Nonlinear physics…

The approximate dynamics of many physical phenomena, including turbulence, can be represented by dissipative systems of ordinary differential equations. One often turns to numerical integration to solve them. There is an incompatibility, however, between the answers it can produce (i.e., specific solution trajectories) and the questions one might wish to ask (e.g., what behavior would be typical in the laboratory?) To determine its outcome, numerical integration requires more detailed initial conditions than a laboratory could normally provide. In place of initial conditions, experiments stipulate how tests should be carried out: only under statistically stationary conditions, for example, or only during asymptotic approach to a final state. Stipulations such as these, rather than initial conditions, are what determine outcomes in the laboratory. This theoretical study examines whether the points of view can be reconciled: What is the relationship between one's statistical stipulations for how an experiment should be carried out--stationarity or asymptotic approach--and the expected results? How might those results be determined without invoking initial conditions explicitly? To answer these questions, stationarity and asymptotic approach conditions are analyzed in detail. Each condition is treated as a statistical constraint on the system--a restriction on the probability density of states that might be occupied when measurements take place. For stationarity, this reasoning leads to a singular, invariant probability density which is already familiar from dynamical systems theory. For asymptotic approach, it leads to a new, more regular probability density field. A conjecture regarding what appears to be a limit relationship between the two densities is presented. By making use of the new probability densities, one can derive output statistics directly, avoiding the need to create or manipulate initial data, and thereby avoiding the conceptual incompatibility mentioned above. This approach also provides a clean way to derive reduced-order models, complete with local and global error estimates, as well as a way to compare existing reduced-order models objectively. The new approach is explored in the context of five separate test problems: a trivial one-dimensional linear system, a damped unforced linear oscillator in two dimensions, the isothermal Rayleigh-Plesset equation, Lorenz's equations, and the Stokes limit of Burgers' equation in one space dimension. In each case, various output statistics are deduced without recourse to initial conditions. Further, reduced-order models are constructed for asymptotic approach of the damped unforced linear oscillator, the isothermal Rayleigh-Plesset system, and Lorenz's equations, and for stationarity of Lorenz's equations.

There Can Be Only One: Reversible Cellular Automata and the Conservation of Genki

A method is presented for the solution of ordinary linear differential equations with constant coefficients. This method stays wholly in the time domain and obtains a general solution in a compact form including the effects of initial conditions. The differential equation as treated involves a driving function made up of the sum of terms involving the system input and its derivatives. The general solution is also given in terms of weighting functions operating on the input (in the driving function) and including outside the weighted integral, terms involving initial conditions of the input as well as the dependent variable.

The response at time t (R(t)) of a (causal linear time invariant) system to an input A(t) is represented by: [equation: see text] where K(t) is called the unit impulse response function of the system, and the integration on the right side of the equation (above) is called the convolution (from the latin cum volvere: to interwine) of A(t) and K(t). The system described by this equation is at zero (initial conditions) when t = 0. Although it does not even begin to describe the incredible variety of possible responses of biological systems to inputs, this representation has large applicability in biology. One of the most frequently used applications is known as deconvolution: to deinterwine R(t) given a known K(t) (or A(t)) and observations of R(t), to obtain A(t) (or K(t)). In this paper attention is focused on a greater variety of aspects associated with the use of linear systems to describe biological systems. In particular I define causal linear time-invariant systems and their properties and review the most important classes of methods to solve the deconvolution problem, address. The problem of model selection, the problem of obtaining statistics and in particular confidence bands for the estimated A(t) (and K(t)), and the problem of deconvolution in a population context is also addressed, and so is the application of linear system analysis to determine fraction of input absorbed (bioavailability). A general model to do so in a multiinput-site linear system is presented. Finally the application of linear system analysis to control a biological system, and in particular to target a desired response level, is described, and a general method to do so is presented. Applications to simulated, endocrinology, and pharmacokinetics data are reported.

On the Proper Orthogonal Modes and Normal Modes of Continuous Vibration Systems

Submerged breakwaters are a very common structural solution employed for shore erosion control. Predicting wave climate leeward the barriers is indeed the main task to the functional design of such structures; this because it rules the distribution of wave thrusts in the protected area that, in turn, governs hydrodynamic and morphodynamic processes. In this regard it should be noticed that the interaction between waves and submerged breakwaters is a strongly non-linear process. In fact, when water waves come across the structure, multiple-frequency harmonic wave components arise, generating a transmitted wave field very different from the incident one. The super-harmonics consist of phase-locked waves to the fundamental one plus free waves that travels with their own phase speed in the shadow zone. These components interact leading all kinematic and dynamic quantities to spatially fluctuate behind the structure. The aim of this work is to investigate the main characteristics of the non-linear interaction between waves and submerged barriers for conditions of both breaking and no-breaking waves over the structure crown. For this purpose a long series of laboratory experiments have been carried out at the small scale wave flume of the Hydraulic, Geotechnical and Environmental Engineering Department (DIGA) of University of Naples “Federico II”. Models of rubble-mound submerged breakwaters with different geometries and porosity have been employed under the action of regular waves with different height and period. The free surface elevation process has been acquired at several positions leeward of structure models. Experimental results clearly show that the wave shape in the sheltered area is non-permanent leading the profile asymmetry coefficients spatially fluctuate as well as the twice frequency Fourier modes. This behavior, due to the interaction between phase-locked waves and free waves, leads the variance profile (i.e. the spectrum area) to fluctuate in the same way making the definition itself of transmitted wave height quite uncertain. As a further result it has been noticed that higher harmonics transmitted rate reaches a maximum value when waves start breaking over the structure and then, with increasing of dissipation process, to be reduced lying in a saturation curve.

In a recent paper Carrier & Greenspan (1958) showed that, within the framework of the non-linear shallow-water theory, there exist waves which do not break as they climb a sloping beach. The formation of a shock or bore is dependent on a variety of factors (wave shape, particle velocity, etc.) and, as yet, no general criteria for breaking have been found. In this paper, we consider waves which propagate shoreward into quiescent water; it is shown that any compressive wave (a wave of positive amplitude) which has a non-zero slope at the wave-front eventually breaks before reaching the coastline. In fact, an explicit relation is obtained between the initial conditions and the position where breaking occurs.

1 Introduction The question raised in this article is about how to theoretically and methodologically apprehend public policy controversies (1). More precisely, the role of ideas during public policy controversies between different policy coalitions is put into perspective. The starting point is the will to analytically construct ideas as dependent variables. Public policy scholars have underlined the confusion that lies at the heart of public policy analysis, where the status of ideas as causes or as variables is often blurred (Hassenteufel/Smith 2002: 60). When ideas are considered as independent variables, the analysis is likely to be more descriptive than explicative. In order to construct ideas as dependent variables, the focus has to be on the progressive formation of belief through the coalitions' adversarial activities. Hence, what will be questioned here is the process by which individuals do engage--or not-in public policy debate regarding harm reduction. We will not consider actors' engagement in a cause as unilaterally deriving from their beliefs. This line of questioning is closely linked to the ontological and epistemological choices. We assume that a processualist ontological stance is well-fitted to avoid the tautology induced by considering ideas as deja-la. By viewing militant commitment as a social and dynamic activity, a processualist focus highlights how ideas occur in the course of action (Fillieule 2001: 199-200). From this point of view, mobilized groups are by no means seen as preexisting entities driven by fixed ideas (Offerle 1994). This is particularly salient in the case of highly emotional controversies, where actors' ideas are often considered as if they had always existed. Furthermore, actors engaged in the same struggle do not necessarily share homogenous beliefs. Analyzing the heterogeneity of investments enables to understand the dynamics of the collective action (Mathieu 2004: 19). Similarly, the focus on the process of idea formation during policy controversies provides an opportunity to attain a deep understanding of the dynamics of the controversy. Analyzing ideas as dependent variables has important methodological repercussions. It shifts the location to where the explicative factors -and hence the data- are searched. The present reflection on public policy controversies is based on a case study research on harm reduction policies in matters of drug addiction in two Swiss cantons, Vaud and Geneva. The theoretical starting point of this study is the Advocacy Coalition Framework (ACF). Daniel Kubler (2000) has convincingly illustrated that this particular subject is well captured with the help of ACF's theoretical framework. The formulation of harm reduction policies are regularly the scene of sharp confrontations between policy coalitions, and ACF's concepts are particularly well-suited to the analysis of the Swiss multilevel politico-administrative system. Meanwhile, the benefits of using concepts from the social movement theory (Kubler 2001: 623) or the political sociology (Hassenteufel/Smith 2002: 63) in the public policy analysis has been asserted. The paper is structured as follows: In the first section, we review the contributions of the ACF to public policy analysis, and point out two theoretical points that appeared to be worth specification in the light of our case studies. These two points, already identified in the literature, concern the study of the long-term coalitions' birth and structuring process (Kubler 2001: 623; Schlager 1995), and the analysis of the specific arenas where the controversy takes place (Hassenteufel/Smith 2002:70-71; Muller 2006: 52). The second section is a theoretical discussion where we detail the concepts drawn from political sociology we used in our study. In the third section, we discuss the methodological operationalization of these concepts. We finally turn to the case studies on harm reduction policies. 2 Advocacy Coalition Framework By founding the Advocacy Coalition Framework at the end of the 1980s, Paul A. …

A numerical model is proposed for the flow in a deep turbulent boundary layer over water waves. The momentum equations are closed by the use of an isotropic eddy viscosity and the turbulent energy equation. For small amplitudes the results are similar to those of Townsend's (1972) linear model, but nonlinear effects become important as the ratio of wave height to wavelength increases. With uniform surface roughness zo, the predicted fractional rate of energy input per radian advance in phase, ζ, decreases slightly with increasing amplitude and is of the same order of magnitude as in Miles’ (1957, 1959) and Townsend's linear theories. If zo is allowed to vary with position along the wave, however, the fractional rate of energy input can be significantly increased for small amplitude waves. If the variation in zo is half the mean value and the maximum wave slope zak is 0.01, we find ζ ≈ 60 (ρair/ρwater) (uo/c)2, where uo is the friction velocity and c the wave phase speed. Comparison is also made with recent laboratory and field data.

There is no doubt that chaotic sequences attracted much attention in recent decades due to its distinctive properties. These distinct properties give the availability for the chaotic sequences to be applied in different fields. One of the fields that exploit the chaos features is the encryption field, in which chaotic sequences can be used to encrypt texts or images. The randomization attitude of the chaotic sequences can be affected by the initial condition, or by the code length. As these parameters optimized, the performance of the chaotic sequences enhanced and improved from many points of view. This paper presents study, analysis and evaluation for different chaotic sequences from the security viewpoint. The study and analysis include the standard NIST statistical test suite for random and pseudo random number generators for cryptography applications. The presented sequences are the traditional and enhanced version of chaotic sequences based on the logistic map. The new contribution of the paper is to perform the statistical tests over a range of the code lengths, and initial conditions in order to study the effect of each parameter, then optimize its values. Moreover, the proposed sequences are evaluated and applied to image encryption to verify its robustness.

This chapter treats the theory of ordinary differential equations, both linear and nonlinear.Sections 1–4 establish existence and uniqueness theorems for ordinary differential equations. The first section gives some examples of first-order equations, mostly nonlinear, to illustrate certain kinds of behavior of solutions. The second section shows, in the presence of continuity for a vectorvalued F satisfying a “Lipschitz condition,” that the first-order system y′ = F(t, y) has a unique local solution satisfying an initial condition y (t0) = y0. Since higher-order equations can always be reduced to first-order systems, these results address existence and uniqueness for nth-order equations as a special case. Section 3 shows that the solutions to a system depend well on the initial condition and on any parameters that are present in F. Section 4 applies these results to existence of integral curves for a vector field and to construction of coordinate systems from families of integral curves.Sections 5–8 concern linear systems. Section 5 shows that local solutions of linear systems may be extended to global solutions and that in the homogeneous case the vector space of global solutions has dimension equal to the size of the system. The method of variation of parameters reduces the solution of any linear system to the solution of a homogeneous linear system. Sections 6–7 identify explicit solutions to nth-order linear equations and first-order linear systems. The “Jordan canonical form” of a square matrix plays a role in the case of a system. Section 8 discusses power-series solutions to second-order homogeneous linear equations whose coefficients are given by convergent power series, as well as solutions that arise in the case of regular singular points. Two kinds of special functions are mentioned that result from this study-Legendre polynomials and Bessel functions.KeywordsOrdinary Differential EquationUniqueness TheoremDiagonal EntryLipschitz ConditionIntegral CurfThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

In this paper, we study the Poincare recurrence phenomenon in linear dynamical systems, that is, dynamical systems whose trajectories return infinitely often to neighborhoods of their initial condition. Specifically, we provide several equivalent notions of Poincare recurrence and review sufficient conditions for nonlinear dynamical systems that ensure that the system exhibits Poincare recurrence. Furthermore, we establish necessary and sufficient conditions for Poincare recurrence in linear dynamical systems. Finally, we show that in the case of linear systems the absence of volume-preservation is equivalent to the absence of Poincare recurrence implying irreversibility of a dynamical system

Анализируется скоростная чувствительность семейства диаграмм деформирования, порождаемых физически линейным определяющим соотношением вязкоупругости Больцмана-Вольтерры с произвольной функцией релаксации в одноосных испытаниях с постоянными скоростями деформации. Выведено общее выражение для показателя скоростной чувствительности (скоростного упрочнения) и аналитически исследованы его общие качественные свойства: зависимость от деформации, скорости деформации и характеристик функции релаксации, диапазон значений, интервалы монотонности и существование точек экстремума, предельные значения при стремлении скорости деформации к нулю или бесконечности, способы определения по диаграммам деформирования или по кривым релаксации. Установлено, что (в рамках линейной теории вязкоупругости) этот показатель зависит не от двух независимых аргументов (деформации и скорости деформации), а только от их отношения, что он выражается через отношение касательного модуля к секущему и может быть вычислен по одной диаграмме деформирования с произвольной скоростью деформации, и что по заданной (или измеренной в испытаниях) функции скоростной чувствительности можно однозначно восстановить функцию релаксации. Доказано, что значения показателя скоростной чувствительности всегда лежат в интервале от нуля до единицы (т.е. линейное определяющее соотношение описывает только псевдопластические среды и не может описывать дилатантные) и могут быть сколь угодно близки к единице (верхней границе для псевдопластических сред), что как функция скорости он не только может монотонно возрастать или убывать, но может иметь точки экстремума, в частности точку максимума (при малообременительных ограничениях на функцию релаксации). Тем самым обнаружена неожиданная способность линейной теории вязкоупругости не только порождать семейство диаграмм деформирования с выраженными участками течения при практически постоянном напряжении, но и качественно описывать «сигмоидальную» форму зависимости напряжения от скорости деформации (в логарифмических осях) и очень высокую скоростную чувствительность, характерные для режима сверхпластического деформирования материалов. Установленные свойства показателя скоростной чувствительности и его характерные особенности проиллюстрированы на примерах классических регулярных, сингулярных и фрактальных моделей вязкоупругости (Максвелла, Фойгта, Кельвина, Зенера, Бюргерса, Скотт-Блэра) и их параллельных соединений.

Previous investigations have studied the robustness of agility metrics to variations in initial conditions and uncertainties in physical characteristics and coefficients. In each of these investigations, the selected agility metrics were evaluated in a Cartesian coordinate system. It is known that values of calculated parameters can be sensitive to the particular form of the equation used, which in turn results from the choice of coordinate system. Although simple and intuitive for quantifying agility, Cartesian coordinates are not necessarily the best coordinate system for studying agility. Best is defined here as the form of agility equation, resulting from choice of coordinate system, which is most robust to variations in initial conditions and uncertainties in physical characteristics and coefficients. This paper seeks to establish the relationship between form of agility equations, choice of coordinate system, and the measured agility of high performance aircraft. Cartesian based agility metrics consisting of time to roll through bank angle metric, average pitch rate metric, and power onset/loss parameter metric, are compared to the Beck metrics which are based in the Frenet coordinate system. Each group of metrics is evaluated with initial condition errors and parametric uncertainties of physical constants using linear error theory to validate the use of linear approximations to propagate the errors, l^. is shown that the Beck metrics are less sensitive to initial condition errors and parametric uncertainties for the longitudinal and axial axes. The time to roll through bank angle metric is less sensitive to these variations than the Beck metrics for evaluating agility in the lateral axis.

: A systematic study of two-dimensional, turbulent, small-deficit wakes investigated the structure and universality of their self-preserving states. Various wake generators, including circular cylinders, a symmetrical airfoil, a flat plate, and an assortment of screens of varying solidity, were studied for a wide range of downstream distances. The drag coefficient of most generators, and therefore the momentum thickness, was tailored to be identical, permitting comparison at identical Reynolds numbers and aspect ratios. The flat plate and airfoil and had a small, trailing-edge flap which could be externally driven to introduce forced sinuous oscillations into the wake. Results indicate that the normalized characteristics velocity and length scales depend on the initial conditions, while the shape of the normalized mean velocity profile is independent of initial conditions or nature of the generator. The normalized distributions of the longitudinal turbulence intensity, however, are dependent on the initial conditions. Linear inviscid stability theory, in which the divergence of the mean flow was taken into account, predicts quite well the amplification and transverse distributions of amplitudes and phases of externally imposed sinuous waves on a fully developed turbulent wake generated by a flat plate. There is a strong indication that the large structures observed in the unforced wake are related to the two-dimensional instability modes and therefore can be modelled by linear stability theory. Interaction of the two possible modes of instability may be responsible for the vortex street-type pattern observed visually in the small-deficit, turbulent wake.