Robert Roussarie's Books

Canard Cycles

From Birth to Transition

By: Peter De Maesschalck,Freddy Dumortier,Robert Roussarie

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Canard Cycles

From Birth to Transition

By: Peter De Maesschalck,Freddy Dumortier,Robert Roussarie

This book offers the first systematic account of canard cycles, an intriguing phenomenon in the study of ordinary differential equations. The canard cycles are treated in the general context of slow-fast families of two-dimensional vector fields. The central question of controlling the limit cycles is addressed in detail and strong results are presented with complete proofs. In particular, the book provides a detailed study of the structure of the transitions near the critical set of non-isolated singularities. This leads to precise results on the limit cycles and their bifurcations, including the so-called canard phenomenon and canard explosion. The book also provides a solid basis for the use of asymptotic techniques. It gives a clear understanding of notions like inner and outer solutions, describing their relation and precise structure. The first part of the book provides a thorough introduction to slow-fast systems, suitable for graduate students. The second and third parts will be of interest to both pure mathematicians working on theoretical questions such as Hilbert's 16th problem, as well as to a wide range of applied mathematicians looking for a detailed understanding of two-scale models found in electrical circuits, population dynamics, ecological models, cellular (FitzHugh–Nagumo) models, epidemiological models, chemical reactions, mechanical oscillators with friction, climate models, and many other models with tipping points.

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Page Count

408

Publisher

Springer Nature

Published Date

ISBN 10

3030792331

ISBN 13

9783030792336

Bifurcations of Planar Vector Fields

Nilpotent Singularities and Abelian Integrals

By: Freddy Dumortier,Robert Roussarie,Jorge Sotomayor,Henryk Zoladek

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Bifurcations of Planar Vector Fields

Nilpotent Singularities and Abelian Integrals

By: Freddy Dumortier,Robert Roussarie,Jorge Sotomayor,Henryk Zoladek

The book reports on recent work by the authors on the bifurcation structure of singular points of planar vector fields whose linear parts are nilpotent. The bifurcation diagrams of the most important codimension-three cases are studied in detail. The results presented reach the limits of what is currently known on the bifurcation theory of planar vector fields. While the treatment is geometric, special analytical tools using abelian integrals are needed, and are explicitly developed. The rescaling and normalization methods are improved for application here. The reader is assumed to be familiar with the elements of Bifurcation and Dynamical Systems Theory. The book is addressed to researchers and graduate students working in Ordinary Differential Equations and Dynamical Systems, as well as anyone modelling complex multiparametric phenomena.

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Page Count

234

Publisher

Springer

Published Date

ISBN 10

3540384332

ISBN 13

9783540384335

Bifurcations of Planar Vector Fields and Hilbert's Sixteenth Problem

By: Robert Roussarie

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Bifurcations of Planar Vector Fields and Hilbert's Sixteenth Problem

By: Robert Roussarie

In a coherent, exhaustive and progressive way, this book presents the tools for studying local bifurcations of limit cycles in families of planar vector fields. A systematic introduction is given to such methods as division of an analytic family of functions in its ideal of coefficients, and asymptotic expansion of non-differentiable return maps and desingularisation. The exposition moves from classical analytic geometric methods applied to regular limit periodic sets to more recent tools for singular limit sets. The methods can be applied to theoretical problems such as Hilbert's 16th problem, but also for the purpose of establishing bifurcation diagrams of specific families as well as explicit computations. - - - The book as a whole is a well-balanced exposition that can be recommended to all those who want to gain a thorough understanding and proficiency in the recently developed methods. The book, reflecting the current state of the art, can also be used for teaching special courses. (Mathematical Reviews)

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Page Count

215

Publisher

Springer Science & Business Media

Published Date

ISBN 10

303480718X

ISBN 13

9783034807180

Canard Cycles and Center Manifolds

By: Freddy Dumortier,Robert H. Roussarie

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Canard Cycles and Center Manifolds

By: Freddy Dumortier,Robert H. Roussarie

In this book, the ``canard phenomenon'' occurring in Van der Pol's equation $\epsilon \ddot x+(x^2+x)\dot x+x-a=0$ is studied. For sufficiently small $\epsilon >0$ and for decreasing $a$, the limit cycle created in a Hopf bifurcation at $a = 0$ stays of ``small size'' for a while before it very rapidly changes to ``big size'', representing the typical relaxation oscillation. The authors give a geometric explanation and proof of this phenomenon using foliations by center manifolds and blow-up of unfoldings as essential techniques. The method is general enough to be useful in the study of other singular perturbation problems.

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Page Count

117

Publisher

American Mathematical Soc.

Published Date

ISBN 10

082180443X

ISBN 13

9780821804438

Historical Developments in Singular Perturbations

By: Robert E. O'Malley

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Historical Developments in Singular Perturbations

By: Robert E. O'Malley

This engaging text describes the development of singular perturbations, including its history, accumulating literature, and its current status. While the approach of the text is sophisticated, the literature is accessible to a broad audience. A particularly valuable bonus are the historical remarks. These remarks are found throughout the manuscript. They demonstrate the growth of mathematical thinking on this topic by engineers and mathematicians. The book focuses on detailing how the various methods are to be applied. These are illustrated by a number and variety of examples. Readers are expected to have a working knowledge of elementary ordinary differential equations, including some familiarity with power series techniques, and of some advanced calculus. Dr. O'Malley has written a number of books on singular perturbations. This book has developed from many of his works in the field of perturbation theory.

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Page Count

263

Publisher

Springer

Published Date

ISBN 10

3319119249

ISBN 13

9783319119243

Speech Rhythm in Varieties of English

Evidence from Educated Indian English and British English

By: Robert Fuchs

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Speech Rhythm in Varieties of English

Evidence from Educated Indian English and British English

By: Robert Fuchs

This book addresses the question whether Educated Indian English is more syllable-timed than British English from two standpoints: production and perception. Many post-colonial varieties of English, which are mostly spoken as a second language in countries such as India, Nigeria and the Philippines, are thought to have a syllable-timed rhythm, whereas first language varieties such as British English are characterized as being stress-timed. While previous studies mostly relied on a single acoustic correlate of speech rhythm, usually duration, the author proposes a multidimensional approach to the production of speech rhythm that takes into account various acoustic correlates. The results reveal that the two varieties differ with regard to a number of dimensions, such as duration, sonority, intensity, loudness, pitch and glottal stop insertion. The second part of the study addresses the question whether the difference in speech rhythm between Indian and British English is perceptually relevant, based on intelligibility and dialect discrimination experiments. The results reveal that speakers generally find the rhythm of their own variety more intelligible and that listeners can identify which variety a speaker is using on the basis of differences in speech rhythm.

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Page Count

240

Publisher

Springer

Published Date

ISBN 10

3662478188

ISBN 13

9783662478189

Castaway

The extraordinary survival story of Narcisse Pelletier, a young French cabin boy shipwrecked on Cape York in 1858

By: Robert Macklin

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Castaway

The extraordinary survival story of Narcisse Pelletier, a young French cabin boy shipwrecked on Cape York in 1858

By: Robert Macklin

In 1858, 14-year-old Narcisse Pelletier sailed from Marseilles in the French trader Saint-Paul. With a cargo of Bordeaux wine, they stopped in Bombay, then Hong Kong, and from there they set sail with more than 300 Chinese prospectors bound for the goldfields of Ballarat and Bendigo. Around the eastern tip of New Guinea, however, the ship became engulfed in fog, struck reefs and ran aground. Scrambling aboard a longboat, the survivors undertook a perilous voyage, crossing almost 1000 kilometres of the Coral Sea before reaching the shores of the Daintree region in far north Queensland, where, abandoned by his shipmates and left for dead, Narcisse was rescued by the local Aboriginal people. For seventeen years he lived with them, growing to manhood and participating fully in their world - until in 1875 he was discovered by the crew of a pearling lugger and wrenched from his Aboriginal family. Taken back to his 'real' life in France, he became a lighthouse keeper, married and had another family, all the while dreaming of what he had left behind... Drawing from firsthand interviews with Narcisse after his return to France and other contemporary accounts of exploration and survival, and documenting the spread of European settlement in Queensland and the brutal frontier wars that followed, Robert Macklin weaves an unforgettable tale of a young man caught between two cultures in a time of transformation and upheaval.

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