Three-Dimensional Flows

In this book, the authors present the elements of a general theory for flows on three-dimensional compact boundaryless manifolds, encompassing flows with equilibria accumulated by regular orbits.

The book aims to provide a global perspective of this theory and make it easier for the reader to digest the growing literature on this subject. This is not the first book on the subject of dynamical systems, but there are distinct aspects which together make this book unique.

Firstly, this book treats mostly continuous time dynamical systems, instead of its discrete counterpart, exhaustively treated in some other texts. Secondly, this book treats all the subjects from a mathematical perspective with proofs of most of the results included. Thirdly, this book is meant to be an advanced graduate textbook and not just a reference book or monograph on the subject. This aspect is reflected in the way the cover material is presented, with careful and complete proofs, and precise references to topics in the book.

• Lyapunov stability
• partial hyperbolicity
• robust transitivity
• singular-attractors
• singular-hyperbolicity
• ordinary differential equations

From the reviews:

“The text is well organized and very well presented and certainly this book will be a major reference in this field. Moreover, it is largely self-contained and also the authors are quite careful to present several simplified proofs … .  the book is enriched with numerous figures that illustrate the highly geometric content of this beautiful topic. In conclusion, this book lies at forefront of knowledge in the field and for this reason researchers and students are encouraged … to extend the results explored here.” (Mário Bessa, Mathematical Reviews, Issue 2011 h)

“The present research monograph considers continuous dynamical systems on three-dimensional compact manifolds. … Moreover, several illustrations (partly in color) are present to enhance understanding of the text. In the main body the authors survey the recent results on robustness for 3-flows including both hyperbolic systems and Lorenz-type systems. … As a consequence, it can be recommended as a valuable source for any reader with an advanced background in hyperbolic dynamics.” (G. Teschl, Internationale Mathematische Nachrichten, Issue 217, August, 2011)

“This book presents in a coherent way the results of a long sequence of papers leading to a deep understanding of the behavior of flows on compact 3-manifolds, in particular for the non-conservative setting. It is clearly presented, with a subjective but pertinent and coherent point of view. … it will be the natural reference for those studying the qualitative behavior of vector fields on compact manifolds.” (Christian Bonatti, Jahresbericht der Deutschen Mathematiker-Vereinigung, Vol. 111, 2012)

“This book deals with the general theory of flows on three-dimensional compact manifolds. … the book under consideration is interesting and presents many results on the dynamical behavior of three dimensional flows with a lot of graphical illustrations which make it morereadable.” (Angela Slavova, Zentralblatt MATH, Vol. 1202, 2011)

• Rio de Janeiro (UFRJ), Inst. Matematica, Universidade Federal do, Rio de Janeiro, Brazil

  Vítor Araújo

• Instituto de Matemática, Depto. Métodos Matemáticos, Universidade Federal do Rio de Janeiro, Rio de Janeiro, Brazil

  Maria José Pacifico

Both authors are active researchers in the field of Dynamical Systems and Ergodic Theory. One is very young and the other well established in the field, being a coauthor of the main results in the theory.

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