New delay-dependent stability criteria for uncertain neutral systems with mixed time-varying delays and nonlinear perturbations

The problem of stability analysis for a class of neutral systems with mixed time-varying neutral, discrete and distributed delays and nonlinear parameter perturbations is addressed. By introducing a novel Lyapunov-Krasovskii functional and combining the descriptor model transformation, the Leibniz-Newton formula, some free-weighting matrices, and a suitable change of variables, new sufficient conditions are established for the stability of the considered system, which are neutral-delay-dependent, discrete-delay-range dependent, and distributeddelay-dependent. The conditions are presented in terms of linear matrix inequalities (LMIs) and can be efficiently solved using convex programming techniques. Two numerical examples are given to illustrate the efficiency of the proposed method

Comparing meta-analysis and ecological-longitudinal analysis in time-series studies: a case study of the effects of air pollution on mortality in three Spanish cities

The objective of this paper is to introduce a diVerent approach, called the ecological-longitudinal, to carrying out pooled analysis in time series ecological studies. Because it gives a larger number of data points and, hence, increases the statistical power of the analysis, this approach, unlike conventional ones, allows the complementation of aspects such as accommodation of random effect models, of lags, of interaction between pollutants and between pollutants and meteorological variables, that are hardly implemented in conventional approaches. Design—The approach is illustrated by providing quantitative estimates of the short-termeVects of air pollution on mortality in three Spanish cities, Barcelona,Valencia and Vigo, for the period 1992–1994. Because the dependent variable was a count, a Poisson generalised linear model was first specified. Several modelling issues are worth mentioning. Firstly, because the relations between mortality and explanatory variables were nonlinear, cubic splines were used for covariate control, leading to a generalised additive model, GAM. Secondly, the effects of the predictors on the response were allowed to occur with some lag. Thirdly, the residual autocorrelation, because of imperfect control, was controlled for by means of an autoregressive Poisson GAM. Finally, the longitudinal design demanded the consideration of the existence of individual heterogeneity, requiring the consideration of mixed models. Main results—The estimates of the relative risks obtained from the individual analyses varied across cities, particularly those associated with sulphur dioxide. The highest relative risks corresponded to black smoke in Valencia. These estimates were higher than those obtained from the ecological-longitudinal analysis. Relative risks estimated from this latter analysis were practically identical across cities, 1.00638 (95% confidence intervals 1.0002, 1.0011) for a black smoke increase of 10 μg/m3 and 1.00415 (95% CI 1.0001, 1.0007) for a increase of 10 μg/m3 of sulphur dioxide. Because the statistical power is higher than in the individual analysis more interactions were statistically significant,especially those among air pollutants and meteorological variables. Conclusions—Air pollutant levels were related to mortality in the three cities of the study, Barcelona, Valencia and Vigo. These results were consistent with similar studies in other cities, with other multicentric studies and coherent with both, previous individual, for each city, and multicentric studies for all three cities

Enhancing dominant modes in nonstationary time series by means of the symbolic resonance analysis

We present the symbolic resonance analysis (SRA) as a viable method for addressing the problem of enhancing a weakly dominant mode in a mixture of impulse responses obtained from a nonlinear dynamical system. We demonstrate this using results from a numerical simulation with Duffing oscillators in different domains of their parameter space, and by analyzing event-related brain potentials (ERPs) from a language processing experiment in German as a representative application. In this paradigm, the averaged ERPs exhibit an N400 followed by a sentence final negativity. Contemporary sentence processing models predict a late positivity (P600) as well. We show that the SRA is able to unveil the P600 evoked by the critical stimuli as a weakly dominant mode from the covering sentence final negativity. (c) 2007 American Institute of Physics. (c) 2007 American Institute of Physics.

The application of the Hilbert spectrum to the analysis of electromyographic signals

This paper investigates the application of the Hilbert spectrum (HS), which is a recent tool for the analysis of nonlinear and nonstationary time-series, to the study of electromyographic (EMG) signals. The HS allows for the visualization of the energy of signals through a joint time-frequency representation. In this work we illustrate the use of the HS in two distinct applications. The first is for feature extraction from EMG signals. Our results showed that the instantaneous mean frequency (IMNF) estimated from the HS is a relevant feature to clinical practice. We found that the median of the IMNF reduces when the force level of the muscle contraction increases. In the second application we investigated the use of the HS for detection of motor unit action potentials (MUAPs). The detection of MUAPs is a basic step in EMG decomposition tools, which provide relevant information about the neuromuscular system through the morphology and firing time of MUAPs. We compared, visually, how MUAP activity is perceived on the HS with visualizations provided by some traditional (e.g. scalogram, spectrogram, Wigner-Ville) time-frequency distributions. Furthermore, an alternative visualization to the HS, for detection of MUAPs, is proposed and compared to a similar approach based on the continuous wavelet transform (CWT). Our results showed that both the proposed technique and the CWT allowed for a clear visualization of MUAP activity on the time-frequency distributions, whereas results obtained with the HS were the most difficult to interpret as they were extremely affected by spurious energy activity. (c) 2008 Elsevier Inc. All rights reserved.

Dynamic integrated system optimization and parameter estimation for discrete time optimal control of nonlinear systems

An algorithm for solving nonlinear discrete time optimal control problems with model-reality differences is presented. The technique uses Dynamic Integrated System Optimization and Parameter Estimation (DISOPE), which achieves the correct optimal solution in spite of deficiencies in the mathematical model employed in the optimization procedure. A version of the algorithm with a linear-quadratic model-based problem, implemented in the C+ + programming language, is developed and applied to illustrative simulation examples. An analysis of the optimality and convergence properties of the algorithm is also presented.

A titration model: an analysis and solution

The Newton‐Raphson method is proposed for the solution of the nonlinear equation arising from a theoretical model of an acid/base titration. It is shown that it is necessary to modify the form of the equation in order that the iteration is guaranteed to converge. A particular example is considered to illustrate the analysis and method, and a BASIC program is included that can be used to predict the pH of any weak acid/weak base titration.

On reliability analysis of multi-categorical forecasts

Reliability analysis of probabilistic forecasts, in particular through the rank histogram or Talagrand diagram, is revisited. Two shortcomings are pointed out: Firstly, a uniform rank histogram is but a necessary condition for reliability. Secondly, if the forecast is assumed to be reliable, an indication is needed how far a histogram is expected to deviate from uniformity merely due to randomness. Concerning the first shortcoming, it is suggested that forecasts be grouped or stratified along suitable criteria, and that reliability is analyzed individually for each forecast stratum. A reliable forecast should have uniform histograms for all individual forecast strata, not only for all forecasts as a whole. As to the second shortcoming, instead of the observed frequencies, the probability of the observed frequency is plotted, providing and indication of the likelihood of the result under the hypothesis that the forecast is reliable. Furthermore, a Goodness-Of-Fit statistic is discussed which is essentially the reliability term of the Ignorance score. The discussed tools are applied to medium range forecasts for 2 m-temperature anomalies at several locations and lead times. The forecasts are stratified along the expected ranked probability score. Those forecasts which feature a high expected score turn out to be particularly unreliable.

Isomap nonlinear dimensionality reduction and bimodality of Asian monsoon convection

It is known that the empirical orthogonal function method is unable to detect possible nonlinear structure in climate data. Here, isometric feature mapping (Isomap), as a tool for nonlinear dimensionality reduction, is applied to 1958–2001 ERA-40 sea-level pressure anomalies to study nonlinearity of the Asian summer monsoon intraseasonal variability. Using the leading two Isomap time series, the probability density function is shown to be bimodal. A two-dimensional bivariate Gaussian mixture model is then applied to identify the monsoon phases, the obtained regimes representing enhanced and suppressed phases, respectively. The relationship with the large-scale seasonal mean monsoon indicates that the frequency of monsoon regime occurrence is significantly perturbed in agreement with conceptual ideas, with preference for enhanced convection on intraseasonal time scales during large-scale strong monsoons. Trend analysis suggests a shift in concentration of monsoon convection, with less emphasis on South Asia and more on the East China Sea.

A low-order model investigation of the analysis of gravity waves in the ensemble Kalman filter.

The behavior of the ensemble Kalman filter (EnKF) is examined in the context of a model that exhibits a nonlinear chaotic (slow) vortical mode coupled to a linear (fast) gravity wave of a given amplitude and frequency. It is shown that accurate recovery of both modes is enhanced when covariances between fast and slow normal-mode variables (which reflect the slaving relations inherent in balanced dynamics) are modeled correctly. More ensemble members are needed to recover the fast, linear gravity wave than the slow, vortical motion. Although the EnKF tends to diverge in the analysis of the gravity wave, the filter divergence is stable and does not lead to a great loss of accuracy. Consequently, provided the ensemble is large enough and observations are made that reflect both time scales, the EnKF is able to recover both time scales more accurately than optimal interpolation (OI), which uses a static error covariance matrix. For OI it is also found to be problematic to observe the state at a frequency that is a subharmonic of the gravity wave frequency, a problem that is in part overcome by the EnKF.However, error in themodeled gravity wave parameters can be detrimental to the performance of the EnKF and remove its implied advantages, suggesting that a modified algorithm or a method for accounting for model error is needed.

The troposphere-to-stratosphere transition in kinetic energy spectra and nonlinear spectral fluxes as seen in ECMWF analyses

Global horizontal wavenumber kinetic energy spectra and spectral fluxes of rotational kinetic energy and enstrophy are computed for a range of vertical levels using a T799 ECMWF operational analysis. Above 250 hPa, the kinetic energy spectra exhibit a distinct break between steep and shallow spectral ranges, reminiscent of dual power-law spectra seen in aircraft data and high-resolution general circulation models. The break separates a large-scale ‘‘balanced’’ regime in which rotational flow strongly dominates divergent flow and a mesoscale ‘‘unbalanced’’ regime where divergent energy is comparable to or larger than rotational energy. Between 230 and 100 hPa, the spectral break shifts to larger scales (from n 5 60 to n 5 20, where n is spherical harmonic index) as the balanced component of the flow preferentially decays. The location of the break remains fairly stable throughout the stratosphere. The spectral break in the analysis occurs at somewhat larger scales than the break seen in aircraft data. Nonlinear spectral fluxes defined for the rotational component of the flow maximize between about 300 and 200 hPa. Large-scale turbulence thus centers on the extratropical tropopause region, within which there are two distinct mechanisms of upscale energy transfer: eddy–eddy interactions sourcing the transient energy peak in synoptic scales, and zonal mean–eddy interactions forcing the zonal flow. A well-defined downscale enstrophy flux is clearly evident at these altitudes. In the stratosphere, the transient energy peak moves to planetary scales and zonal mean–eddy interactions become dominant.

Metabifurcation analysis of a mean field model of the cortex

Mean field models (MFMs) of cortical tissue incorporate salient, average features of neural masses in order to model activity at the population level, thereby linking microscopic physiology to macroscopic observations, e.g., with the electroencephalogram (EEG). One of the common aspects of MFM descriptions is the presence of a high-dimensional parameter space capturing neurobiological attributes deemed relevant to the brain dynamics of interest. We study the physiological parameter space of a MFM of electrocortical activity and discover robust correlations between physiological attributes of the model cortex and its dynamical features. These correlations are revealed by the study of bifurcation plots, which show that the model responses to changes in inhibition belong to two archetypal categories or “families”. After investigating and characterizing them in depth, we discuss their essential differences in terms of four important aspects: power responses with respect to the modeled action of anesthetics, reaction to exogenous stimuli such as thalamic input, and distributions of model parameters and oscillatory repertoires when inhibition is enhanced. Furthermore, while the complexity of sustained periodic orbits differs significantly between families, we are able to show how metamorphoses between the families can be brought about by exogenous stimuli. We here unveil links between measurable physiological attributes of the brain and dynamical patterns that are not accessible by linear methods. They instead emerge when the nonlinear structure of parameter space is partitioned according to bifurcation responses. We call this general method “metabifurcation analysis”. The partitioning cannot be achieved by the investigation of only a small number of parameter sets and is instead the result of an automated bifurcation analysis of a representative sample of 73,454 physiologically admissible parameter sets. Our approach generalizes straightforwardly and is well suited to probing the dynamics of other models with large and complex parameter spaces.

Lateral boundary contributions to wave-activity invariants and nonlinear stability theorems for balanced dynamics

The slow advective-timescale dynamics of the atmosphere and oceans is referred to as balanced dynamics. An extensive body of theory for disturbances to basic flows exists for the quasi-geostrophic (QG) model of balanced dynamics, based on wave-activity invariants and nonlinear stability theorems associated with exact symmetry-based conservation laws. In attempting to extend this theory to the semi-geostrophic (SG) model of balanced dynamics, Kushner & Shepherd discovered lateral boundary contributions to the SG wave-activity invariants which are not present in the QG theory, and which affect the stability theorems. However, because of technical difficulties associated with the SG model, the analysis of Kushner & Shepherd was not fully nonlinear. This paper examines the issue of lateral boundary contributions to wave-activity invariants for balanced dynamics in the context of Salmon's nearly geostrophic model of rotating shallow-water flow. Salmon's model has certain similarities with the SG model, but also has important differences that allow the present analysis to be carried to finite amplitude. In the process, the way in which constraints produce boundary contributions to wave-activity invariants, and additional conditions in the associated stability theorems, is clarified. It is shown that Salmon's model possesses two kinds of stability theorems: an analogue of Ripa's small-amplitude stability theorem for shallow-water flow, and a finite-amplitude analogue of Kushner & Shepherd's SG stability theorem in which the ‘subsonic’ condition of Ripa's theorem is replaced by a condition that the flow be cyclonic along lateral boundaries. As with the SG theorem, this last condition has a simple physical interpretation involving the coastal Kelvin waves that exist in both models. Salmon's model has recently emerged as an important prototype for constrained Hamiltonian balanced models. The extent to which the present analysis applies to this general class of models is discussed.

Nonlinear stability of multilayer quasi-geostrophic flow

New nonlinear stability theorems are derived for disturbances to steady basic flows in the context of the multilayer quasi-geostrophic equations. These theorems are analogues of Arnol’d's second stability theorem, the latter applying to the two-dimensional Euler equations. Explicit upper bounds are obtained on both the disturbance energy and disturbance potential enstrophy in terms of the initial disturbance fields. An important feature of the present analysis is that the disturbances are allowed to have non-zero circulation. While Arnol’d's stability method relies on the energy–Casimir invariant being sign-definite, the new criteria can be applied to cases where it is sign-indefinite because of the disturbance circulations. A version of Andrews’ theorem is established for this problem, and uniform potential vorticity flow is shown to be nonlinearly stable. The special case of two-layer flow is treated in detail, with particular attention paid to the Phillips model of baroclinic instability. It is found that the short-wave portion of the marginal stability curve found in linear theory is precisely captured by the new nonlinear stability criteria.

A spectral view of nonlinear fluxes and stationary-transient interaction in the atmosphere

Nonlinear spectral transfers of kinetic energy and enstrophy, and stationary-transient interaction, are studied using global FGGE data for January 1979. It is found that the spectral transfers arise primarily from a combination, in roughly equal measure, of pure transient and mixed stationary-transient interactions. The pure transient interactions are associated with a transient eddy field which is approximately locally homogeneous and isotropic, and they appear to be consistently understood within the context of two-dimensional homogeneous turbulence. Theory based on spatial wale separation concepts suggests that the mixed interactions may be understood physically, to a first approximation, as a process of shear-induced spectral transfer of transient enstrophy along lines of constant zonal wavenumber. This essentially conservative enstrophy transfer generally involves highly nonlocal stationary-transient energy conversions. The observational analysis demonstrates that the shear-induced transient enstrophy transfer is mainly associated with intermediate-scale (zonal wavenumber m > 3) transients and is primarily to smaller (meridional) scales, so that the transient flow acts as a source of stationary energy. In quantitative terms, this transient-eddy rectification corresponds to a forcing timescale in the stationary energy budget which is of the same order of magnitude as most estimates of the damping timescale in simple stationary-wave models (5 to 15 days). Moreover, the nonlinear interactions involved are highly nonlocal and cover a wide range of transient scales of motion.