Tim O’Riordan, Tim Lenton, and Ian Christie
- Published in print:
- 2013
- Published Online:
- January 2014
- ISBN:
- 9780197265536
- eISBN:
- 9780191760327
- Item type:
- chapter
- Publisher:
- British Academy
- DOI:
- 10.5871/bacad/9780197265536.003.0001
- Subject:
- Political Science, Environmental Politics
Tipping points are metaphors of sudden change, of fear, of falling, of foreboding, and of failure. Tipping points are thresholds of tolerance, of bifurcation, and of transformation which are built ...
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Tipping points are metaphors of sudden change, of fear, of falling, of foreboding, and of failure. Tipping points are thresholds of tolerance, of bifurcation, and of transformation which are built into complex systems of transformation. Sudden change can arise from earth system phase changes (for example in the condition of ice, ocean acidity, drying of the tropical forests and the onset of monsoons). But they can also depict rapid shifts in geopolitics, local and regional conflicts, and in economic performance with implications for the well-being of societies all over the globe. The patterns of suddenness and aftermath of physical and socio-economic systems vary greatly. Tipping points can lead to unintended worsening, to induced vulnerabilities, to chaos and confusion in communication, and to the scope for restorative redirection. The scope for benign transformation is an intrinsic aspect of the tipping point metaphor.Less
Tipping points are metaphors of sudden change, of fear, of falling, of foreboding, and of failure. Tipping points are thresholds of tolerance, of bifurcation, and of transformation which are built into complex systems of transformation. Sudden change can arise from earth system phase changes (for example in the condition of ice, ocean acidity, drying of the tropical forests and the onset of monsoons). But they can also depict rapid shifts in geopolitics, local and regional conflicts, and in economic performance with implications for the well-being of societies all over the globe. The patterns of suddenness and aftermath of physical and socio-economic systems vary greatly. Tipping points can lead to unintended worsening, to induced vulnerabilities, to chaos and confusion in communication, and to the scope for restorative redirection. The scope for benign transformation is an intrinsic aspect of the tipping point metaphor.
Kevin S. McCann
- Published in print:
- 2011
- Published Online:
- October 2017
- ISBN:
- 9780691134178
- eISBN:
- 9781400840687
- Item type:
- chapter
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691134178.003.0002
- Subject:
- Biology, Ecology
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 ...
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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.Less
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.
Paul A. David
- Published in print:
- 2004
- Published Online:
- September 2007
- ISBN:
- 9780199269426
- eISBN:
- 9780191710179
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199269426.003.0011
- Subject:
- Business and Management, Organization Studies
This chapter looks at the development of electric lighting and power supply networks in terms of the battles waged around 1887-1892 between proponents of direct and alternative current systems of ...
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This chapter looks at the development of electric lighting and power supply networks in terms of the battles waged around 1887-1892 between proponents of direct and alternative current systems of electrical supply in order to raise questions about technical progress as a continuous flow. Utilizing the physics concept of hysteresis as the persistence of an altered state when the force that caused alteration ceases, the chapter concentrates on the critical moments or ‘points of bifurcation’ in the dynamic of technical change that are prone to appear at the early stages of an incremental process. The chapter concludes from its reading of the ultimate victory of alternative over direct currents of electrical supply that innovation implies not so much the work of unique creative attitudes in the manner of the classic Schumpeterian entrepreneur, as it does the occupation of a pivotal situation during comparatively brief moments of industrial development when the balance between choices can go either way.Less
This chapter looks at the development of electric lighting and power supply networks in terms of the battles waged around 1887-1892 between proponents of direct and alternative current systems of electrical supply in order to raise questions about technical progress as a continuous flow. Utilizing the physics concept of hysteresis as the persistence of an altered state when the force that caused alteration ceases, the chapter concentrates on the critical moments or ‘points of bifurcation’ in the dynamic of technical change that are prone to appear at the early stages of an incremental process. The chapter concludes from its reading of the ultimate victory of alternative over direct currents of electrical supply that innovation implies not so much the work of unique creative attitudes in the manner of the classic Schumpeterian entrepreneur, as it does the occupation of a pivotal situation during comparatively brief moments of industrial development when the balance between choices can go either way.
Mark Dykman (ed.)
- Published in print:
- 2012
- Published Online:
- September 2012
- ISBN:
- 9780199691388
- eISBN:
- 9780191742255
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199691388.001.0001
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics
The book elucidates the physics of classical and quantum fluctuations in nonlinear oscillators and provides a unifying insight into fluctuation phenomena in a variety of vibrational systems of ...
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The book elucidates the physics of classical and quantum fluctuations in nonlinear oscillators and provides a unifying insight into fluctuation phenomena in a variety of vibrational systems of current interest. The considered systems are mesoscopis: they are small, so that fluctuations play an important role, but can be individually accessed. The book consists of chapters written by leading experts in the field. The chapters are self-contained and complement each other. They describe major types of nonlinear mesoscopic vibrational systems and the new quantum and classical physics learned using these systems. Also described are new approaches to quantum and classical measurements. The discussed topics include nonlinear dynamics, bistability, and quantum control of microwave cavity modes coupled to qubits; measurements with bifurcation amplifiers based on modulated vibrational systems and new types of such amplifiers; switching rate scaling and a new quantum mechanism of switching in modulated systems; nonlinear wave mixing, parametric excitation, and amplification in the quantum regime; collective phenomena in coupled modulated vibrational systems and the interaction-induced breaking of the time-translation symmetry; quantum back-action in strongly coupled electron-vibrational systems and the effect on the vibrations of the shot noise from spin current; new mechanisms of vibrational relaxation; and the quantum-classical correspondence in the strongly nonlinear regime. The specific systems considered in the book include Josephson junctions, microwave cavities containing qubits or other devices based on Josephson junctions, nano- and micro-mechanical systems, carbon nanotubes, cold atoms, and nano-magnetic oscillators.Less
The book elucidates the physics of classical and quantum fluctuations in nonlinear oscillators and provides a unifying insight into fluctuation phenomena in a variety of vibrational systems of current interest. The considered systems are mesoscopis: they are small, so that fluctuations play an important role, but can be individually accessed. The book consists of chapters written by leading experts in the field. The chapters are self-contained and complement each other. They describe major types of nonlinear mesoscopic vibrational systems and the new quantum and classical physics learned using these systems. Also described are new approaches to quantum and classical measurements. The discussed topics include nonlinear dynamics, bistability, and quantum control of microwave cavity modes coupled to qubits; measurements with bifurcation amplifiers based on modulated vibrational systems and new types of such amplifiers; switching rate scaling and a new quantum mechanism of switching in modulated systems; nonlinear wave mixing, parametric excitation, and amplification in the quantum regime; collective phenomena in coupled modulated vibrational systems and the interaction-induced breaking of the time-translation symmetry; quantum back-action in strongly coupled electron-vibrational systems and the effect on the vibrations of the shot noise from spin current; new mechanisms of vibrational relaxation; and the quantum-classical correspondence in the strongly nonlinear regime. The specific systems considered in the book include Josephson junctions, microwave cavities containing qubits or other devices based on Josephson junctions, nano- and micro-mechanical systems, carbon nanotubes, cold atoms, and nano-magnetic oscillators.
Wolfgang Götze
- Published in print:
- 2008
- Published Online:
- May 2009
- ISBN:
- 9780199235346
- eISBN:
- 9780191715600
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199235346.003.0004
- Subject:
- Physics, Condensed Matter Physics / Materials
In this chapter, mode-coupling equations of motion for a correlation-function description of the dynamics of simple liquids and colloids are derived and their mathematical properties are analysed. ...
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In this chapter, mode-coupling equations of motion for a correlation-function description of the dynamics of simple liquids and colloids are derived and their mathematical properties are analysed. The central concepts are expressions for the fluctuating-force kernels as mode-coupling polynomials of the density-fluctuation-correlation functions. The arrested parts of the latter are solutions of a fixed-point equation, which exhibits bifurcation singularities. The simplest ones are generic and degenerate fold bifurcations, which describe liquid–glass transitions. Schematic models are introduced in order to exemplify by elementary calculations different scenarios for the correlation arrest. The transitions in hard-sphere systems and in square-well systems are explained. Evolutions of the dynamics due to the approach of control parameters towards the critical values for the bifurcation points are analysed in order to show that the theoretical results are similar to those observed for the glassy dynamics of liquids.Less
In this chapter, mode-coupling equations of motion for a correlation-function description of the dynamics of simple liquids and colloids are derived and their mathematical properties are analysed. The central concepts are expressions for the fluctuating-force kernels as mode-coupling polynomials of the density-fluctuation-correlation functions. The arrested parts of the latter are solutions of a fixed-point equation, which exhibits bifurcation singularities. The simplest ones are generic and degenerate fold bifurcations, which describe liquid–glass transitions. Schematic models are introduced in order to exemplify by elementary calculations different scenarios for the correlation arrest. The transitions in hard-sphere systems and in square-well systems are explained. Evolutions of the dynamics due to the approach of control parameters towards the critical values for the bifurcation points are analysed in order to show that the theoretical results are similar to those observed for the glassy dynamics of liquids.
Nils Berglund and Barbara Gentz
- Published in print:
- 2009
- Published Online:
- February 2010
- ISBN:
- 9780199235070
- eISBN:
- 9780191715778
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199235070.003.0003
- Subject:
- Mathematics, Biostatistics
Some models of action potential generation in neurons like the Fitzhugh–Nagumo and the Morris–Lecar model are given by slow–fast differential equations. We outline a general theory allowing us to ...
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Some models of action potential generation in neurons like the Fitzhugh–Nagumo and the Morris–Lecar model are given by slow–fast differential equations. We outline a general theory allowing us to quantify the effect of noise on such equations. The method combines local analyses around stable and unstable equilibria, and around bifurcation points. We discuss in particular two different mechanisms of excitability, which lead to different types of interspike statistics.Less
Some models of action potential generation in neurons like the Fitzhugh–Nagumo and the Morris–Lecar model are given by slow–fast differential equations. We outline a general theory allowing us to quantify the effect of noise on such equations. The method combines local analyses around stable and unstable equilibria, and around bifurcation points. We discuss in particular two different mechanisms of excitability, which lead to different types of interspike statistics.
Baltazar D. Aguda and Avner Friedman
- Published in print:
- 2008
- Published Online:
- September 2008
- ISBN:
- 9780198570912
- eISBN:
- 9780191718717
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198570912.003.0003
- Subject:
- Physics, Soft Matter / Biological Physics
This chapter reviews chemical kinetics to illustrate the formulation of model equations for a given reaction mechanism. For spatially uniform systems, these model equations are usually ordinary ...
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This chapter reviews chemical kinetics to illustrate the formulation of model equations for a given reaction mechanism. For spatially uniform systems, these model equations are usually ordinary differential equations; but coupling of chemical reactions to physical processes such as diffusion requires the formulation of partial differential equations to describe the spatiotemporal evolution of the system. Mathematical analysis of the dynamical models involves basic concepts from ordinary and partial differential equations. Computational methods, including stochastic simulations and sources of computer software programs available free on the internet are also summarized.Less
This chapter reviews chemical kinetics to illustrate the formulation of model equations for a given reaction mechanism. For spatially uniform systems, these model equations are usually ordinary differential equations; but coupling of chemical reactions to physical processes such as diffusion requires the formulation of partial differential equations to describe the spatiotemporal evolution of the system. Mathematical analysis of the dynamical models involves basic concepts from ordinary and partial differential equations. Computational methods, including stochastic simulations and sources of computer software programs available free on the internet are also summarized.
Swapan Dasgupta and Tapan Mitra
- Published in print:
- 2011
- Published Online:
- September 2012
- ISBN:
- 9780198073970
- eISBN:
- 9780199081615
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198073970.003.0004
- Subject:
- Economics and Finance, Microeconomics
The theory of optimal forest management is a key component of the economic theory of natural resources due to the fact that forests constitute a major renewable resource. It also constitutes one of ...
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The theory of optimal forest management is a key component of the economic theory of natural resources due to the fact that forests constitute a major renewable resource. It also constitutes one of the key examples of vintage capital theory, making it an important factor in understanding the general theory of intertemporal allocation. This chapter explores the theory of optimal forest management, focusing on the forester's (optimal) policy function. Whereas the literature places an (almost exclusive) emphasis on long-run behaviour of optimally managed forests, the chapter focuses on the optimal harvesting and replanting decisions that should be implemented currently, given any inherited forest. Using bifurcation analysis, it examines how the optimal policy function changes in response to variations in two key parameters of the forestry model: the growth rate of trees and the planner's discount rate.Less
The theory of optimal forest management is a key component of the economic theory of natural resources due to the fact that forests constitute a major renewable resource. It also constitutes one of the key examples of vintage capital theory, making it an important factor in understanding the general theory of intertemporal allocation. This chapter explores the theory of optimal forest management, focusing on the forester's (optimal) policy function. Whereas the literature places an (almost exclusive) emphasis on long-run behaviour of optimally managed forests, the chapter focuses on the optimal harvesting and replanting decisions that should be implemented currently, given any inherited forest. Using bifurcation analysis, it examines how the optimal policy function changes in response to variations in two key parameters of the forestry model: the growth rate of trees and the planner's discount rate.
Robert C. Hilborn
- Published in print:
- 2000
- Published Online:
- January 2010
- ISBN:
- 9780198507239
- eISBN:
- 9780191709340
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198507239.003.0003
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics
The description of nonlinear behaviour using state space techniques is introduced in this chapter by examining systems with one or two dynamical variables, that is, the state spaces have one or two ...
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The description of nonlinear behaviour using state space techniques is introduced in this chapter by examining systems with one or two dynamical variables, that is, the state spaces have one or two dimensions. The important notions of attractors, fixed points, limit cycles, bifurcations, and stability are introduced using simple models. For systems described by differential equations, the behaviours in one and two-dimensional state spaces are quite limited due to the ‘no-intersection theorem’ for deterministic systems. Models described in this chapter are the (linear) simple harmonic oscillator and the Brusselator chemical reaction model. Taylor series expansions allow us to characterize the stability of behaviour near fixed points and limit cycles. Poincare sections are introduced as a way of reducing the effective dimensionality of systems displaying limit cycles. Bifurcation theory characterizes the sudden changes in behaviour that can occur for nonlinear systems as their parameters are changed.Less
The description of nonlinear behaviour using state space techniques is introduced in this chapter by examining systems with one or two dynamical variables, that is, the state spaces have one or two dimensions. The important notions of attractors, fixed points, limit cycles, bifurcations, and stability are introduced using simple models. For systems described by differential equations, the behaviours in one and two-dimensional state spaces are quite limited due to the ‘no-intersection theorem’ for deterministic systems. Models described in this chapter are the (linear) simple harmonic oscillator and the Brusselator chemical reaction model. Taylor series expansions allow us to characterize the stability of behaviour near fixed points and limit cycles. Poincare sections are introduced as a way of reducing the effective dimensionality of systems displaying limit cycles. Bifurcation theory characterizes the sudden changes in behaviour that can occur for nonlinear systems as their parameters are changed.
Nasr M. Ghoniem and Daniel D. Walgraef
- Published in print:
- 2008
- Published Online:
- May 2008
- ISBN:
- 9780199298686
- eISBN:
- 9780191720222
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199298686.003.0007
- Subject:
- Physics, Condensed Matter Physics / Materials
This chapter focuses on the mathematical structures underlying the notions of stability, bifurcation, and instabilities in complex nonlinear dynamical systems described by sets of ordinary or partial ...
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This chapter focuses on the mathematical structures underlying the notions of stability, bifurcation, and instabilities in complex nonlinear dynamical systems described by sets of ordinary or partial differential equations. It begins by presenting the basic ideas of stability analysis in systems described by ordinary differential equations, introducing Lyapunov functions and their utilization in stability analysis. The stability of systems described by partial differential equations is discussed, emphasizing some of the basic ideas that allow quantitative descriptions of patterns. Specific models illustrating the concepts behind Hopf and Turing instabilities are also discussed.Less
This chapter focuses on the mathematical structures underlying the notions of stability, bifurcation, and instabilities in complex nonlinear dynamical systems described by sets of ordinary or partial differential equations. It begins by presenting the basic ideas of stability analysis in systems described by ordinary differential equations, introducing Lyapunov functions and their utilization in stability analysis. The stability of systems described by partial differential equations is discussed, emphasizing some of the basic ideas that allow quantitative descriptions of patterns. Specific models illustrating the concepts behind Hopf and Turing instabilities are also discussed.
Nasr M. Ghoniem and Daniel D. Walgraef
- Published in print:
- 2008
- Published Online:
- May 2008
- ISBN:
- 9780199298686
- eISBN:
- 9780191720222
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199298686.003.0008
- Subject:
- Physics, Condensed Matter Physics / Materials
This chapter discusses instabilities in reaction-diffusion dynamics. Topics covered include stability in reaction-diffusion dynamics, instabilities in reaction-transport dynamics, Turing instability, ...
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This chapter discusses instabilities in reaction-diffusion dynamics. Topics covered include stability in reaction-diffusion dynamics, instabilities in reaction-transport dynamics, Turing instability, and Hopf bifurcation.Less
This chapter discusses instabilities in reaction-diffusion dynamics. Topics covered include stability in reaction-diffusion dynamics, instabilities in reaction-transport dynamics, Turing instability, and Hopf bifurcation.
Raffaella De Rosa
- Published in print:
- 2010
- Published Online:
- February 2010
- ISBN:
- 9780199570379
- eISBN:
- 9780191722455
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199570379.003.0007
- Subject:
- Philosophy, Philosophy of Mind, History of Philosophy
This final chapter addresses some possible objections to the descriptivist causal‐account of the representationality of Cartesian sensations offered in Chapter 5. Examples of these objections are: is ...
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This final chapter addresses some possible objections to the descriptivist causal‐account of the representationality of Cartesian sensations offered in Chapter 5. Examples of these objections are: is the very notion of Cartesian sensations as qualia (shunned in Chapter 2) reintroduced from the back door in a descriptivist‐causal account? Or does a descriptivist‐causal account reintroduce the bifurcation reading, according to which Descartes drew a wedge between the pure sensing of secondary qualities and the intellectual perception of primary qualities? The chapter addresses and dismisses these objections by emphasizing that a descriptivist‐causal account of Cartesian sensations implies that for Descartes there are no pure sensations.Less
This final chapter addresses some possible objections to the descriptivist causal‐account of the representationality of Cartesian sensations offered in Chapter 5. Examples of these objections are: is the very notion of Cartesian sensations as qualia (shunned in Chapter 2) reintroduced from the back door in a descriptivist‐causal account? Or does a descriptivist‐causal account reintroduce the bifurcation reading, according to which Descartes drew a wedge between the pure sensing of secondary qualities and the intellectual perception of primary qualities? The chapter addresses and dismisses these objections by emphasizing that a descriptivist‐causal account of Cartesian sensations implies that for Descartes there are no pure sensations.
David P. Feldman
- Published in print:
- 2012
- Published Online:
- December 2013
- ISBN:
- 9780199566433
- eISBN:
- 9780191774966
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199566433.001.0001
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics
This book provides an elementary introduction to chaos and fractals. It introduces the key phenomena of chaos — aperiodicity, sensitive dependence on initial conditions, bifurcations — via simple ...
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This book provides an elementary introduction to chaos and fractals. It introduces the key phenomena of chaos — aperiodicity, sensitive dependence on initial conditions, bifurcations — via simple iterated functions. Fractals are introduced as self-similar geometric objects and analysed with the self-similarity and box-counting dimensions. After a brief discussion of power laws, subsequent chapters explore Julia sets and the Mandelbrot set. The last part of the book examines two-dimensional dynamical systems, strange attractors, cellular automata, and chaotic differential equations.Less
This book provides an elementary introduction to chaos and fractals. It introduces the key phenomena of chaos — aperiodicity, sensitive dependence on initial conditions, bifurcations — via simple iterated functions. Fractals are introduced as self-similar geometric objects and analysed with the self-similarity and box-counting dimensions. After a brief discussion of power laws, subsequent chapters explore Julia sets and the Mandelbrot set. The last part of the book examines two-dimensional dynamical systems, strange attractors, cellular automata, and chaotic differential equations.
Klaus Böhmer
- Published in print:
- 2010
- Published Online:
- January 2011
- ISBN:
- 9780199577040
- eISBN:
- 9780191595172
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199577040.003.0009
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics
Chapter 9 studies S. Dahlke studies wavelet methods. This is the first presentation in this generality and is appropriate for nonlinear problems and bifurcation for elliptic PDEs. As a consequence of ...
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Chapter 9 studies S. Dahlke studies wavelet methods. This is the first presentation in this generality and is appropriate for nonlinear problems and bifurcation for elliptic PDEs. As a consequence of the difficulties with evaluating nonlinear functionals and operators with wavelet arguments, general quasilinear, and fully nonlinear problems are limited, and excluded, respectively. With this exception, the whole spectrum of corresponding wavelet methods is shown to be stable and convergent. Again the corresponding linearized operator has to be boundedly invertible. This chapter finishes with adaptive wavelet methods. In contrast to Chapter 6, nonlinear approximation, and wavelet matrix compression are employed for adaptive descent iterations.Less
Chapter 9 studies S. Dahlke studies wavelet methods. This is the first presentation in this generality and is appropriate for nonlinear problems and bifurcation for elliptic PDEs. As a consequence of the difficulties with evaluating nonlinear functionals and operators with wavelet arguments, general quasilinear, and fully nonlinear problems are limited, and excluded, respectively. With this exception, the whole spectrum of corresponding wavelet methods is shown to be stable and convergent. Again the corresponding linearized operator has to be boundedly invertible. This chapter finishes with adaptive wavelet methods. In contrast to Chapter 6, nonlinear approximation, and wavelet matrix compression are employed for adaptive descent iterations.
Wolfgang Götze
- Published in print:
- 2008
- Published Online:
- May 2009
- ISBN:
- 9780199235346
- eISBN:
- 9780191715600
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199235346.001.0001
- Subject:
- Physics, Condensed Matter Physics / Materials
The book presents a self-contained exposition of the mode-coupling theory for the evolution of glassy dynamics in liquids. This theory is based on polynomial expressions for the correlations of force ...
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The book presents a self-contained exposition of the mode-coupling theory for the evolution of glassy dynamics in liquids. This theory is based on polynomial expressions for the correlations of force fluctuations in terms of those of density fluctua-tions. These mode-coupling polynomials are motivated as descriptions of the cage-effect-induced transient localization of particles in condensed matter. It is proven that the implied regular mode-coupling equations of motion determine uniquely models for a correlation-function description of the dynamics. This holds for all choices of the polynomial coefficients, which serve as coupling constants. The arrested parts of the correlations are solutions of fixed-point equations. They exhibit spontaneous singularities, which are equivalent to the bifurcation singularities of the real roots of real polynomials. They deal with idealized liquid-glass and glass-glass transitions. Driving the coupling constants towards their critical values, the correlation functions exhibit the evolution of complex dynamics. Its subtleties are due to the interplay of nonlinearities and divergent retardation effects. The book discusses that the relaxation features are similar to those observed in experimental and molecular-dynamics-simulation studies of con-ventional liquids and colloids. Asymptotic expansions are derived for the mode-coupling-theory functions for small frequencies and small separations of the coupling constants from the transition values. The leading-order asymptotic contributions provide an understanding of the essential facets of the scenarios. The leading-asymptotic corrections are deduced and applied to quantify the evolution of the leading-order description.Less
The book presents a self-contained exposition of the mode-coupling theory for the evolution of glassy dynamics in liquids. This theory is based on polynomial expressions for the correlations of force fluctuations in terms of those of density fluctua-tions. These mode-coupling polynomials are motivated as descriptions of the cage-effect-induced transient localization of particles in condensed matter. It is proven that the implied regular mode-coupling equations of motion determine uniquely models for a correlation-function description of the dynamics. This holds for all choices of the polynomial coefficients, which serve as coupling constants. The arrested parts of the correlations are solutions of fixed-point equations. They exhibit spontaneous singularities, which are equivalent to the bifurcation singularities of the real roots of real polynomials. They deal with idealized liquid-glass and glass-glass transitions. Driving the coupling constants towards their critical values, the correlation functions exhibit the evolution of complex dynamics. Its subtleties are due to the interplay of nonlinearities and divergent retardation effects. The book discusses that the relaxation features are similar to those observed in experimental and molecular-dynamics-simulation studies of con-ventional liquids and colloids. Asymptotic expansions are derived for the mode-coupling-theory functions for small frequencies and small separations of the coupling constants from the transition values. The leading-order asymptotic contributions provide an understanding of the essential facets of the scenarios. The leading-asymptotic corrections are deduced and applied to quantify the evolution of the leading-order description.
Roger White, Guy Engelen, and Inge Uljee
- Published in print:
- 2015
- Published Online:
- May 2016
- ISBN:
- 9780262029568
- eISBN:
- 9780262331371
- Item type:
- chapter
- Publisher:
- The MIT Press
- DOI:
- 10.7551/mitpress/9780262029568.003.0004
- Subject:
- Political Science, Environmental Politics
The structure of a system of retail centres as described by their size, composition, and location, is a result of competition among the centres for customers. The evolution of the system is described ...
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The structure of a system of retail centres as described by their size, composition, and location, is a result of competition among the centres for customers. The evolution of the system is described by a set of cost and revenue equations. The revenue equations include a distance decay parameter. When this parameter is below a critical value, retail activity tends to agglomerate in a major, centrally located centre; otherwise, it tends to be dispersed among a number of similar centres. This fundamental bifurcation appears in actual retail systems. It underlies such phenomena as itinerant medieval trade fairs, the historical migration of the major retail centre of cities like London and New York, and innovations like the department store, the regional mall, and power centres. Since a lower distance decay parameter is associated with higher energy densities, a direct link is established between spatial structure, energy, and technology.Less
The structure of a system of retail centres as described by their size, composition, and location, is a result of competition among the centres for customers. The evolution of the system is described by a set of cost and revenue equations. The revenue equations include a distance decay parameter. When this parameter is below a critical value, retail activity tends to agglomerate in a major, centrally located centre; otherwise, it tends to be dispersed among a number of similar centres. This fundamental bifurcation appears in actual retail systems. It underlies such phenomena as itinerant medieval trade fairs, the historical migration of the major retail centre of cities like London and New York, and innovations like the department store, the regional mall, and power centres. Since a lower distance decay parameter is associated with higher energy densities, a direct link is established between spatial structure, energy, and technology.
Kaloshin Vadim and Zhang Ke
- Published in print:
- 2020
- Published Online:
- May 2021
- ISBN:
- 9780691202525
- eISBN:
- 9780691204932
- Item type:
- chapter
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691202525.003.0008
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics
This chapter defines cohomology of Aubry-Mather type and explains why it implies one of the diffusion mechanisms, after a generic perturbation. The definition of Aubry-Mather type includes a much ...
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This chapter defines cohomology of Aubry-Mather type and explains why it implies one of the diffusion mechanisms, after a generic perturbation. The definition of Aubry-Mather type includes a much simpler case, that is when the Aubry set is a hyperbolic periodic orbit, still contained in a normally hyperbolic invariant cylinder. This definition says that each of the two local components of the Aubry set is of Aubry-Mather type. There is another type of bifurcation in which one component of the Aubry set is of Aubry-Mather type with an invariant cylinder and another is a hyperbolic periodic orbit. This can be called the asymmetric bifurcation. This case appears at double resonance, when the shortest loop is simple non-critical.Less
This chapter defines cohomology of Aubry-Mather type and explains why it implies one of the diffusion mechanisms, after a generic perturbation. The definition of Aubry-Mather type includes a much simpler case, that is when the Aubry set is a hyperbolic periodic orbit, still contained in a normally hyperbolic invariant cylinder. This definition says that each of the two local components of the Aubry set is of Aubry-Mather type. There is another type of bifurcation in which one component of the Aubry set is of Aubry-Mather type with an invariant cylinder and another is a hyperbolic periodic orbit. This can be called the asymmetric bifurcation. This case appears at double resonance, when the shortest loop is simple non-critical.
Kaloshin Vadim and Zhang Ke
- Published in print:
- 2020
- Published Online:
- May 2021
- ISBN:
- 9780691202525
- eISBN:
- 9780691204932
- Item type:
- chapter
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691202525.003.0009
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics
This chapter proves Aubry-Mather type in the single-resonance regime. It proves that for a single-resonance normal form system which satisfies the non-degeneracy conditions, every c in the resonance ...
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This chapter proves Aubry-Mather type in the single-resonance regime. It proves that for a single-resonance normal form system which satisfies the non-degeneracy conditions, every c in the resonance curve is of either Aubry-Mather or bifurcation Aubry-Mather type. The main results are Theorems 9.3 and 9.5, which restate Propositions 3.9 and 3.10. The chapter then proves that the conditions hold on an open and dense set of Hamiltonians. It discusses bifurcations in the double maxima case, as well as hyperbolic coordinates. The chapter also examines normally hyperbolic invariant cylinder, the localization of the Aubry and Mañé sets, and the genericity of the single-resonance conditions.Less
This chapter proves Aubry-Mather type in the single-resonance regime. It proves that for a single-resonance normal form system which satisfies the non-degeneracy conditions, every c in the resonance curve is of either Aubry-Mather or bifurcation Aubry-Mather type. The main results are Theorems 9.3 and 9.5, which restate Propositions 3.9 and 3.10. The chapter then proves that the conditions hold on an open and dense set of Hamiltonians. It discusses bifurcations in the double maxima case, as well as hyperbolic coordinates. The chapter also examines normally hyperbolic invariant cylinder, the localization of the Aubry and Mañé sets, and the genericity of the single-resonance conditions.
Susan C. Stokes
- Published in print:
- 1995
- Published Online:
- May 2012
- ISBN:
- 9780520086173
- eISBN:
- 9780520916234
- Item type:
- chapter
- Publisher:
- University of California Press
- DOI:
- 10.1525/california/9780520086173.003.0006
- Subject:
- Anthropology, Latin American Cultural Anthropology
This chapter draws on survey data to explore the bifurcated mentalities and practices associated with local political culture among Independencia's mass voting population, addressing two sets of ...
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This chapter draws on survey data to explore the bifurcated mentalities and practices associated with local political culture among Independencia's mass voting population, addressing two sets of questions. The first set concerns the relevance to the mass population of patterns among leaders. Does survey analysis confirm the existence of a rift in popular political culture in Peru between “radicals” and “clients”? The second set of questions concerns the link between contemporary and historical patterns. Do data about individuals confirm the historical sequence of changes leading to the rise of a new social movement among the urban poor?Less
This chapter draws on survey data to explore the bifurcated mentalities and practices associated with local political culture among Independencia's mass voting population, addressing two sets of questions. The first set concerns the relevance to the mass population of patterns among leaders. Does survey analysis confirm the existence of a rift in popular political culture in Peru between “radicals” and “clients”? The second set of questions concerns the link between contemporary and historical patterns. Do data about individuals confirm the historical sequence of changes leading to the rise of a new social movement among the urban poor?
Robert C. Hilborn
- Published in print:
- 2000
- Published Online:
- January 2010
- ISBN:
- 9780198507239
- eISBN:
- 9780191709340
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198507239.003.0004
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics
This chapter extends state space descriptions to systems with three (or more) dimensions. For systems described by differential equations, higher state space dimensions allow for more complex ...
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This chapter extends state space descriptions to systems with three (or more) dimensions. For systems described by differential equations, higher state space dimensions allow for more complex behaviour including quasi-periodic behaviour and chaotic behaviour. Chaotic behaviour is characterized by strange attractors in state space. Intermittency and crises as new types of bifurcations are described. Stable manifolds, unstable manifolds, and homoclinic and heteroclinic orbits are used to describe three-dimensional state space behaviour. These ideas are used to understand some of the dynamics of the famous Lorenz model and the Smale horseshoe map. Lyapunov exponents provide a quantitative measure of the degree of chaos.Less
This chapter extends state space descriptions to systems with three (or more) dimensions. For systems described by differential equations, higher state space dimensions allow for more complex behaviour including quasi-periodic behaviour and chaotic behaviour. Chaotic behaviour is characterized by strange attractors in state space. Intermittency and crises as new types of bifurcations are described. Stable manifolds, unstable manifolds, and homoclinic and heteroclinic orbits are used to describe three-dimensional state space behaviour. These ideas are used to understand some of the dynamics of the famous Lorenz model and the Smale horseshoe map. Lyapunov exponents provide a quantitative measure of the degree of chaos.
