Slow Trajectory Research Articles

Duck Trajectories of Three-Dimensional Singularly Perturbed Systems

Abstract For the singularly perturbed system of three equations with one fast variable and two slow ones the problem of the emergence of duck trajectories is considered in the case with two different slow motion trajectories intersecting in a general manner.

Global Marine Biodiversity Trends

▪ Abstract Marine biodiversity encompasses all levels of complexity of life in the sea, from within species to across ecosystems. At all levels, marine biodiversity has naturally exhibited a general, slow trajectory of increase, punctuated by mass extinctions at the evolutionary scale and by disturbances at the ecological scale. In historical times, a synergy of human threats, including overfishing, global warming, biological introductions, and pollution, has caused a rapid decline in global marine biodiversity, as measured by species extinctions, population depletions, and community homogenization. The consequences of this biodiversity loss include changes in ecosystem function and a reduction in the provision of ecosystem services. Global biodiversity loss will continue and likely accelerate in the future, with potentially more frequent ecological collapses and community-wide shifts. However, the timing and magnitude of these catastrophic events are probably unpredictable.

SAR Sensor Trajectory Deviations: Fourier Domain Formulation and Extended Scene Simulation of Raw Signal

Synthetic aperture radar (SAR) raw signal simulation is a useful tool for SAR system design, mission planning, processing algorithm testing, and inversion algorithm design. A two-dimensional (2-D) Fourier domain SAR raw signal simulator, exploiting the efficiency of fast Fourier transform algorithms, has been presented some years ago and is able to generate the raw signal corresponding to extended scenes. However, it cannot account for the effects of sensor trajectory deviations with respect to the nominal straight-line path. This paper explores the possibility of extending the efficient Fourier domain simulation approach to the case of sensor trajectory deviations, which is more realistic for airborne SAR systems. We first of all obtain a general and compact Fourier domain formulation of the SAR raw signal in the presence of arbitrary trajectory deviations, and show that in this general case no efficient simulation scheme can be devised. However, we demonstrate that, if a narrow beam and slow trajectory deviation assumption is made, a full 2-D Fourier domain simulation can be used. This approach can be applied only to some SAR systems and/or trajectory deviations, but it has the advantage that processing time is practically not increased with respect to the nominal trajectory case. The validity limits of the approach are analytically evaluated. Some simulation results are finally presented in order to verify the effectiveness of the proposed simulation scheme. In another paper, which is the second part of this work, it will be shown that the narrow beam-slow deviation assumption can be relaxed, at the expense of computation efficiency, if a one-dimensional azimuth Fourier domain processing followed by a range time-domain integration is used

Band-Limited Trajectory Planning and Tracking for Certain Dynamically Stabilized Mobile Systems

In this paper, a general framework for trajectory planning and tracking is formulated for dynamically stabilized mobile systems, e.g., inverted wheeled pendulums and autonomous helicopters. Within this framework, the system state is divided into slow and fast substates. The fast substate is used as a pseudocontrol for tracking a desired slow substate trajectory. First, an exponential fast substate controller is designed to track a fast substate reference trajectory. This fast substate reference trajectory is, in turn, planned so that the slow substate follows its desired trajectory. To ensure that the fast substate reference trajectory is feasible for the exponential controller, it is designed using band-limited “Sinc” functions whose maximum frequency is less than the inverse of the time constant of the exponential controller. To illustrate the procedure, the dynamic model of an inverted wheeled pendulum is reformulated by a partial feedback linearization such that it is amenable to the separation into slow and fast components. The planning and tracking controller design is explained using simulation results. This technique is shown to be easily embedded inside a modified nonlinear model predictive control framework for the slow subsystem. This framework tries to explicitly take the computational delay into account. The computation time required for this technique is encouraging from a real-world implementation perspective.

The association of birthweight and contemporary size with insulin resistance among children from Estonia and Denmark: findings from the European Youth Heart Study

To assess the associations of birthweight, contemporary body mass index and height with insulin resistance in children. Cross-sectional study. From Estonia (n = 1174) and Denmark (n = 1018), 2192 school children aged 9 and 15 years were randomly selected. Insulin resistance (homeostasis model assessment), triglyceride levels, high-density lipoprotein cholesterol and systolic blood pressure. There was an inverse association between birthweight and insulin resistance and a positive association between contemporary body mass index and insulin resistance. With adjustment for maternal and paternal educational level, income, smoking and body mass index, an increase of one unit of sex, age and country standardized body mass index z-score was associated with a 5% (95% CI: 2, 7%) increase in homeostasis model assessment (HOMA) score and a one-unit z-score increase in birthweight with a 2% (95% CI: 0, 5%) decrease in HOMA score. In the 9-year-old age group, height was positively associated with insulin resistance [for a one-unit increase in height z-score HOMA score increased by 30% (95% CI: 14, 50%)], but in the 15-year-old age group there was no association between height and insulin resistance (4% (95% CI: -5, 14%), P for interaction with age group = 0.001). For both ages, those in the lowest third of the birthweight distribution and highest third of the body mass index distribution were most insulin resistant and, among 9-year olds, those in the lowest third of the birthweight distribution and highest third of the height distribution were most insulin resistant. Birthweight was only inversely associated with systolic blood pressure when adjustment was made for either contemporary body mass index or height and there was no association between birthweight and high-density lipoprotein or triglyceride concentrations. Taken together, these results suggest that a slow intrauterine growth trajectory and/or a fast post-natal growth trajectory is associated with greater insulin resistance in childhood.

On the relevance of resonances

For a chain transitive flow, the averages produced by the invariant probability measures are approximated by multi-valued averages of an inflated flow. The results are applied to singularly perturbed differential equations with slow and fast trajectories. Here, the multi-valued character of the averaged system is due to the presence of many invariant probability measures, often caused by resonances. Hence, the inflation of a chain transitive flow gives information on the relevance of resonances. The Takens equation modelling strong elastic constraints is treated in detail.

APPROXIMATED OPTIMAL CONTROL OF SINGULARLY PERTURBED SYSTEMS VIA HAAR WAVELETS

This paper deals with the implementation of Haar wavelet to the optimal control of linear singularly perturbed systems. The approximated composite control and the slow and fast trajectories with respect to a quadratic cost function by solving only the linear algebraic equations are calculated. The results are illustrated with a simple example.

Analysis of a Canard Mechanism by Which Excitatory Synaptic Coupling Can Synchronize Neurons at Low Firing Frequencies

A population of oscillatory Hodgkin--Huxley (HH) model neurons is shown numerically to exhibit a behavior in which the introduction of excitatory synaptic coupling synchronizes and dramatically slows firing. This effect contrasts with the standard theory that recurrent synaptic excitation promotes states of rapid, sustained activity, independent of intrinsic neuronal dynamics. The observed behavior is not due to simple depolarization block nor to standard elliptic bursting, although it is related to these phenomena. We analyze this effect using a reduced model for a single, self-coupled HH oscillator. The mechanism explained here involves an extreme form of delayed bifurcation in which the development of a vortex structure through interaction of fast and slow subsystems pins trajectories near a surface that consists of unstable equilibria of a certain reduced system, in a canard-like manner. Using this vortex structure, a new passage time calculation is used to approximate the interspike time interval. We...

Approaches for dynamic modelling of flexible manipulator systems

An investigation into the dynamic modelling of flexible manipulator systems with different mechanical structures and actuation mechanisms is presented. Dynamic models of the systems are developed based on the assumed modes method considering linear displacements, quadratic displacements and the finite element method. Simulation and experimental exercises are performed with bang-bang torque and smooth displacement driven manoeuvres. To investigate the effects of centrifugal forces, slow and fast trajectories are implemented. Simulation and experimental results of the response of the flexible manipulators are presented in the time and frequency domains. Moreover, model validations are carried out by comparing the results to assess the performance of the modelling approaches.

Feedbacks between canopy composition and seedling regeneration in mixed conifer broad‐leaved forests

To address the role of canopy‐seedling feedbacks in the structure and dynamics of mixed conifer broad‐leaved forests in the eastern US, we monitored seedling regeneration patterns and environmental conditions in the understorey of stands dominated by either hemlock (Tsuga canadensis) or red oak (Quercus rubra) for three years. Hemlock seedlings were favoured over other species’ seedlings in hemlock stands (a true positive feedback), due to a combination of high seed inputs, high seedling emergence and relatively high seedling survival during the growing season, which allowed hemlock to remain dominant under its own canopy. Red oak stands favoured a suite of mid‐successional broad‐leaved species over hemlock. A more even age structure of broad‐leaved species in red oak stands revealed that high seedling survival in such stands were driving this feedback. Canopy‐mediated variations in both understorey light availability (1.5% for hemlock vs 3.5% for red oak) and soil pH (3.9 for hemlock vs 4.4 for red oak) were found to be the primary correlates of stand‐level differences in seedling regeneration dynamics. In mixed temperate forests in the eastern US, canopy‐seedling feedbacks could act to slow successional trajectories and contribute to the maintenance of a stable landscape structure over many generations.

Alternative approaches to analyzing individual differences in the rate of early vocabulary development

This paper examines individual differences in the rate of early lexical development with a specific interest in gender differences. Twenty-six children were assessed monthly from either 8, 9, or 10 months of age through 14 months of age, using the MacArthur Communicative Development Inventory: Words and Gestures. Individual differences in developmental trajectories of vocabulary comprehension and production were explored using two analytic approaches. The first involved traditional parametric statistics, while the latter utilized classification procedures. Both techniques demonstrated that the lexical development of girls outpaced that of boys. The inductive approach also revealed the presence of distinctive “fast” and “slow” trajectories for both comprehension and production that were not exclusively segregated by gender. Cases exhibiting fast trajectories were predominantly girls, but several boys also followed this developmental pattern. The opposite pattern emerged for the slow trajectories. There was strong correspondence between production and comprehension, but a few cases clustered into the fast development group on one measure and the slow group on the other. The identification of these outliers may offer an important tool for exploring mechanisms of language development. Validation of the clustering results was based on the prospective prediction of an external criterion variable, namely, lexical development at 21 months, and by replication on an independent sample.

Growth processes in teeth distinguish modern humans from Homo erectus and earlier hominins.

A modern human-like sequence of dental development, as a proxy for the pace of life history, is regarded as one of the diagnostic hallmarks of our own genus Homo. Brain size, age at first reproduction, lifespan and other life-history traits correlate tightly with dental development. Here we report differences in enamel growth that show the earliest fossils attributed to Homo do not resemble modern humans in their development. We used daily incremental markings in enamel to calculate rates of enamel formation in 13 fossil hominins and identified differences in this key determinant of tooth formation time. Neither australopiths nor fossils currently attributed to early Homo shared the slow trajectory of enamel growth typical of modern humans; rather, both resembled modern and fossil African apes. We then reconstructed tooth formation times in australopiths, in the approximately 1.5-Myr-old Homo erectus skeleton from Nariokotome, Kenya, and in another Homo erectus specimen, Sangiran S7-37 from Java. These times were shorter than those in modern humans. It therefore seems likely that truly modern dental development emerged relatively late in human evolution.

On the rise of an ellipsoidal bubble in water: oscillatory paths and liquid-induced velocity

This work is an experimental study of the rise of an air bubble in still water. For the bubble diameter considered, path oscillations develop in the absence of shape oscillations and the effect of surfactants is shown to be negligible. Both the three-dimensional motion of the bubble and the velocity induced in the liquid are investigated. After the initial acceleration stage, the bubble shape remains constant and similar to an oblate ellipsoid with its symmetry axis parallel to the bubble-centre velocity, and with constant velocity magnitude. The bubble motion combines path oscillations with slow trajectory displacements. (These displacements, which consist of horizontal drift and rotation about a vertical axis, are shown to have no influence on the oscillations). The bubble dynamics involve two unstable modes which have the same frequency and are π/2 out of phase. The primary mode develops first, leading to a plane zigzag trajectory. The secondary mode then grows, causing the trajectory to progressively change into a circular helix. Liquid-velocity measurements are taken up to 150 radii behind the bubble. The nature of the liquid flow field is analysed from systematic comparisons with potential theory and direct numerical simulations. The flow is potential in front of the bubble and a long wake develops behind. The wake structure is controlled by two mechanisms: the development of a quasi-steady wake that spreads around the non-rectilinear bubble trajectory; and the wake instability that generates unsteady vortices at the bubble rear. The velocities induced by the wake vortices are small compared to the bubble velocity and, except in the near wake, the flow is controlled by the quasi-steady wake.

Gas and Aerosol Loadings at Holme Moss 1996–1998 Measurements and Analysis Using Back-Trajectory Clustering

A continuous two-year atmospheric datasetcomprising gas and aerosol loadings from amountain site in northern England (Holme Moss,W1°51′30′′ N53°32′0′′) is presented. The data are analysed with respect to three-dayback-trajectories that are grouped according to aclustering technique that allows speed, directionand curvature of the airmass track to beconsidered. The technique is successful inseparating the data into chemically distinctsubsets. Up to 29% of the variance in the datais explained by back-trajectory clusters. Slow trajectories are associated with highloadings especially for the oxides of nitrogen,which may imply a local source for much of thesuspended pollutant. The data suggest thatproduction of nitrate and sulphate is limited byoxidant availability at least in the winter. Itmay be possible to optimise the analytical powerof the technique by increasing the importance ofrecent airmass track in determining clusterallocation. This applies especially to the totalsulphur loading.

Limits of nonlinear discrete-time control systems with fast subsystems

We investigate the asymptotic properties of nonlinear discrete-time control systems with fast subsystems. In particular, we use the averaging method in order to construct a limiting system for the slow subsystem. We show continuous dependence of the slow trajectories on the perturbation parameter measuring the ratio of the different step sizes. As a byproduct we get an exponential formula for the reachable set of nonlinear continuous time control systems with finite control range.

The adoption of life-cycle approaches by industry: patterns and impacts

Firms are the primary organisers and drivers of resources flows through and emissions from developed economies. Sustainable industrial development is a process in which these flows are modified. This includes both a reduction of materials and energy intensity, increasing cyclicity of materials fluxes, and a reduction of dissipative uses of toxic materials. The role of firms in achieving these changes is discussed in this paper. Life cycle assessment activities in large European firms in six industrial sectors are assessed. Market and regulatory pressures on firms to adopt life cycle approaches differ markedly between industrial sectors and across national markets. Producers of final products are most likely to find opportunities for improvements in the life cycle environmental performance of products while making gains in dynamic competitiveness at the same time. Upstream producers of commodity products have tended to use life cycle approaches defensively against negative environmental claims and in seeking to influence the policy process. Despite widespread adoption of life cycle approaches, changes in product system ecoprofiles are likely to be slow given technological trajectories, infrastructural and resource endowments, and the fragmentation of environmental responsibility across the product chain.

Microcinematographic studies of flow patterns in the excised rabbit aorta and its major branches

Arterial fluid mechanics may play a role as a localizing factor for early atherosclerosis. Flow patterns in natural rabbit aortas rendered transparent were studied using a microcinematographic visualization technique. The aortic arch exhibited a single cell of clockwise-rotating helical secondary flow along the ventral and inner walls. Flow separation occurred proximal to the two arch branches with flow reversal proximal to the brachiocephalic artery. Sinusoidal flow rendered the helical motion more pronounced in systole, while the reverse flow zone periodically expanded and contracted. Steady flow in the abdominal aorta revealed streamlines which follow slow looping trajectories lateral to ostia before tracing helical paths into the branches. Flow separation was present along the dorsal wall of the aorta opposite the superior mesenteric artery. With the exception of the left renal artery, steady flow wall shear stresses were higher distal to ostia than proximal. Spatial gradients of wall shear stress were larger around branches than elsewhere. Similar to observed flow patterns, sites of enhanced macromolecular permeability, as observed previously in the normal rabbit aorta, follow a clockwise helical pattern in the arch and exhibit a distribution around ostia that correlates to some degree with regions of elevated shear stress gradients.

Time-optimal series capacitor control for damping interarea modes in interconnected power systems

Controllable series capacitors have been proposed to damp power swings resulting from large transient disturbances in two-machine power systems. In this paper, the authors propose a time-optimal series capacitor control to damp interarea modes in multi-machine power systems. To find the switching line, they apply the slow coherency transformation to the full power system and project a slow subsystem trajectory found from the trajectory reversing method onto the predominant interarea mode. The method is illustrated by the time simulation of a six-machine power system with two interarea modes and is shown to give the desired result.

Increased dependence upon visual information of movement performance during visuo-motor tracking in cerebellar disorders

The effect of temporarily suppressing the visual display of either the target (desired) trajectory or the actually movement trajectory upon the accuracy of visuo-motor tracking was studied in 6 patients with cerebellar syndromes and 6 healthy subjects. Subjects made extension and flexion movements of the wrist to superimpose a cursor displaying their movement (movement cursor) upon one indicating target (target cursor) on a VDU screen. The target trajectory consisted of a sawtooth pattern of slow (4 deg/sec) ramp extension and instantaneous flexion return phases. Following practice, the tracking of cerebellar patients was significantly less accurate than that of healthy subjects for each phase ( P = 0.02). Temporary suppression of the movement cursor during both the mid-section of the ramp phase ( P = 0.05) and around the reversal phase ( P = 0.04) caused a significant increase in tracking errors in the patients whereas suppression of the target cursor did not alter their performance. Suppression of neither cursor altered the tracking accuracy of healthy subjects during the ramp extensions whilst suppression of either caused reduced ( P = 0.02) performance for the reversal phase. We interpret the increased dependence of patients upon visual information of their movements during slow trajectories as indicating an impairment of proprioceptive guidance.

On slow motions in chaotic systems

A chaotic dynamical system with a separation of the time scales is considered. An asymptotic method is developed which predicts the slow trajectory and the distribution of fluctuations. It is shown that generally the latter converge to no limit as ϵ goes to zero.