The Convergence Problem for Dissipative Autonomous Systems: Classical Methods and Recent Advances (SpringerBriefs in Mathematics) - Tapa blanda

Libro 59 de 155: SpringerBriefs in Mathematics

Haraux, Alain

 
9783319234069: The Convergence Problem for Dissipative Autonomous Systems: Classical Methods and Recent Advances (SpringerBriefs in Mathematics)
Tapa blanda
ISBN 10: 3319234064 ISBN 13: 9783319234069
Editorial: Springer, 2015

Sinopsis

The book investigates classical and more recent methods of study for the asymptotic behavior of dissipative continuous dynamical systems with applications to ordinary and partial differential equations, the main question being convergence (or not) of the solutions to an equilibrium. After reviewing the basic concepts of topological dynamics and the definition of gradient-like systems on a metric space, the authors present a comprehensive exposition of stability theory relying on the so-called linearization method. For the convergence problem itself, when the set of equilibria is infinite, the only general results that do not require very special features of the non-linearities are presently consequences of a gradient inequality discovered by S. Lojasiewicz. The application of this inequality jointly with the so-called Liapunov-Schmidt reduction requires a rigorous exposition of Semi-Fredholm operator theory and the theory of real analytic maps on infinite dimensional Banach spaces, which cannot be found anywhere in a readily applicable form. The applications covered in this short text are the simplest, but more complicated cases are mentioned in the final chapter, together with references to the corresponding specialized papers.

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Críticas

“The book ... is ‘a snapshot of a hot or emerging topic’ and ‘a presentation of core concepts that students must understand in order to make independent contributions’. ... it is a very pleasant and reader-friendly text with small surprises in different sections.” (Alp O. Eden, Mathematical Reviews, April, 2016)

Reseña del editor

The book investigates classical and more recent methods of study for the asymptotic behavior of dissipative continuous dynamical systems with applications to ordinary and partial differential equations, the main question being convergence (or not) of the solutions to an equilibrium. After reviewing the basic concepts of topological dynamics and the definition of gradient-like systems on a metric space, the authors present a comprehensive exposition of stability theory relying on the so-called linearization method. For the convergence problem itself, when the set of equilibria is infinite, the only general results that do not require very special features of the non-linearities are presently consequences of a gradient inequality discovered by S. Lojasiewicz. The application of this inequality jointly with the so-called Liapunov-Schmidt reduction requires a rigorous exposition of Semi-Fredholm operator theory and the theory of real analytic maps on infinite dimensional Banach spaces, which cannot be found anywhere in a readily applicable form. The applications covered in this short text are the simplest, but more complicated cases are mentioned in the final chapter, together with references to the corresponding specialized papers.

"Sobre este título" puede pertenecer a otra edición de este libro.

Editorial
Springer
Año de publicación
2015
Idioma
Inglés
ISBN 10 
3319234064
ISBN 13 
9783319234069
Encuadernación
Tapa blanda
Número de edición
1
Número de páginas
156
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