62 results
Clinical predictors of conversion to bipolar disorder in a prospective longitudinal familial high-risk sample: focus on depressive features
- Andrew Frankland, Gloria Roberts, Ellen Holmes-Preston, Tania Perich, Florence Levy, Rhoshel Lenroot, Dusan Hadzi-Pavlovic, Michael Breakspear, Philip B. Mitchell
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- Psychological Medicine / Volume 48 / Issue 10 / July 2018
- Published online by Cambridge University Press:
- 07 November 2017, pp. 1713-1721
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Background
Identifying clinical features that predict conversion to bipolar disorder (BD) in those at high familial risk (HR) would assist in identifying a more focused population for early intervention.
MethodIn total 287 participants aged 12–30 (163 HR with a first-degree relative with BD and 124 controls (CONs)) were followed annually for a median of 5 years. We used the baseline presence of DSM-IV depressive, anxiety, behavioural and substance use disorders, as well as a constellation of specific depressive symptoms (as identified by the Probabilistic Approach to Bipolar Depression) to predict the subsequent development of hypo/manic episodes.
ResultsAt baseline, HR participants were significantly more likely to report ⩾4 Probabilistic features (40.4%) when depressed than CONs (6.7%; p < .05). Nineteen HR subjects later developed either threshold (n = 8; 4.9%) or subthreshold (n = 11; 6.7%) hypo/mania. The presence of ⩾4 Probabilistic features was associated with a seven-fold increase in the risk of ‘conversion’ to threshold BD (hazard ratio = 6.9, p < .05) above and beyond the fourteen-fold increase in risk related to major depressive episodes (MDEs) per se (hazard ratio = 13.9, p < .05). Individual depressive features predicting conversion were psychomotor retardation and ⩾5 MDEs. Behavioural disorders only predicted conversion to subthreshold BD (hazard ratio = 5.23, p < .01), while anxiety and substance disorders did not predict either threshold or subthreshold hypo/mania.
ConclusionsThis study suggests that specific depressive characteristics substantially increase the risk of young people at familial risk of BD going on to develop future hypo/manic episodes and may identify a more targeted HR population for the development of early intervention programs.
Contributors
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- By Mitchell Aboulafia, Frederick Adams, Marilyn McCord Adams, Robert M. Adams, Laird Addis, James W. Allard, David Allison, William P. Alston, Karl Ameriks, C. Anthony Anderson, David Leech Anderson, Lanier Anderson, Roger Ariew, David Armstrong, Denis G. Arnold, E. J. Ashworth, Margaret Atherton, Robin Attfield, Bruce Aune, Edward Wilson Averill, Jody Azzouni, Kent Bach, Andrew Bailey, Lynne Rudder Baker, Thomas R. Baldwin, Jon Barwise, George Bealer, William Bechtel, Lawrence C. Becker, Mark A. Bedau, Ernst Behler, José A. Benardete, Ermanno Bencivenga, Jan Berg, Michael Bergmann, Robert L. Bernasconi, Sven Bernecker, Bernard Berofsky, Rod Bertolet, Charles J. Beyer, Christian Beyer, Joseph Bien, Joseph Bien, Peg Birmingham, Ivan Boh, James Bohman, Daniel Bonevac, Laurence BonJour, William J. Bouwsma, Raymond D. Bradley, Myles Brand, Richard B. Brandt, Michael E. Bratman, Stephen E. Braude, Daniel Breazeale, Angela Breitenbach, Jason Bridges, David O. Brink, Gordon G. Brittan, Justin Broackes, Dan W. Brock, Aaron Bronfman, Jeffrey E. Brower, Bartosz Brozek, Anthony Brueckner, Jeffrey Bub, Lara Buchak, Otavio Bueno, Ann E. Bumpus, Robert W. Burch, John Burgess, Arthur W. Burks, Panayot Butchvarov, Robert E. Butts, Marina Bykova, Patrick Byrne, David Carr, Noël Carroll, Edward S. Casey, Victor Caston, Victor Caston, Albert Casullo, Robert L. Causey, Alan K. L. Chan, Ruth Chang, Deen K. Chatterjee, Andrew Chignell, Roderick M. Chisholm, Kelly J. Clark, E. J. Coffman, Robin Collins, Brian P. Copenhaver, John Corcoran, John Cottingham, Roger Crisp, Frederick J. Crosson, Antonio S. Cua, Phillip D. Cummins, Martin Curd, Adam Cureton, Andrew Cutrofello, Stephen Darwall, Paul Sheldon Davies, Wayne A. Davis, Timothy Joseph Day, Claudio de Almeida, Mario De Caro, Mario De Caro, John Deigh, C. F. Delaney, Daniel C. Dennett, Michael R. DePaul, Michael Detlefsen, Daniel Trent Devereux, Philip E. Devine, John M. Dillon, Martin C. Dillon, Robert DiSalle, Mary Domski, Alan Donagan, Paul Draper, Fred Dretske, Mircea Dumitru, Wilhelm Dupré, Gerald Dworkin, John Earman, Ellery Eells, Catherine Z. Elgin, Berent Enç, Ronald P. Endicott, Edward Erwin, John Etchemendy, C. Stephen Evans, Susan L. Feagin, Solomon Feferman, Richard Feldman, Arthur Fine, Maurice A. Finocchiaro, William FitzPatrick, Richard E. Flathman, Gvozden Flego, Richard Foley, Graeme Forbes, Rainer Forst, Malcolm R. Forster, Daniel Fouke, Patrick Francken, Samuel Freeman, Elizabeth Fricker, Miranda Fricker, Michael Friedman, Michael Fuerstein, Richard A. Fumerton, Alan Gabbey, Pieranna Garavaso, Daniel Garber, Jorge L. A. Garcia, Robert K. Garcia, Don Garrett, Philip Gasper, Gerald Gaus, Berys Gaut, Bernard Gert, Roger F. Gibson, Cody Gilmore, Carl Ginet, Alan H. Goldman, Alvin I. Goldman, Alfonso Gömez-Lobo, Lenn E. Goodman, Robert M. Gordon, Stefan Gosepath, Jorge J. E. Gracia, Daniel W. Graham, George A. Graham, Peter J. Graham, Richard E. Grandy, I. Grattan-Guinness, John Greco, Philip T. Grier, Nicholas Griffin, Nicholas Griffin, David A. Griffiths, Paul J. Griffiths, Stephen R. Grimm, Charles L. Griswold, Charles B. Guignon, Pete A. Y. Gunter, Dimitri Gutas, Gary Gutting, Paul Guyer, Kwame Gyekye, Oscar A. Haac, Raul Hakli, Raul Hakli, Michael Hallett, Edward C. Halper, Jean Hampton, R. James Hankinson, K. R. Hanley, Russell Hardin, Robert M. Harnish, William Harper, David Harrah, Kevin Hart, Ali Hasan, William Hasker, John Haugeland, Roger Hausheer, William Heald, Peter Heath, Richard Heck, John F. Heil, Vincent F. Hendricks, Stephen Hetherington, Francis Heylighen, Kathleen Marie Higgins, Risto Hilpinen, Harold T. Hodes, Joshua Hoffman, Alan Holland, Robert L. Holmes, Richard Holton, Brad W. Hooker, Terence E. Horgan, Tamara Horowitz, Paul Horwich, Vittorio Hösle, Paul Hoβfeld, Daniel Howard-Snyder, Frances Howard-Snyder, Anne Hudson, Deal W. Hudson, Carl A. Huffman, David L. Hull, Patricia Huntington, Thomas Hurka, Paul Hurley, Rosalind Hursthouse, Guillermo Hurtado, Ronald E. Hustwit, Sarah Hutton, Jonathan Jenkins Ichikawa, Harry A. Ide, David Ingram, Philip J. Ivanhoe, Alfred L. Ivry, Frank Jackson, Dale Jacquette, Joseph Jedwab, Richard Jeffrey, David Alan Johnson, Edward Johnson, Mark D. Jordan, Richard Joyce, Hwa Yol Jung, Robert Hillary Kane, Tomis Kapitan, Jacquelyn Ann K. Kegley, James A. Keller, Ralph Kennedy, Sergei Khoruzhii, Jaegwon Kim, Yersu Kim, Nathan L. King, Patricia Kitcher, Peter D. Klein, E. D. Klemke, Virginia Klenk, George L. Kline, Christian Klotz, Simo Knuuttila, Joseph J. Kockelmans, Konstantin Kolenda, Sebastian Tomasz Kołodziejczyk, Isaac Kramnick, Richard Kraut, Fred Kroon, Manfred Kuehn, Steven T. Kuhn, Henry E. Kyburg, John Lachs, Jennifer Lackey, Stephen E. Lahey, Andrea Lavazza, Thomas H. Leahey, Joo Heung Lee, Keith Lehrer, Dorothy Leland, Noah M. 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Contributors
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- By Peter J. D. Andrews, Sandeep Ankolekar, Issam A. Awad, Omar Ayoub, Philip Bath, Jürgen Bardutzky, Alexander Beck, Patrícia Canhão, J. Ricardo Carhuapoma, Winward Choy, Mahua Dey, Rajat Dhar, Michael C. Diringer, Arnd Dörfler, Joshua R. Dusick, Justin A. Dye, Corina Epple, José M. Ferro, Reiner Fietkau, Anthony Frattalone, Philippe Gailloud, Oliver Ganslandt, Anil Gholkar, Philipp Gölitz, Barbara A. Gregson, Daniel Hanley, Thomas M. Hemmen, Dan Holmes, Hagen B. Huttner, Jennifer Jaffe, Olav Jansen, Eric Jüttler, Karl L. Kiening, Martin Köhrmann, Rainer Kollmar, Kara L. Krajewski, Joji B. Kuramatsu, Perttu J. Lindsberg, Andrew Losiniecki, Patrick Lyden, Neil A. Martin, Heinrich P. Mattle, A. David Mendelow, Patrick Mitchell, Daniel T. Nagasawa, Neeraj S. Naval, Jan-Oliver Neumann, Tim Nowe, Berk Orakcioglu, Soenke Peters, Sara Pitoni, François Proust, Adnan I. Qureshi, Martin Radvany, Elise Rowan, Tiina Sairanen, Oliver W. Sakowitz, Edgar Santos, Peter D. Schellinger, Stefan Schwab, Günter Seidel, Sabine Semrau, Louise Sinclair, Dimitre Staykov, Thorsten Steiner, Jeanne Teitelbaum, Wondwossen G. Tekle, Andreas W. Unterberg, Katayoun Vahedi, H. Bart van der Worp, Paul M. Vespa, Raghu Vindlacheruvu, Jens Witsch, Isaac Yang, Wendy C. Ziai, Mario Zuccarello, Klaus Zweckberger
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- By Chris Beyers, Amy Branam, Alan Brown, Mark Canada, Patricia A. Cunningham, Satwik Dasgupta, David Dowling, John Evelev, Michael J. Everton, Benjamin F. Fisher, Paul Fisher, Meghan A. Freeman, Christopher Gair, Andrea Goulet, Jonathan Hartmann, Kevin J. Hayes, Gregory Hays, Alvin Holm, Lindsey Hursh, James M. Hutchisson, Paul Christian Jones, Katherine Kim, Nathaniel Lewis, Bruce Mills, Travis Montgomery, Tara Moore, Bran Nicol, Philip Edward Phillips, Anne Boyd Rioux, Therese M. Rizzo, Kathryn K. Shinn, Heidi Silcox, Peter Swirski, Jonathan Taylor, John Tresch, Lois Davis Vines, Jeffrey Andrew Weinstock, Brett Zimmerman
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12 - Some other fluid problems
- from PART FOUR - OTHER APPLICATIONS AND RELATED WORK
- Philip Holmes, Princeton University, New Jersey, John L. Lumley, Cornell University, New York, Gahl Berkooz, Clarence W. Rowley, Princeton University, New Jersey
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Summary
The methods developed in this book can be applied rather generally to model the dynamics of coherent structures in spatially extended systems. They are gaining acceptance in many areas in addition to fluid mechanics, including mechanical vibrations, laser dynamics, nonlinear optics, and chemical processes. They are even being applied to studies of neural activity in the human brain. Numerous studies of closed flow systems have been done using empirical eigenfunctions, some of which were discussed in Section 3.7. A considerable amount of work has also been done on model PDEs for weakly nonlinear waves, such as the Ginzburg–Landau and Kuramoto–Sivashinsky equations, but this work falls largely outside the scope of this book. We do not have the abilities (or space) to provide a survey of these multifarious applications, but we do wish to draw the reader's attention to some of the other recent work on open fluid flows.
We restrict ourselves to studies in which empirical eigenfunctions are used to construct low-dimensional models and some attempt is made to analyze their dynamical behavior. There is an enormous amount of work in which the POD is applied and its results assessed in a “static,” averaged fashion. Some of this we have reviewed in Section 3.7. Yet even thus restricted, our survey cannot pretend to be complete: new applications to fluid flows are appearing at an increasing rate. We have selected five problems on which a reasonable amount of work has been done, the first of which (the jet) is a “strongly” turbulent flow.
References
- Philip Holmes, Princeton University, New Jersey, John L. Lumley, Cornell University, New York, Gahl Berkooz, Clarence W. Rowley, Princeton University, New Jersey
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Acknowledgements
- Philip Holmes, Princeton University, New Jersey, John L. Lumley, Cornell University, New York, Gahl Berkooz, Clarence W. Rowley, Princeton University, New Jersey
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PART TWO - DYNAMICAL SYSTEMS
- Philip Holmes, Princeton University, New Jersey, John L. Lumley, Cornell University, New York, Gahl Berkooz, Clarence W. Rowley, Princeton University, New Jersey
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PART ONE - TURBULENCE
- Philip Holmes, Princeton University, New Jersey, John L. Lumley, Cornell University, New York, Gahl Berkooz, Clarence W. Rowley, Princeton University, New Jersey
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13 - Review: prospects for rigor
- from PART FOUR - OTHER APPLICATIONS AND RELATED WORK
- Philip Holmes, Princeton University, New Jersey, John L. Lumley, Cornell University, New York, Gahl Berkooz, Clarence W. Rowley, Princeton University, New Jersey
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Summary
As we near the end of our story, the reader will now appreciate that there are many steps in the process of reducing the Navier–Stokes equations to a low-dimensional model for the dynamics of coherent structures. Some of these involve purely mathematical issues, but most require an interplay among physical considerations, judgement, and mathematical tractability. While our development of a general strategy for constructing low-dimensional models has been based on theoretical developments such as the POD and dynamical systems methods, the general theory is still sketchy and, in specific applications, many details remain unresolved.
The mathematical techniques we have drawn on lie primarily in probability and dynamical systems theory. In this closing chapter we review some aspects of the reduction process and attempt to put them into context. Some prospects for rigor in the reduction process are also mentioned. This is by no means a comprehensive review or discussion of future work; instead, we have chosen to highlight a few applications of dynamical and probabilistic ideas to illustrate lines along which a general theory might be further developed.
We start by discussing some desirable properties for low-dimensional models, and criteria by which they might be judged. We then outline in Section 13.2 an a-priori short-term tracking estimate which describes, in a probabilistic context, how rapidly typical solutions of the model equations are expected to diverge from those of the full Navier–Stokes equations restricted to the model domain. Here and in the following section we view low-dimensional models as perturbations of the full evolution equations. Section 13.3 also addresses reproduction of statistics by low-dimensional models.
10 - Low-dimensional models
- from PART THREE - THE BOUNDARY LAYER
- Philip Holmes, Princeton University, New Jersey, John L. Lumley, Cornell University, New York, Gahl Berkooz, Clarence W. Rowley, Princeton University, New Jersey
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Summary
In the preceding nine chapters we have developed our basic tools and techniques. In this chapter and the next we illustrate their use in the derivation and analysis of low-dimensional models of the wall region of a turbulent boundary layer. First, the Navier– Stokes equations are rewritten in a form that highlights the dynamics of the coherent structures (CS) and their interaction with the mean flow. To do this, both the neglected (high) wavenumber modes and the mean flow must be modeled, unlike a large eddy simulation (LES), in which only the neglected high modes are modeled. Second, using physical considerations, we select a family of empirical subspaces upon which to project the equations. Galerkin projection is then carried out. In doing this, we restrict ourselves to a small physical flow domain, and so the response of the (quasi)local mean flow to the coherent structures must also be modeled. This chapter describes each step of the process in some detail, drawing on material presented in Chapters 2, 3, and 4. After deriving the family of low-dimensional models, in the last three sections we discuss in more depth the validity of assumptions used in their derivation. In Chapter 11 we describe use of the dynamical systems ideas presented in Chapters 6 through 9 in the analysis of these models, and interpret their solutions in terms of the dynamical behavior of the fluid flow.
Our presentation is based on a series of papers, beginning with [22] and including [24, 43, 44, 158, 161]. We have selected the boundary layer as our main illustrative example largely because we are most familiar with it.
Turbulence, Coherent Structures, Dynamical Systems and Symmetry
- 2nd edition
- Philip Holmes, John L. Lumley, Gahl Berkooz, Clarence W. Rowley
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Turbulence pervades our world, from weather patterns to the air entering our lungs. This book describes methods that reveal its structures and dynamics. Building on the existence of coherent structures – recurrent patterns – in turbulent flows, it describes mathematical methods that reduce the governing (Navier–Stokes) equations to simpler forms that can be understood more easily. This second edition contains a new chapter on the balanced proper orthogonal decomposition: a method derived from control theory that is especially useful for flows equipped with sensors and actuators. It also reviews relevant work carried out since 1995. The book is ideal for engineering, physical science and mathematics researchers working in fluid dynamics and other areas in which coherent patterns emerge.
Contents
- Philip Holmes, Princeton University, New Jersey, John L. Lumley, Cornell University, New York, Gahl Berkooz, Clarence W. Rowley, Princeton University, New Jersey
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2 - Coherent structures
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- Philip Holmes, Princeton University, New Jersey, John L. Lumley, Cornell University, New York, Gahl Berkooz, Clarence W. Rowley, Princeton University, New Jersey
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3 - Proper orthogonal decomposition
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- Philip Holmes, Princeton University, New Jersey, John L. Lumley, Cornell University, New York, Gahl Berkooz, Clarence W. Rowley, Princeton University, New Jersey
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Summary
The proper orthogonal decomposition (POD) provides a basis for the modal decomposition of an ensemble of functions, such as data obtained in the course of experiments. Its properties suggest that it is the preferred basis to use in various applications. The most striking of these is optimality: it provides the most efficient way of capturing the dominant components of an infinite-dimensional process with only finitely many, and often surprisingly few, “modes.”
The POD was introduced in the context of turbulence by Lumley in [220]. In other disciplines the same procedure goes by the names: Karhunen–Loève decomposition, principal components analysis, singular systems analysis, and singular value decomposition. The basis functions it yields are variously called: empirical eigenfunctions, empirical basis functions, and empirical orthogonal functions. According to Yaglom (see [221]), the POD was introduced independently by numerous people at different times, including Kosambi [197], Loève [215], Karhunen [183], Pougachev [285], and Obukhov [272]. Lorenz [216], whose name we have already met in another context, suggested its use in weather prediction. The procedure has been used in various disciplines other than fluid mechanics, including random variables [275], image processing [313], signal analysis [5], data compression [7], process identification and control in chemical engineering [118,119], and oceanography [286]. Computational packages based on the POD are now readily available (an early example appeared in [11]).
In the bulk of these applications, the POD is used to analyze experimental data with a view to extracting dominant features and trends – in particular coherent structures. In the context of turbulence and other complex spatio-temporal fields, these will typically be patterns in space and time. However, our goal is somewhat different.
Preface to the first edition
- Philip Holmes, Princeton University, New Jersey, John L. Lumley, Cornell University, New York, Gahl Berkooz, Clarence W. Rowley, Princeton University, New Jersey
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Summary
On physical grounds there is no doubt that the Navier–Stokes equations provide an excellent model for fluid flow as long as shock waves are relatively thick (in terms of mean free paths), and in such conditions of temperature and pressure that we can regard the fluid as a continuum. The incompressible version is restricted, of course, to lower speeds and more moderate temperatures and pressures. There are some mathematical difficulties – indeed, we still lack a satisfactory existence-uniqueness theory in three dimensions – but these do not appear to compromise the equations’ validity. Why then is the “problem of turbulence” so difficult? We can, of course, solve these nonlinear partial differential equations numerically for given boundary and initial conditions, to generate apparently unique turbulent solutions, but this is the only useful sense in which they are soluble, save for certain non-turbulent flows having strong symmetries and other simplifications. Unfortunately, numerical solutions do not bring much understanding.
However, three fairly recent developments offer some hope for improved understanding: (1) the discovery, by experimental fluid mechanicians, of coherent structures in certain fully developed turbulent flows; (2) the suggestion that strange attractors and other ideas from finite-dimensional dynamical systems theory might play a rôle in the analysis of the governing equations; and (3) the introduction of the statistical technique of Karhunen– Loève or proper orthogonal decomposition. This book introduces these developments and describes how the three threads can be drawn together to weave low-dimensional models that address the rôle of coherent structures in turbulence generation.
5 - Balanced proper orthogonal decomposition
- from PART ONE - TURBULENCE
- Philip Holmes, Princeton University, New Jersey, John L. Lumley, Cornell University, New York, Gahl Berkooz, Clarence W. Rowley, Princeton University, New Jersey
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Summary
As we shall see in Parts III and IV, the techniques of proper orthogonal decomposition and Galerkin projection can be powerful tools for obtaining low-order models that capture the qualitative behavior of complex, high-dimensional systems. However, for certain systems, the resulting models can perform poorly: even if a large fraction of energy (over 99%) is captured by the modes used for projection, the resulting low-order models may still have completely different qualitative behavior. The transients may be poorly captured, and the stability types of equilibria can even be different.
In this chapter, we present a method which can dramatically outperform projection onto traditional energy-based empirical eigenfunctions described in Chapter 3.We focus primarily (though not exclusively) on linear systems, for several reasons. Many of the pitfalls of traditional proper orthogonal decomposition can be demonstrated for linear systems, without the additional complexity of nonlinearities. Furthermore, for linear systems, one can use operator norms to quantify the difference between a detailed model and its reduced order approximation. Most importantly, for linear systems, there are established tools for performing model reduction, for instance using balanced truncation, which is described in Section 5.1. In contrast, while some modest extensions to nonlinear systems have been attempted, model reduction of nonlinear systems is still an active area of research.
The techniques described in this chapter also differ from those in Chapters 3 and 4 in that they are formulated for input–output systems. The inputs represent the external influences on the system, for instance from external disturbances, or from actuators in a flow-control setting.
Preface to the second edition
- Philip Holmes, Princeton University, New Jersey, John L. Lumley, Cornell University, New York, Gahl Berkooz, Clarence W. Rowley, Princeton University, New Jersey
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- Book:
- Turbulence, Coherent Structures, Dynamical Systems and Symmetry
- Published online:
- 05 June 2012
- Print publication:
- 23 February 2012, pp xiii-xiv
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Summary
Much work has been done on low-dimensional models of turbulence and fluid systems in the 16 years since the first edition of this book appeared. In preparing the second edition, we have not attempted a comprehensive review: indeed, we doubt that this is possible, or even desirable. Rather, we have added one chapter and several sections and subsections on some new developments that are most closely related to material in our first edition. We have also made minor corrections and clarifications throughout, and added comments in several places, as well as correcting a number of errors that readers have pointed out. Here, to orient the reader, we outline the major changes.
Clancy Rowley (the new member of our team) has contributed a chapter on balanced truncation, a technique from linear control theory that chooses bases that optimally align inputs and outputs. Over the past ten years this has led to the method of balanced proper orthogonal decomposition (BPOD), which is especially useful for systems equipped with sensors and actuators. Since low-dimensional models provide a computational means for studying control of turbulence, we feel that BPOD has considerable potential. This new chapter (5) now closes the first part of the book (readers familiar with the first edition must therefore remember to add 1 to correctly identify the following eight chapters). The only other entirely new sections are 7.5, a discussion of traveling modes in translation-invariant systems, 12.6, a review of work on coherent structures in internal combustion engines, and 12.7, which gathers a miscellany of recent results.
6 - Qualitative theory
- from PART TWO - DYNAMICAL SYSTEMS
- Philip Holmes, Princeton University, New Jersey, John L. Lumley, Cornell University, New York, Gahl Berkooz, Clarence W. Rowley, Princeton University, New Jersey
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- Book:
- Turbulence, Coherent Structures, Dynamical Systems and Symmetry
- Published online:
- 05 June 2012
- Print publication:
- 23 February 2012, pp 155-189
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Summary
This chapter and the following one provide a review of some aspects of the qualitative theory of dynamical systems that we need in our analyses of low-dimensional models derived from the Navier–Stokes equations. Dynamical systems theory is a broad and rapidly growing field which, in its more megalomaniacal forms, might be claimed to encompass all of differential equations (ordinary, partial, and functional), iterations of mappings (real and complex), devices such as cellular automata and neural networks, as well as large parts of analysis and differential topology. Here our aim is merely the modest one of introducing, with simple examples, some tools for analysis of nonlinear ordinary differential equations that may not be as familiar as, say, perturbation and asymptotic methods.
The viewpoint of dynamical systems theory is geometric, and invariant manifolds play a central rôle, but we do not assume or require familiarity with differential topology. In the same way, symmetries are crucial in determining the behavior, and permitting the analysis, of the low-dimensional models of interest, but we avoid appeals to the subtleties of group theory in our introduction to symmetric bifurcations. Thus, it should be clear that these two chapters cannot substitute for a serious course (or, more likely, courses) in dynamical systems theory. The makings of such a course can be found in the books of Arnold [15,17], Guckenheimer and Holmes [144], Arrowsmith and Place [18], or Glendinning [129], and in other references cited below. In particular we omit entirely any discussion of partial differential equations, which may seem scandalous, since this book ostensibly treats turbulence as described by the Navier–Stokes equations.
1 - Introduction
- from PART ONE - TURBULENCE
- Philip Holmes, Princeton University, New Jersey, John L. Lumley, Cornell University, New York, Gahl Berkooz, Clarence W. Rowley, Princeton University, New Jersey
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- Book:
- Turbulence, Coherent Structures, Dynamical Systems and Symmetry
- Published online:
- 05 June 2012
- Print publication:
- 23 February 2012, pp 3-16
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Summary
Turbulence
Turbulence is the last great unsolved problem of classical physics. Although temporarily abandoned by much of the community in favor of particle physics, the current popularity of chaos and dynamical systems theory (as well as funding problems in particle physics) is now drawing the physicists back. During the interim and up to the present, turbulence has been avidly pursued by engineers.
Turbulence has enormous intellectual fascination for physicists, engineers, and mathematicians alike. This scientific appeal stems in part from its inherent difficulty – most of the approaches that can be used on other problems in fluid mechanics are useless in turbulence. Turbulence is usually approached as a stochastic problem, yet the simplifications that can be used in statistical mechanics are not applicable – turbulence is characterized by strong dependency in space and in time, so that not much can be modeled usefully as a simple Markov process, for example. The nonlinearity of turbulence is essential – linearization destroys the problem. Many problems in fluid mechanics can be approached by supposing that the flow is irrotational – that is, that the vorticity is zero everywhere. In turbulence, the presence of vorticity is essential to the dynamics. In fact, the nonlinearity, rotationality, and the dimensionality interact dynamically to feed the turbulence – hence, to suppose that a realization of the flow is two-dimensional also destroys the problem. There is more, but this is enough to make it clear that one faces the turbulence problem stripped of the usual arsenal of techniques, reduced to hand-to-hand combat. One is forced to find unexpected chinks in its armor almost by necromancy, and to fabricate new approaches from whole cloth. This is its fascination.