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The majority of physics problems are impossible to solve by analytic means. Various strategies have been developed to cope with the highly non-linear nature of many of these problems. Dimensional analysis provides a powerful tool for addressing many complex problems, suggesting the form the solutions must have. Examples include the non-linear pendulum, explosions, flow at high Reynolds number and the law of corresponding states. The study of chaotic phenomena became feasible with the development of high speed computers and revealed regularities despite the apparent unpredictability of the systems. Scaling laws for extremely complex and non-linear problems lead to the concept of self-organised criticality, illustrated by the model computations for rice and sand piles.
This is the first of five chapters on ecological modelling and presents basic homogenous (absence of spatial structure and variability among individuals) and deterministic (absence of stochasticity and randomness) population models. The first model describes unlimited exponential growth, which is followed by the introduction of intra-specific competition and density dependent growth. Two types of dynamics are considered: continuous in time and discrete in time. It is demonstrated that the time-discrete formulation can lead to chaotic population dynamics which in ecological models can be caused by scramble competiton where individuals share limited resources rather evenly so that all individuals suffer from the lack of resources. Opposed to this is contest competition where the share of the resources is uneven, so a few winners reproduce and/or survive, and the emerging population dynamics are not chaotic.
In this chapter, the author explores the book of Job for a perspective on the modern Darwinian Problem of evil. He concurs with recent scholars who reject the commonplace reading of Job, i.e., that God refuses to answer Job’s question: how can his suffering be just? He concurs with Carol Newsom that in the divine speeches at the end, God answers Job indirectly in the form of carefully crafted symbolic poetics, the rhetorical structure and imagery of which radically reconstruct Deuteronomic tradition on God and suffering. The author proposes that the treatment of God and wild animals in Job makes the Darwinian configuration of animal suffering more plausible on canonical theism than commonly supposed. He further proposes that Job provides grounds for belief that the Jewish/Christian God will defeat evils for animals and include them in the messianic eschatological realm. Job offers a religiously framed aesthetic perspective on Darwinian evil that helps us to recover the “theistic sight” in nature that Darwinian discoveries have obscured.
The general conclusion wraps up the main findings of the book highlighting its innovative approach to considering how Madagascar functions in order to understand the country’s trajectory since independence. It investigates the hybrid nature of contemporary Malagasy society, a mix of external and internal influences in fateful combination (a chimera), despite the conclusion that endogenous dynamics are leading the dance. Drawing on the most recent trends, the chapter elaborates three potential scenarios for the years to come. Chaos is not the least probable. The second is the road to a more positive societal change, by establishing sustainable coalitions among elites and more organised top-down patron-client system. This scenario would establish a stable social order, but would mean the abandonment of the democratic process. The third scenario called democratic consolidation would be achieved by means of an inclusive social, political and economic dynamic. Perhaps the most ambitious, it would build on Madagascar most positive assets: the institutional capacity for regulation demonstrated in the past, the taboo on violence and the democratic aspirations of the population. The only avenue is for Madagascar to consolidate countervailing institutions and develop intermediary bodies, and giving more voice to the people.
On-again/off-again relationships challenge the standard dichotomous definition of relationship stability (i.e., whether the relationship remains intact or dissolves). This chapter reviews the various conceptualizations of stability. Although on-off partners report less relationship stability when using subjective, one-time assessments (e.g., perceived stability, sense of security, or persistence in the relationship), a process-oriented assessment of stability is advocated in which relationship dynamics are measured over time. As argued by chaos theory, fluctuations over time could indicate a stable pattern. For example, research suggests certain fluctuations are associated with greater stability in on-off relationships. A process-oriented perspective could thus provide a more nuanced assessment of relationship stability for both on-off and non-cyclical relationships. Additional considerations for future research are also offered.
This chapter provides an in-depth discussion on the complexities associated with the Ex-Utero Intrapartum Therapy; EXIT procedure. The authors provide a thorough analysis of patient and procedural considerations from the maternal and fetal aspects. The perioperative approach for these procedures is reviewed in detail with respect to fetal and maternal anesthetic goals.
In an acoustic cavity with a heat source, such as a flame in a gas turbine, the thermal energy of the heat source can be converted into acoustic energy, which may generate a loud oscillation. If uncontrolled, these nonlinear acoustic oscillations, also known as thermoacoustic instabilities, can cause large vibrations up to structural failure. Numerical and experimental studies showed that thermoacoustic oscillations can be chaotic. It is not yet known, however, how to minimise such chaotic oscillations. We propose a strategy to analyse and minimise chaotic acoustic oscillations, for which traditional stability and sensitivity methods break down. We investigate the acoustics of a nonlinear heat source in an acoustic resonator. First, we propose covariant Lyapunov analysis as a tool to calculate the stability of chaotic acoustics, making connections with eigenvalue and Floquet analyses. We show that covariant Lyapunov analysis is the most general flow stability tool. Second, covariant Lyapunov vector analysis is applied to a chaotic system. The time-averaged acoustic energy is investigated by varying the heat-source parameters. Thermoacoustic systems can display both hyperbolic and non-hyperbolic chaos, as well as discontinuities in the time-averaged acoustic energy. Third, we embed sensitivities of the time-averaged acoustic energy in an optimisation routine. This procedure achieves a significant reduction in acoustic energy and identifies the bifurcations to chaos. The analysis and methods proposed enable the reduction of chaotic oscillations in thermoacoustic systems by optimal passive control. The techniques presented can be used in other unsteady fluid-dynamics problems with virtually no modification.
Although the whip is a common tool that has been used for thousands of years, there have been very few studies on its dynamic behavior. With the advance of modern technology, designing and building softbody robot whips has become feasible. This paper presents a study on the modeling and experimental testing of a robot whip. The robot whip is modeled using a Pseudo-Rigid-Body Model (PRBM). The PRBM consists of a number of pseudo-rigid-links and pseudo-revolute-joints just like a multi-linkage pendulum. Because of its large number of degrees of freedom (DOF) and inherited underactuation, the robot whip exhibits prominent transient chaotic behavior. In particular, depending on the initial driving force, the chaos may start sooner or later, but will die down because of the gravity and air damping. The dynamic model is validated by experiments. It is interesting to note that with the same amount of force, the robot whip can generate a velocity more than 3 times and an acceleration up to 43 times faster than that of its rigid counterpart. This gives the robot whip some potential applications, such as whipping, wrapping and grabbing. This study also helps to develop other soft-body robots that involve nonlinear dynamics.
To improve global human capital, an understanding of the interplay of endowment across the full range of socioeconomic status (SES) is needed. Relevant data, however, are absent in the nations with the most abject poverty (Tucker-Drob & Bates, 2016), where the lowest heritability and strong effects of SES are predicted. Here we report the first study of biopsychosocial gene–environment interaction in extreme poverty. In a sub-Saharan sample of early teenage twins (N = 3192), we observed substantial (~30–40%) genetic influence on cognitive abilities. Surprisingly, shared environmental influences were similar to those found in adolescents growing in Western affluent countries (25–28%). G × SES moderation was estimated at aˋ = .06 (p = .355). Family chaos did not moderate genetic effects but did moderate shared environment influence. Heritability of cognitive abilities in extreme poverty appears comparable to Western data. Reduced family chaos may be a modifiable factor promoting cognitive development.
In this paper we introduce the multivariate Brownian semistationary (BSS) process and study the joint asymptotic behaviour of its realised covariation using in-fill asymptotics. First, we present a central limit theorem for general multivariate Gaussian processes with stationary increments, which are not necessarily semimartingales. Then, we show weak laws of large numbers, central limit theorems, and feasible results for BSS processes. An explicit example based on the so-called gamma kernels is also provided.
Recently, there has been increased focus on sub-threshold stages of mental disorders, with attempts to model and predict progression to full-threshold disorder. Given this considerable research attention and clinical significance, it is timely to analyse the assumptions of theoretical models in the field. Research into predicting onset of mental disorder has shown an overreliance on one-off sampling of cross-sectional data (i.e., a "snapshot" of clinical state and other risk markers) and may benefit from taking dynamic changes into account. Cross-disciplinary approaches to complex system structures and changes, such as dynamical systems theory, network theory, instability mechanisms, chaos theory and catastrophe theory, offer potent models that can be applied to emergence (or decline) of psychopathology, including psychosis prediction and transdiagnostic symptom emergence. Staging provides a useful framework to research dynamic prediction in psychiatry. Psychiatric research may benefit from approaching psychopathology as a system rather than a category, identifying dynamics of system change (e.g., abrupt/gradual psychosis onset), factors to which these systems are most sensitive (e.g., interpersonal dynamics, neurochemical change), and individual variability in system architecture and change. The next generation of prediction studies may more accurately model the highly dynamic nature of psychopathology and system change, with treatment implications, such as introducing a means of identifying critical risk periods for mental state deterioration.
A fundamental aim of diagnosis is to guide treatment planning and predict illness course. Yet for too long psychiatric diagnosis, grounded on traditional silo-based approaches, has lacked clinical utility. This chapter explores the purpose of diagnosis and classification as well as the inability to validate diagnosis in psychiatry. It is proposed that new testable models are needed to improve the utility of diagnosis and support more personalised and sequential treatment selection. A number of new approaches have been put forward, including hierarchical and network-based methods, however at present, these offer limited value in guiding treatment selection. Clinical staging offers a viable solution. Clinical staging in psychiatry recognises that mental disorders are not static and discretely defined entities, but rather they are syndromes that overlap and develop in stages. The model ensures that interventions are proportional to both need and the risk of progressing to later stages and more established syndromes, which are likely to be comorbid, persistent, recurrent and disabling. Ultimately, it advocates a transdiagnostic approach to intervention, with a pre-emptive focus, that is based on risk-benefit considerations and patient needs. Clinical staging also provides a framework in which underlying biological mechanisms can be linked to each stage, to build a personalised and pre-emptive psychiatry.
There is much research on the dynamical complexity on irregular sets and level sets of ergodic average from the perspective of density in base space, the Hausdorff dimension, Lebesgue positive measure, positive or full topological entropy (and topological pressure), etc. However, this is not the case from the viewpoint of chaos. There are many results on the relationship of positive topological entropy and various chaos. However, positive topological entropy does not imply a strong version of chaos, called DC1. Therefore, it is non-trivial to study DC1 on irregular sets and level sets. In this paper, we will show that, for dynamical systems with specification properties, there exist uncountable DC1-scrambled subsets in irregular sets and level sets. Meanwhile, we prove that several recurrent level sets of points with different recurrent frequency have uncountable DC1-scrambled subsets. The major argument in proving the above results is that there exists uncountable DC1-scrambled subsets in saturated sets.
We employ lattice Boltzmann simulation to numerically investigate the two-dimensional incompressible flow inside a right-angled isosceles triangular enclosure driven by the tangential motion of its hypotenuse. While the base flow, directly evolved from creeping flow at vanishing Reynolds number, remains stationary and stable for flow regimes beyond
, chaotic motion is nevertheless observed from as low as
. Chaotic dynamics is shown to arise from the destabilisation, following a variant of the classic Ruelle–Takens route, of a secondary solution branch that emerges at a relatively low
and appears to bear no connection to the base state. We analyse the bifurcation sequence that takes the flow from steady to periodic and then quasi-periodic and show that the invariant torus is finally destroyed in a period-doubling cascade of a phase-locked limit cycle. As a result, a strange attractor arises that induces chaotic dynamics.
We examine the motion in a shear flow at zero Reynolds number of particles with two planes of symmetry. We show that in most cases the rotational motion is qualitatively similar to that of a non-axisymmetric ellipsoid, and characterised by a combination of chaotic and quasiperiodic orbits. We use Kolmogorov–Arnold–Moser (KAM) theory and related ideas in dynamical systems to elucidate the underlying mathematical structure of the motion and thence to explain why such a large class of particles all rotate in essentially the same manner. Numerical simulations are presented for curved spheroids of varying centreline curvature, which are found to drift persistently across the streamlines of the flow for certain initial orientations. We explain the origin of this migration as the result of a lack of symmetries of the particle’s orientation orbit.
We construct a multiply Xiong chaotic set with full Hausdorff dimension everywhere that is contained in some multiply proximal cell for the full shift over finite symbols and the Gauss system, respectively.
Turbulent flows respond to bounding walls with a predominant spanwise heterogeneity – that is, a heterogeneity parallel to the prevailing transport direction – with formation of Reynolds-averaged turbulent secondary flows. Prior experimental and numerical work has determined that these secondary rolls occur in a variety of arrangements, contingent only upon the existence of a spanwise heterogeneity (i.e. from complex, multiscale roughness with a predominant spanwise heterogeneity, to canonical step changes, to different roughness elements). These secondary rolls are known to be a manifestation of Prandtl’s secondary flow of the second kind: driven and sustained by the existence of spatial heterogeneities in the Reynolds (turbulent) stresses, all of which vanish in the absence of spanwise heterogeneity. Herein, we show results from a suite of large-eddy simulations and complementary experimental measurements of flow over spanwise-heterogeneous surfaces. Although the resultant secondary cell location is clearly correlated with the surface characteristics, which ultimately dictates the Reynolds-averaged flow patterns, we show the potential for instantaneous sign reversals in the rotational sense of the secondary cells. This is accomplished with probability density functions and conditional sampling. In order to address this further, a base flow representing the streamwise rolls is introduced. The base flow intensity – based on a leading-order Galerkin projection – is allowed to vary in time through the introduction of time-dependent parameters. Upon substitution of the base flow into the streamwise momentum and streamwise vorticity transport equations, and via use of a vortex forcing model, we are able to assess the phase-space evolution (orbit) of the resulting system of ordinary differential equations. The system resembles the Lorenz system, but the forcing conditions differ intrinsically. Nevertheless, the system reveals that chaotic, non-periodic trajectories are possible for sufficient inertial conditions. Poincaré projection is used to assess the conditions needed for chaos, and to estimate the fractal dimension of the attractor. Its simplicity notwithstanding, the propensity for chaotic, non-periodic trajectories in the base flow model suggests similar dynamics is responsible for the large-scale reversals observed in the numerical and experimental datasets.
Unsteady spatially localized states such as puffs, slugs or spots play an important role in transition to turbulence. In plane Couette flow, steady versions of these states are found on two intertwined solution branches describing homoclinic snaking (Schneider et al., Phys. Rev. Lett., vol. 104, 2010, 104501). These branches can be used to generate a number of spatially localized initial conditions whose transition can be investigated. From the low Reynolds numbers where homoclinic snaking is first observed (
) to transitional ones (
), these spatially localized states traverse various regimes where their relaminarization time and dynamics are affected by the dynamical structure of phase space. These regimes are reported and characterized in this paper for a
-periodic domain in the streamwise direction as a function of the two remaining variables: the Reynolds number and the width of the localized pattern. Close to the snaking, localized states are attracted by spatially localized periodic orbits before relaminarizing. At larger values of the Reynolds number, the flow enters a chaotic transient of variable duration before relaminarizing. Very long chaotic transients (
) can be observed without difficulty for relatively low values of the Reynolds number (
We describe how to approximate fractal transformations generated by a one-parameter family of dynamical systems
constructed from a pair of monotone increasing diffeomorphisms
. An algorithm is provided for determining the unique parameter value such that the closure of the symbolic attractor
is symmetrical. Several examples are given, one in which the
are affine and two in which the
are nonlinear. Applications to digital imaging are also discussed.
In large storage systems, files are often coded across several servers to improve reliability and retrieval speed. We study load balancing under the batch sampling routeing scheme for a network of n servers storing a set of files using the maximum distance separable (MDS) code (cf. Li (2016)). Specifically, each file is stored in equally sized pieces across L servers such that any k pieces can reconstruct the original file. When a request for a file is received, the dispatcher routes the job into the k-shortest queues among the L for which the corresponding server contains a piece of the file being requested. We establish a law of large numbers and a central limit theorem as the system becomes large (i.e. n → ∞), for the setting where all interarrival and service times are exponentially distributed. For the central limit theorem, the limit process take values in ℓ2, the space of square summable sequences. Due to the large size of such systems, a direct analysis of the n-server system is frequently intractable. The law of large numbers and diffusion approximations established in this work provide practical tools with which to perform such analysis. The power-of-d routeing scheme, also known as the supermarket model, is a special case of the model considered here.