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This chapter introduces the reader to some of the main conceptual ideas behind dynamical systems theory from the perspective of an experimentalist. It first considers the qualitative approaches used to study complex problems before discussing dynamical systems and bifurcations. In particular, it examines the use of time series to represent solutions and dynamics in the phase space, phase space respresentations of equilibrium and nonequilibrium steady states, the qualitative analysis of steady states, and some of the mechanics of local stability analysis for an equilibrium using the Lotka–Volterra model for an equilibrium steady state. It also explores the relationship between the type of model dynamics and the geometry of the underlying mathematical functions. Finally, it presents an empirical example from ecology, Hopf bifurcation in an aquatic microcosm, to illustrate the main concepts of dynamical systems theory and shows that the mathematics of dynamical systems underlies the dynamics of real ecological systems.

Dynamical systems thinking can provide metaphors that help ask new questions, generate new experimental paradigms and measures, and lead to new kinds of explanations. This chapter reviews dynamical systems theory (DST) as a set of concepts that formalizes such metaphors and thus becomes a scientific theory of considerable rigor. It considers five concepts: (1) Behavioral patterns resist change; that is, they are stable. This may be mathematically characterized by considering behavioral patterns as the attractor states of a dynamical system. (2) Behavioral change is brought about by a loss of stability. (3) Representations possess stability properties, as well, and can be understood as the attractor states of dynamic fields, that is, of continuous distributions of neuronal activation. (4) Cognitive processes emerge from instabilities of dynamic fields. (5) Learning occurs as changes in behavioral or field dynamics that shift the behavioral and environmental context in which these instabilities occur.

This chapter discusses old-fashioned cognitive development from the point of view of two modern approaches, connectionism and nonlinear dynamical systems theory. The main assertion in both connectionist and dynamic systems approaches is that higher cognitive functioning is largely based on nonsymbolic, graded, and dynamic properties, of which these same approaches provide the best account. The chapter argues that the claim concerning nonsymbolic higher order cognition is overstated and explains its position by focusing on sudden transitions in cognitive development.

This chapter provides a historical introduction to X-ray dynamical diffraction. It starts with an account of Ewald's thesis on the dispersion of light and of the famous experiment of the diffraction of X-rays by crystals by M. Laue, W. Friedrich, and P. Knipping. The successive steps in the development of the theory of X-ray diffraction are then summarized: Laue's and Darwin's geometrical theories; Darwin's, Ewald's, and Laue's dynamical theories; early experimental proofs, the notion of extinction and the mosaic crystal model, observation in the fifties and sixties of the fundamental properties of the X-ray wavefields in crystals (anomalous absorption and the Borrmann effect, double refraction, Pendellösung, bent trajectories in deformed crystals), extension of the dynamical theory to the case of deformed crystals, modern applications for the characterization of crystal defects and X-ray optics for synchrotron radiation.

This chapter provides a brief introduction to concepts from systems biology; tools from differential equations and control theory; and approaches to the modeling, analysis, and design of biomolecular feedback systems. It begins with a discussion of the role of modeling, analysis, and feedback in biological systems. This is followed by a short review of key concepts and tools from control and dynamical systems theory, which is intended to provide insight into the main methodology described in this volume. Finally, this chapter gives another brief introduction—this time to the field of synthetic biology, which is the primary topic of the latter portion of this book.

This chapter spells out the embedded view and the principles of modeling normally associated with it. The embedded approach aims to minimize the amount of computation and representation included in cognitive models, thereby explaining how the cognitive system carries out its cognitive tasks in a timely fashion; the general strategy for doing so employs locally valid heuristics. The chapter then focuses on the role of computation in the embedded view, advancing the following contentions: that the embedded view has been most successful when set within a computational framework, that simplifying computation does not amount to eliminating computation—even where explicit rules are eliminated—and that dynamical systems theory does not model the subtle timing of the organism's interaction with the environment in a way inherently superior to computationalist strategies.

This chapter provides a review of recently-developed Dynamical Mean-Field Theory (DMFT) approaches to the general problem of strongly correlated electronic systems with disorder. The chapter first describes the standard DMFT approach, which is exact in the limit of large coordination, and explain why in its simplest form it cannot capture either Anderson localization or the glassy behavior of electrons. Various extensions of DMFT are then described, including statistical DMFT, typical medium theory, and extended DMFT, methods specifically designed to overcome the limitations of the original formulation. The chapter provides an overview of the results obtained using these approaches, including the formation of electronic Griffiths phases, the self-organized criticality of the Coulomb glass, and the two-fluid behavior near Mott-Anderson transitions. Finally, the chapter outlines research directions that may provide a route to bridge the gap between the DMFT-based theories and the complementary diffusion-mode approaches to the metal-insulator

This chapter explains the most important notions of entropy and clarifies their interrelations. It furthermore explores what notions of probabilities are at work when entropy is defined in terms of probability. Entropies from thermodynamics, information theory, statistical mechanics, dynamical systems theory, and fractal geometry are surveyed. Notions of entropy from statistical mechanics such as varieties of the Boltzmann and Gibbs entropies can be traced back to information-theoretic entropy. As the chapter points out, an analytic connection between fine-grained Boltzmann entropy and thermodynamic entropy can be established for ideal gases. No easy results about probabilities follow, since several interpretations are compatible with each definition of entropy.

This chapter concerns highly deformed crystals where the Eikonal approximation is no longer valid. An expression is given for the limit of validity of this approximation. Takagi's equations are extended so as to apply to highly deformed crystals. Their resolution is the discussed and the principle of their numerical integration in an inverted Borrmann triangle given. The ray concept is generalized to the case of strong deformations by noting that new wavefields are generated in the highly strained regions; this is known as the interbranch scattering effect. The last part of the chapter is devoted to an account of the statistical dynamical theories for highly imperfect crystals, with emphasis on Kato's statistical theories. Examples of experimental test of the dynamical theory are also given.

In this chapter I draw on the conceptual tools of dynamical systems theory to conceptualize emotional episodes as self-organizing patterns of the entire organism. I first overview how dynamical systems concepts have been used in “dynamical affective science” to model emotions. Although existing proposals apply at very different levels of description, they all characterize the organism as complex, self-organizing, open, and plastic, realizing emotional episodes that are softly assembled, context dependent, and highly variable, yet patterned and recurrent. I then elaborate on the implications of this dynamical conceptualization for the debate on the nature of the emotions discussed in the previous chapter. This dynamical conceptualization notably posits neither internal causes of emotional episodes and of their different aspects, nor a distinction between alleged basic and nonbasic emotions; rather, it treats all emotions as complex organismic patterns subject to both evolutionary and developmental pressures. I conclude with a discussion of how dynamical systems concepts also help characterize the relationship between emotions and moods.

This chapter considers unimolecular reactions; photo-induced reactions, i.e. true unimolecular reactions; and reactions initiated by collisional activation, i.e. apparent unimolecular reactions where it is assumed that the time scales for activation and subsequent reaction are well separated. Elements of classical and quantum dynamical descriptions are discussed, including Slater theory and the quantum mechanical description of photo-induced reactions. Statistical theories aiming at the calculation of micro-canonical as well as canonical rate constants are discussed, including a detailed discussion of RRKM theory. It concludes with a discussion of femtochemistry, i.e. the observation and control of chemical dynamics using femtosecond pulses of electromagnetic radiation, focusing on the control of unimolecular reactions via the interaction with coherent light, that is, laser control.

Due to the finite number of dynamic behaviours that can be implemented by regulatory systems, it should be possible to enumerate and classify different developmental mechanisms that can achieve the same biological function. For example, there are only a small number of ways by which small regulatory networks can create a stripe of gene expression in a static or growing tissue. By comparing these different mechanisms, we can discover the design principles of stripe-producing regulatory networks. Such a rigorous and mechanistic classification scheme would constitute the basis for a theory of development that characterizes and explains the regularities and recurring motifs observed in organismal morphology. This tackles a central question in biology, which has fascinated numerous researchers since the rational taxonomists first raised it in the 19th century. This chapter introduces and defines a concept of developmental mechanism suitable for this endeavour, based on the conceptual framework of dynamical systems theory, which characterizes the dynamical repertoire of regulatory networks. Equivalent mechanisms are defined as sharing the same topology of their phase portraits: they have the same number and geometrical arrangement of attracting states, saddle points, and basins of attraction, and undergo structurally stable bifurcations as systems parameters change over time. These abstract concepts and their application are illustrated using specific examples such as simulated stripe-forming networks, vertebrate limb development, and the segmentation gene network in insects. This constitutes a first, tentative step towards a more general geometrical theory of developmental mechanisms and the complex map from genotype to phenotype.

This chapter describes the thermodynamic modeling of large-scale interconnected dynamical systems. Using compartmental dynamical system theory, it develops energy flow models possessing energy conservation and energy equipartition principles for large-scale dynamical systems. It then gives a deterministic definition of entropy for a large-scale dynamical system that is consistent with the classical definition of entropy and shows that it satisfies a Clausius-type inequality leading to the law of nonconservation of entropy. It also introduces the notion of ectropy as a measure of the tendency of a dynamical system to do useful work and grow more organized. It demonstrates how conservation of energy in an isolated thermodynamic large-scale system leads to nonconservation of ectropy and entropy. Finally, the chapter uses the system ectropy as a Lyapunov function candidate to show that the large-scale thermodynamic energy flow model has convergent trajectories to Lyapunov stable equilibria determined by the system initial subsystem energies.

This chapter is a final attempt to think about an epistemology and an aesthetics outside of the constraints of dynamical systems theory. It investigates the photographic image and what has been called cinematographic knowledge, explaining that for cinematographic knowledge, time is an independent variable, a parameter of the spatial manifold. The chapter proposes the concept of the discrete photograph and a new image of philosophy.

Psychopathology might be one domain where we can get some empirical perch on the Systematicity debate. In patients with schizophrenia transformational systematicity and other types of systematicity often break down. Systems neuroscience has provided some reason to believe that the best explanation for this break down is in terms of the degradation of key brain subsymbolic network properties such as small-world graphical structures. It is argued that if such systems neuroscience explanations for failures of systematicity in schizophrenia are robust then this is a victory for network approaches over symbol-and-rule approaches that themselves provide little insight into said failures. Finally, there is speculation that the relevant dynamical and graphical relations in such cases extend beyond the brain to include body and environment.

This introductory chapter provides an overview and a brief history of complexity science, which is the study of complex systems. All living systems and all intelligent systems are complex systems. Complexity science is relatively new but already indispensable. Many of the most important problems in engineering, medicine, and public policy are now addressed with the ideas and methods of complexity science. However, there is no agreement about the definition of 'complexity' or 'complex system', nor even about whether a definition is possible or needed. The conceptual foundations of complexity science are disputed, and there are many and diverging views among scientists about what complexity and complex systems are. Even the status of complexity as a discipline can be questioned given that it potentially covers almost everything. The origins of complexity science lie in cybernetics and systems theory, both of which began in the 1950s. Complexity science is related to dynamical systems theory, which matured in the 1970s, and to the study of cellular automata, which were invented at the end of the 1940s. By then computer science had become established as a new scientific discipline.

This chapter is devoted to optical properties of X-rays. In the first section the early measurements of optical properties of X-rays are discussed briefly (specular reflection, refraction, diffraction by a slit). In the ensuing sections the principle of the Ewald and Laue dynamical theories of X-ray diffraction is presented. The special optical properties of X-ray wavefields are described, such as their direction of propagation inside the crystal, anomalous absorption, and standing waves, as well as their application to the Kossel effect and the location of impurities at crystal surfaces. It is shown how the deviation from Bragg’s law, which is due to the effect of refraction, was measured experimentally, and the principle of the double-crystal spectrometer is given. The discovery of the Compton effect is also related, and its consequences as to the nature of X-rays are discussed.

Ch 8 is an analysis of Ørsted's doctoral thesis, which is a revised version (in Latin) of his review of Hauch's textbook. The ‘architecture’ of the thesis follows Kant's Metaphysical Foundations of Natural Science, which is basically a critical survey of Newton's Philosophiae Naturalis Principia Mathematica. Ørsted, however, adheres even more meticulously to Kant's epistemological scheme, which he calls ‘Ariadne's Clew’, than Kant himself. In conclusion he agrees with Kant that Newton's mechanical concepts, although they are conducive to mathematical treatment if subsumed under the quantitative category as matter, forces per se defy mathematization, because they are immeasurable and uncountable. Consequently, we are left with the paradox that corpuscular theory makes sense in mathematical terms, but is epistemologically flawed, while Kant's (and Ørsted's) alternative, dynamical theory, does make sense epistemologically, but lacks the quantifiable concepts necessary for constituting equations and thus the precondition of becoming scientific.