow may be interpreted in terms of wave breaking associated with baroclinic life cycles in lateral shear.

Towards dynamical system models of language-related brain potentials

Event-related brain potentials (ERP) are important neural correlates of cognitive processes. In the domain of language processing, the N400 and P600 reflect lexical-semantic integration and syntactic processing problems, respectively. We suggest an interpretation of these markers in terms of dynamical system theory and present two nonlinear dynamical models for syntactic computations where different processing strategies correspond to functionally different regions in the system's phase space.

Contextual emergence of mental states from neurodynamics

The emergence of mental states from neural states by partitioning the neural phase space is analyzed in terms of symbolic dynamics. Well-defined mental states provide contexts inducing a criterion of structural stability for the neurodynamics that can be implemented by particular partitions. This leads to distinguished subshifts of finite type that are either cyclic or irreducible. Cyclic shifts correspond to asymptotically stable fixed points or limit tori whereas irreducible shifts are obtained from generating partitions of mixing hyperbolic systems. These stability criteria are applied to the discussion of neural correlates of consiousness, to the definition of macroscopic neural states, and to aspects of the symbol grounding problem. In particular, it is shown that compatible mental descriptions, topologically equivalent to the neurodynamical description, emerge if the partition of the neural phase space is generating. If this is not the case, mental descriptions are incompatible or complementary. Consequences of this result for an integration or unification of cognitive science or psychology, respectively, will be indicated.

Response theory for equilibrium and non-equilibrium statistical mechanics: causality and generalized Kramers-Kronig relations

We consider the general response theory recently proposed by Ruelle for describing the impact of small perturbations to the non-equilibrium steady states resulting from Axiom A dynamical systems. We show that the causality of the response functions entails the possibility of writing a set of Kramers-Kronig (K-K) relations for the corresponding susceptibilities at all orders of nonlinearity. Nonetheless, only a special class of directly observable susceptibilities obey K-K relations. Specific results are provided for the case of arbitrary order harmonic response, which allows for a very comprehensive K-K analysis and the establishment of sum rules connecting the asymptotic behavior of the harmonic generation susceptibility to the short-time response of the perturbed system. These results set in a more general theoretical framework previous findings obtained for optical systems and simple mechanical models, and shed light on the very general impact of considering the principle of causality for testing self-consistency: the described dispersion relations constitute unavoidable benchmarks that any experimental and model generated dataset must obey. The theory exposed in the present paper is dual to the time-dependent theory of perturbations to equilibrium states and to non-equilibrium steady states, and has in principle similar range of applicability and limitations. In order to connect the equilibrium and the non equilibrium steady state case, we show how to rewrite the classical response theory by Kubo so that response functions formally identical to those proposed by Ruelle, apart from the measure involved in the phase space integration, are obtained. These results, taking into account the chaotic hypothesis by Gallavotti and Cohen, might be relevant in several fields, including climate research. In particular, whereas the fluctuation-dissipation theorem does not work for non-equilibrium systems, because of the non-equivalence between internal and external fluctuations, K-K relations might be robust tools for the definition of a self-consistent theory of climate change.

A unified theory of balance in the extratropics

Many physical systems exhibit dynamics with vastly different time scales. Often the different motions interact only weakly and the slow dynamics is naturally constrained to a subspace of phase space, in the vicinity of a slow manifold. In geophysical fluid dynamics this reduction in phase space is called balance. Classically, balance is understood by way of the Rossby number R or the Froude number F; either R ≪ 1 or F ≪ 1. We examined the shallow-water equations and Boussinesq equations on an f -plane and determined a dimensionless parameter _, small values of which imply a time-scale separation. In terms of R and F, ∈= RF/√(R^2+R^2 ) We then developed a unified theory of (extratropical) balance based on _ that includes all cases of small R and/or small F. The leading-order systems are ensured to be Hamiltonian and turn out to be governed by the quasi-geostrophic potential-vorticity equation. However, the height field is not necessarily in geostrophic balance, so the leading-order dynamics are more general than in quasi-geostrophy. Thus the quasi-geostrophic potential-vorticity equation (as distinct from the quasi-geostrophic dynamics) is valid more generally than its traditional derivation would suggest. In the case of the Boussinesq equations, we have found that balanced dynamics generally implies hydrostatic balance without any assumption on the aspect ratio; only when the Froude number is not small and it is the Rossby number that guarantees a timescale separation must we impose the requirement of a small aspect ratio to ensure hydrostatic balance.

Whistler mode wave growth and propagation in the prenoon magnetosphere

Pitch-angle scattering of electrons can limit the stably trapped particle flux in the magnetosphere and precipitate energetic electrons into the ionosphere. Whistler-mode waves generated by a temperature anisotropy can mediate this pitch-angle scattering over a wide range of radial distances and latitudes, but in order to correctly predict the phase-space diffusion, it is important to characterise the whistler-mode wave distributions that result from the instability. We use previously-published observations of number density, pitch-angle anisotropy and phase space density to model the plasma in the quiet pre-noon magnetosphere (defined as periods when AE<100nT). We investigate the global propagation and growth of whistler-mode waves by studying millions of growing ray paths and demonstrate that the wave distribution at any one location is a superposition of many waves at different points along their trajectories and with different histories. We show that for observed electron plasma properties, very few raypaths undergo magnetospheric reflection, most rays grow and decay within 30 degrees of the magnetic equator. The frequency range of the wave distribution at large L can be adequately described by the solutions of the local dispersion relation, but the range of wavenormal angle is different. The wave distribution is asymmetric with respect to the wavenormal angle. The numerical results suggest that it is important to determine the variation of magnetospheric parameters as a function of latitude, as well as local time and L-shell.

Non-ergodicity of inviscid two-dimensional flow on a beta-plane and on the surface of a rotating sphere

It is shown that, for a sufficiently large value of β, two-dimensional flow on a doubly-periodic beta-plane cannot be ergodic (phase-space filling) on the phase-space surface of constant energy and enstrophy. A corresponding result holds for flow on the surface of a rotating sphere, for a sufficiently rapid rotation rate Ω. This implies that the higher-order, non-quadratic invariants are exerting a significant influence on the statistical evolution of the flow. The proof relies on the existence of a finite-amplitude Liapunov stability theorem for zonally symmetric basic states with a non-vanishing absolute-vorticity gradient. When the domain size is much larger than the size of a typical eddy, then a sufficient condition for non-ergodicity is that the wave steepness ε < 1, where ε = 2[surd radical]2Z/βU in the planar case and $\epsilon = 2^{\frac{1}{4}} a^{\frac{5}{2}}Z^{\frac{7}{4}}/\Omega U^{\frac{5}{2}}$ in the spherical case, and where Z is the enstrophy, U the r.m.s. velocity, and a the radius of the sphere. This result may help to explain why numerical simulations of unforced beta-plane turbulence (in which ε decreases in time) seem to evolve into a non-ergodic regime at large scales.

Comments on 'Balance and the slow quasimanifold: some explicit results'

The concept of slow vortical dynamics and its role in theoretical understanding is central to geophysical fluid dynamics. It leads, for example, to “potential vorticity thinking” (Hoskins et al. 1985). Mathematically, one imagines an invariant manifold within the phase space of solutions, called the slow manifold (Leith 1980; Lorenz 1980), to which the dynamics are constrained. Whether this slow manifold truly exists has been a major subject of inquiry over the past 20 years. It has become clear that an exact slow manifold is an exceptional case, restricted to steady or perhaps temporally periodic flows (Warn 1997). Thus the concept of a “fuzzy slow manifold” (Warn and Ménard 1986) has been suggested. The idea is that nearly slow dynamics will occur in a stochastic layer about the putative slow manifold. The natural question then is, how thick is this layer? In a recent paper, Ford et al. (2000) argue that Lighthill emission—the spontaneous emission of freely propagating acoustic waves by unsteady vortical flows—is applicable to the problem of balance, with the Mach number Ma replaced by the Froude number F, and that it is a fundamental mechanism for this fuzziness. They consider the rotating shallow-water equations and find emission of inertia–gravity waves at O(F2). This is rather surprising at first sight, because several studies of balanced dynamics with the rotating shallow-water equations have gone beyond second order in F, and found only an exponentially small unbalanced component (Warn and Ménard 1986; Lorenz and Krishnamurthy 1987; Bokhove and Shepherd 1996; Wirosoetisno and Shepherd 2000). We have no technical objection to the analysis of Ford et al. (2000), but wish to point out that it depends crucially on R 1, where R is the Rossby number. This condition requires the ratio of the characteristic length scale of the flow L to the Rossby deformation radius LR to go to zero in the limit F → 0. This is the low Froude number scaling of Charney (1963), which, while originally designed for the Tropics, has been argued to be also relevant to mesoscale dynamics (Riley et al. 1981). If L/LR is fixed, however, then F → 0 implies R → 0, which is the standard quasigeostrophic scaling of Charney (1948; see, e.g., Pedlosky 1987). In this limit there is reason to expect the fuzziness of the slow manifold to be “exponentially thin,” and balance to be much more accurate than is consistent with (algebraic) Lighthill emission.

Implications of model error for numerical climate prediction

Numerical climate models constitute the best available tools to tackle the problem of climate prediction. Two assumptions lie at the heart of their suitability: (1) a climate attractor exists, and (2) the numerical climate model's attractor lies on the actual climate attractor, or at least on the projection of the climate attractor on the model's phase space. In this contribution, the Lorenz '63 system is used both as a prototype system and as an imperfect model to investigate the implications of the second assumption. By comparing results drawn from the Lorenz '63 system and from numerical weather and climate models, the implications of using imperfect models for the prediction of weather and climate are discussed. It is shown that the imperfect model's orbit and the system's orbit are essentially different, purely due to model error and not to sensitivity to initial conditions. Furthermore, if a model is a perfect model, then the attractor, reconstructed by sampling a collection of initialised model orbits (forecast orbits), will be invariant to forecast lead time. This conclusion provides an alternative method for the assessment of climate models.

Reconnection at the high-latitude magnetopause during northward interplanetary magnetic field conditions

The Polar spacecraft had a prolonged encounter with the high-latitude dayside magnetopause on May 29, 1996. This encounter with the magnetopause occurred when the interplanetary magnetic field was directed northward. From the three-dimensional electron and ion distribution functions measured by the Hydra instrument, it has been possible to identify nearly all of the distinct boundary layer regions associated with high-latitude reconnection. The regions that have been identified are (1) the cusp; (2) the magnetopause current layer; (3) magnetosheath field lines that have interconnected in only the Northern Hemisphere; (4) magnetosheath field lines that have interconnected in only the Southern Hemisphere; (5) magnetosheath field lines that have interconnected in both the Northern and Southern Hemispheres; (6) magnetosheath that is disconnected from the terrestrial magnetic field; and (7) high-latitude plasma sheet field lines that are participating in magnetosheath reconnection. Reconnection over this time period was occurring at high latitudes over a broad local-time extent, interconnecting the magnetosheath and lobe and/or plasma sheet field lines in both the Northern and Southern Hemispheres. Newly closed boundary layer field lines were observed as reconnection occur-red first at high latitudes in one hemisphere and then later in the other. These observations establish the location of magnetopause reconnection during these northward interplanetary magnetic field conditions as being at high latitudes, poleward of the cusp, and further reinforce the general interpretation of electron and ion phase space density signatures as indicators of magnetic reconnection and boundary layer formation.

The problem of two fixed centers: bifurcation diagram for positive energies

We give a comprehensive analysis of the Euler-Jacobi problem of motion in the field of two fixed centers with arbitrary relative strength and for positive values of the energy. These systems represent nontrivial examples of integrable dynamics and are analysed from the point of view of the energy-momentum mapping from the phase space to the space of the integration constants. In this setting, we describe the structure of the scattering trajectories in phase space and derive an explicit description of the bifurcation diagram, i.e., the set of critical value of the energy-momentum map.