Variations In Initial Conditions Research Articles

Periodic motions and resonances of impact oscillators

Bilinear oscillators – the oscillators whose springs have different stiffnesses in compression and tension – model a wide range of phenomena. A limiting case of bilinear oscillator with infinite stiffness in compression – the impact oscillator – is studied here. We investigate a special set of impact times – the eigenset, which corresponds to the solution of the homogeneous equation, i.e. the oscillator without the driving force. We found that this set and its subsets are stable with respect to variation of initial conditions. Furthermore, amongst all periodic sets of impact times with the period commensurate with the period of driving force, the eigenset is the only one which can support resonances, in particular the multi-‘harmonic’ resonances. Other resonances should produce non-periodic sets of impact times. This funding indicates that the usual simplifying assumption [e.g., S.W. Shaw, P.J. Holmes, A periodically forced piecewise linear oscillator, Journal of Sound and Vibration 90 (1983) 129–155] that the times between impacts are commensurate with the period of the driving force does not always hold. We showed that for the first sub-‘harmonic resonance’ – the resonance achieved on a half frequency of the main resonance – the set of impact times is asymptotically close to the eigenset. The envelope of the oscillations in this resonance increases as a square root of time, opposite to the linear increase characteristic of multi-‘harmonic’ resonances.

The Topography of Historical Contingency

Abstract Starting with Ben-Menahem’s definition of historical contingency as sensitivity to variations in initial conditions, we suggest that historical events and processes can be thought of as forming a complex landscape of contingency and necessity. We suggest three different ways of extending and elaborating Ben-Menahem’s concepts: (1) By supplementing them with a notion of historical disturbance; (2) by pointing out that contingency and necessity are subject to scaling effects; (3) by showing how degrees of contingency/necessity can change over time. We also argue that further development of Sterelny’s notion of conditional inevitability leads to our conclusion that the topography of historical contingency is something that can change over time.

Robust processes and teleological language

I consider some hitherto unexplored examples of teleological language in the sciences. In explicating these examples, I aim to show (a) that such language is not the sole preserve of the biological sciences, and (b) that not all such talk is reducible to the ascription of functions. In chemistry and biochemistry, scientists explaining molecular rearrangements and protein folding talk informally of molecules rearranging “in order to” maximize stability. Evolutionary biologists, meanwhile, often speak of traits evolving “in order to” optimize some fitness-relevant variable. I argue that in all three contexts such locutions are best interpreted as shorthands for more detailed explanations which, were we to spell them out in full, would show that the relevant process would robustly converge towards the same end-point despite variation in initial conditions. This suggests that, in biology, such talk presupposes a substantial form of adaptationism. The upshot is that such shorthands may be more applicable in the physical sciences than the biological.

Controlling chaos in dynamical systems

Chaos in control systems and controlling chaos in dynamical systems have both attracted increasing attention in recent years. A chaotic system has complex dynamical behaviors that possess some special features, such as being extremely sensitive to tiny variations of initial conditions, having bounded trajectories in the phase space. Controlling chaos has focused on the nonlinear systems such as a Lu’s system.

Growth in post-Soviet Russia: A tale of two transitions

In the early stages of post-Soviet Russia's economic transition, small-scale entrepreneurial activity appeared to be a strong engine of growth. Moreover, striking regional variations in initial conditions and adopted policy reforms appeared useful in accounting statistically for observed regional variations in entrepreneurial activity. Here, we investigate whether these relationships have persisted as Russia's transition has continued to evolve, and find that they have not. We then document that the emergence of bank-issued credit, virtually non-existent outside of Moscow prior to 2000, has been an important engine of growth since 2000. Thus to date, Russia's post-Soviet development appears as a tale of two distinct transition paths.

MODAL DYNAMICS APPROXIMATIONS OF CANTILEVERS UNDER PARTIAL FOLLOWER LOAD

Presented herein is a modified Galerkin discretization procedure for determining the qualitative dynamic behavior of elastic cantilevers with internal damping under partial follower step loading at their tips. For this strong nonlinear nonconservative system, the scheme proposed makes use of basic functions that are a product of nonlinear corrections of approximate linear shape functions. These corrected modes are computed in a way that all the nonlinear nonhomogeneous boundary conditions of the actual problem are satisfied throughout the motion. Numerical results obtained using a two-mode approach are found to be in very good qualitative agreement with the finite element results presented in the literature, not only in the vicinity of the critical states, but also in remote unstable domains. The effect of variation of initial conditions is also investigated and the advantages of the proposed procedure compared with conventional ones are discussed. Further research is required for establishing its capabilities and the range of its applicability for a broader class of nonconservative dynamic problems.

Dependence of star formation on initial conditions and molecular cloud structure

AbstractI review what has been learnt so far regarding the origin of stellar properties from numerical simulations of the formation of groups and clusters of stars. In agreement with observations, stellar properties are found to be relatively robust to variations of initial conditions in terms of molecular cloud structure and kinetics, as long as extreme initial conditions (e.g. strong central condensation, weak or no turbulence) and small-scale driving are avoided, but properties may differ between bound and unbound clouds. Radiative feedback appears crucial for setting stellar masses, even for low-mass stars, while magnetic fields can provide low star formation rates.

Combining climate model output via model correlations

In climate science, collections of climate model output, usually referred to as ensembles, are commonly used devices to study uncertainty in climate model experiments. The ensemble members may reflect variation in initial conditions, different physics implementations, or even entirely different climate models. However, there is a need to deliver a unified product based on the ensemble members that reflects the information contained in whole of the ensemble. We propose a technique for creating linear combinations of ensemble members where the weights are constructed from estimates of variation and correlation both within and between ensemble members. At the heart of this approach is a Bayesian hierarchical model that allows for estimation of the correlation between ensemble members as well as the study of the impact of uncertainty in the parameter estimates of the hierarchical model on the weights. The approach is demonstrated on an ensemble of regional climate model (RCM) output.

Use of multi-model ensembles from global climate models for assessment of climate change impacts

Multi-model ensembles of climate predictions constructed by running several global climate models for a common set of experiments are available for impact assessment of climate change. Multi-model ensembles emphasize the uncertainty in climate predictions resulting from structural differences in the global climate models as well as uncertainty due to variations in initial conditions or model parameterisations. This paper describes a methodology of using multi-model ensembles from global climate models for impact assessments which require local-scale climate scenarios. The approach is based on the use of a weather generator capable of generating the localscale daily climate scenarios used as an input by many process-based impact models. A new version of the LARS-WG weather generator, described in the paper, incorporates climate predictions from 15 climate models from the multi-model ensemble used in the IPCC Fourth Assessment Report (AR4). The use of the AR4 multi-model ensemble allows assessment of the range of uncertainty in the impacts of climate change resulting from the uncertainty in predications of climate. As an example, the impact of climate change on the probability of heat stress during flowering of wheat, which can result in significant yield losses, was assessed using local-scale climate scenarios in conjunction with a wheat simulation model at 4 European locations. The exploitation of much larger perturbed physics ensembles is also discussed.

Increasing model structural complexity inhibits the growth of initial condition errors

One source of inaccuracy in non-linear deterministic ecological models is the growth of initial condition errors. A size-resolved pelagic ecosystem model is used to investigate the effects of changing model structural complexity on the growth rate of initial condition errors. Structural complexity is altered by (1) changing the number of biological size-classes; and (2) changing prey size-ranges which changes the number of linkages for the same number of size-classes. Ensembles of model runs with tiny variations in initial conditions are undertaken and member divergence used to estimate ensemble spread (a measure of the growth of initial condition errors). Increasing prey ranges and therefore the number of linkages greatly reduced the rate of growth of initial condition errors, but ecosystem behaviour is also altered, restricting the generality of the result. At more than 123 size-classes, increasing the number of size-classes while not changing either the model equations or parameters does not alter ecosystem behaviour for over 200 days. In this case, increasing structural complexity through increasing the number of size-classes did not alter the growth of initial condition errors for the first 30 days of the simulations, but afterwards reduced error growth. There are many advantages of parsimonious ecological models with small numbers of classes and linkages, but they are more likely to suffer from the growth of initial condition errors than structurally complex models.

Stability Analysis and Investigation of a Magnetoelastic Beam Subjected to Axial Compressive Load and Transverse Magnetic Field

The interactive behaviors between transverse magnetic fields and axial loads of a magnetoelastic (ME) beam subjected to general boundary conditions are investigated. In particular, the instability criterion for the magneto-mechanical buckling problem is intricately discussed based on the structure characteristics and the initial conditions. The equation of motion for the proposed physical model is introduced according to the Hamilton's principle, and the stability criterion is obtained by using the method of multiple scales implemented on both spatial and time domains. Eventually a so-called Schrodinger equation with cubic nonlinearity (NLS) can be generated by suitably changing the variables; as a result, the stable criterion for the magnetoelastic beam can be acquired after dissecting the nonlinear Schrodinger equation and requiring the imaginary part of the time domain solution to be vanished. Stability criterion curve for the dispersion equation of the ME beam is firstly depicted in order to reveal the magnificent influence of the structure characteristic itself, followed by the instability constraint due to the variation of initial conditions and the observation locations. The results indicate that the prior one actually denotes a parabola, whereas the latter one is sometimes a diamond-like or ellipse-like region spotting along the prior one.

On estimation of earthquake magnitude in Earthquake EarlyWarning systems

Abstract Determination of earthquake magnitude from the initial P-wave portion of ground motion is the key problems for Earthquake Early Warning (EEW). We analyzed performance of scaling relations between one of the frequently used early warning parameters, so-called characteristic period τC and moment magnitude M W in respect of (a) characteristics of datasets accumulated in various regions (earthquake depth, characteristics of network) and (b) variation of initial conditions applying for determination of the parameter τC (length of Pwave window, number of used stations). The used data contain strong-motion records from 110 earthquakes (moment magnitude range 4.4–7.6) occurred in Japan and Taiwan. We show that, although the standard error of regression τC = f (M W) in general becomes smaller with the increase of the length of P-wave window (PL) and the number (N) of averaged observations (stations), the uncertainty in estimation of MW given observed tC does not decrease further for PL > 3–4 seconds and N > 3–4 stations. The information about earthquake depth also plays an important role in reducing the uncertainty of magnitude estimations. The table of confidence limits for estimations of the moment magnitude MW based on observations of the characteristic period τC is provided.

Growth in Post-Soviet Russia: A Tale of Two Transitions

In the early stages of post-Soviet Russia's economic transition, small-scale entrepreneurial activity appeared to be a strong engine of growth. Moreover, striking regional variations in initial conditions and adopted policy reforms appeared useful in accounting statistically for observed regional variations in entrepreneurial activity. Here, we investigate whether these relationships have persisted as Russia's transition has continued to evolve, and find that they have not. We then document that the emergence of bank-issued credit, virtually non-existent outside of Moscow prior to 2000, has been an important engine of growth since 2000. Thus to date, Russia's post-Soviet development appears as a tale of two distinct transition paths.

Investigation of electroconvective flow regimes in liquid dielectrics possessing unipolar injection conductivity

Isothermal electroconvection in a dielectric liquid arising in a plane-parallel electrode system due to unipolar injection of charges from the cathode is considered. Stationary flow regimes formed after the crisis of the equilibrium layer stability loss are numerically modeled. Variations of initial conditions yield two flow regimes providing comparatively higher volume charge densities in the first case near the cathode and in the second case near the anode.

Effects of initial conditions in decaying turbulence generated by passive grids

The effects of initial conditions on grid turbulence are investigated for low to moderate Reynolds numbers. Four grid geometries are used to yield variations in initial conditions and a secondary contraction is introduced to improve the isotropy of the turbulence. The hot-wire measurements, believed to be the most detailed to date for this flow, indicate that initial conditions have a persistent impact on the large-scale organization of the flow over the length of the tunnel. The power-law coefficients, determined via an improved method, also depend on the initial conditions. For example, the power-law exponentmis affected by the various levels of large-scale organization and anisotropy generated by the different grids and the shape of the energy spectrum at low wavenumbers. However, the results show that these effects are primarily related to deviations between the turbulence produced in the wind tunnel and true decaying homogenous isotropic turbulence (HIT). Indeed, when isotropy is improved and the intensity of the large-scale periodicity, which is primarily associated with round-rod grids, is decreased, the importance of initial conditions on both the character of the turbulence andmis diminished. However, even in the case where the turbulence is nearly perfectly isotropic,mis not equal to −1, nor does it show an asymptotic trend inxtowards this value, as suggested by recent analysis. Furthermore, the evolution of the second- and third-order velocity structure functions satisfies equilibrium similarity only approximately.

Chaos: A Very Short Introduction

This book is a new volume of a series designed to introduce the curious reader to anything from ancient Egypt and Indian philosophy to conceptual art and cosmology. Very handy in pocket size, Chaos promises an introduction to fundamental concepts of nonlinear science by using mathematics that is `no more complicated than X=2.' Anyone who ever tried to give a popular science account of research knows that this is a more challenging task than writing an ordinary research article. Lenny Smith brilliantly succeeds to explain in words, in pictures and by using intuitive models the essence of mathematical dynamical systems theory and time series analysis as it applies to the modern world. In a more technical part he introduces the basic terms of nonlinear theory by means of simple mappings. He masterly embeds this analysis into the social, historical and cultural context by using numerous examples, from poems and paintings over chess and rabbits to Olbers' paradox, card games and `phynance'. Fundamental problems of the modelling of nonlinear systems like the weather, sun spots or golf balls falling through an array of nails are discussed from the point of view of mathematics, physics and statistics by touching upon philosophical issues. At variance with Laplace's demon, Smith's 21st century demon makes `real world' observations only with limited precision. This poses a severe problem to predictions derived from complex chaotic models, where small variations of initial conditions typically yield totally different outcomes. As Smith argues, this difficulty has direct implications on decision-making in everyday modern life. However, it also asks for an inherently probabilistic theory, which somewhat reminds us of what we are used to in the microworld. There is little to criticise in this nice little book except that some figures are of poor quality thus not really reflecting the beauty of fractals and other wonderful objects in this field. I feel that occasionally the book is also getting a bit too intricate for the complete layman, and experts may not agree on all details of the more conceptual discussions. Altogether I thoroughly enjoyed reading this book. It was a happy companion while travelling and a nice bedtime literature. It is furthermore an excellent reminder of the `big picture' underlying nonlinear science as it applies to the real world. I will gladly recommend this book as background literature for students in my introductory course on dynamical systems. However, the book will be of interest to anyone who is looking for a very short account on fundamental problems and principles in modern nonlinear science.

Satellite formation flying using along-track thrust

The formation flying of satellites using thrust in along-track direction is explored in the present paper. The system comprising of a leader and a follower satellite is considered and a linear controller is developed to achieve a desired formation. The performance of the proposed controller is tested using numerical simulation of the governing nonlinear system equations of motion considering several factors including variations in initial conditions, formation size, presence of disturbance forces, and orbital eccentricity of the leader satellite. It is found that the controller using only along-track thrust can provide bounded relative position errors with the maximum control acceleration, in m / s 2 , of 0.0718 times the product of square of the orbital rate of the leader satellite and the along-track error. The proposed controller was successful in establishing circular as well as projected circular formations of leader and follower satellites within relative position errors of ± 10 m .

Trends in the Globular Cluster Luminosity Function of Early-Type Galaxies

We present results from a study of the globular cluster luminosity function (GCLF) in a sample of 89 early-type galaxies observed as part of the ACS Virgo Cluster Survey. Using a Gaussian parametrization of the GCLF, we find a highly significant correlation between the GCLF dispersion, sigma, and the galaxy luminosity, M_B, in the sense that the GC systems in fainter galaxies have narrower luminosity functions. The GCLF dispersions in the Milky Way and M31 are fully consistent with this trend, implying that the correlation between sigma and galaxy luminosity is more fundamental than older suggestions that GCLF shape is a function of galaxy Hubble type. We show that the sigma - M_B relation results from a bonafide narrowing of the distribution of (logarithmic) cluster masses in fainter galaxies. We further show that this behavior is mirrored by a steepening of the GC mass function for relatively high masses, M >~ 3 x 10^5 M_sun, a mass regime in which the shape of the GCLF is not strongly affected by dynamical evolution over a Hubble time. We argue that this trend arises from variations in initial conditions and requires explanation by theories of cluster formation. Finally, we confirm that in bright galaxies, the GCLF "turns over" at the canonical mass scale of M_TO ~ 2 x 10^5 M_sun. However, we find that M_TO scatters to lower values (~1-2 x 10^5 M_sun) in galaxies fainter than M_B >~ -18.5, an important consideration if the GCLF is to be used as a distance indicator for dwarf ellipticals.

Entanglement, fidelity and quantum chaos in cavity QED

Nonlinear dynamics in the fundamental interaction between a two-level atom with recoil and a quantized radiation field in a high-quality microcavity is studied. We consider the strongly coupled atom–field system as a quantum–classical hybrid with dynamically coupled quantum and classical degrees of freedom. We show that, even in the absence of any other interaction with environment, the coupling of quantum and classical degrees of freedom provides the emergence of classical dynamical chaos from quantum electrodynamics. Chaos manifests itself in the atomic external degree of freedom as a random walking of an atom inside a cavity with prominent fractal-like behavior and in the quantum atom–field degrees of freedom as a sensitive dependence of atomic inversion on small variations in initial conditions. It is shown that dependences of variance of quantum entanglement and of the maximum Lyapunov exponent on the detuning of the atom–field resonance correlate strongly. It is shown that the Jaynes–Cummings dynamics can be unstable in the regime of chaotic walking of an atom in the quantized field of a standing wave in the absence of any other interaction with environment. Quantum instability manifests itself in strong variations of quantum purity and entropy and in exponential sensitivity of fidelity of quantum states to small variations in the atom–field detuning. It is quantified in terms of the respective classical maximal Lyapunov exponent that can be estimated in appropriate in–out experiments. This result provides a quantum–classical correspondence in a closed physical system.

Online Soil Parameter Estimation Scheme Based on Newton–Raphson Method for Autonomous Excavation

The automation of the excavation process has a huge potential in many industries that require earth removal. The online estimation of soil parameters is an important requirement for developing an impedance controller architecture for automated excavation strategies. This paper presents a fast and robust technique for the experimental identification of soil. This technique, based on the Newton-Raphson method, estimates the unknown parameters of the soil mechanics equations by minimizing the error between measured failure forces and estimated failure forces. The measured failure forces can be obtained by measuring the forces acting on the bucket during the excavation operation, while the estimated failure forces are obtained by an analytical soil model. The Mohr-Coulomb soil model and the Chen and Liu upper bound soil model are employed in this soil parameter estimation scheme. The proposed estimation method is compared to the parameter space intersection method previously employed for parameter estimation and has been tested using four different types of soils (Ticino, Rained Ticino, Nevada Fine, and Glass Beads). The results show that the proposed technique is in good agreement with the parameter space intersection method, but outperforms the latter in terms of computational execution time. A further disadvantage of the parameter space intersection method is that it relies on the search space to be stored in a tabular form. Hence, if high accuracy is needed, the discretization step of the tabular data has to be very small, leading to high memory requirements. Further, in contrast to the method proposed here, the parameter space intersection method suffers from the curse of dimensionality, which leads to exorbitant memory requirements when searching in high-dimensional parameter spaces. The presented results show that the proposed technique has high robustness to initial condition variations and is very insensitive to perturbations of the input signal. The technique presented in this paper is generic and suitable for a real-time soil parameter estimation scheme.