Applied Nonautonomous and Random Dynamical Systems: Applied Dynamical Systems

ISBN-13: 9783319492469 / Angielski / Miękka / 2017 / 108 str.

Tomas Caraballo; Xiaoying Han

Opis

This book offers an introduction to the theory of non-autonomous and stochastic dynamical systems, with a focus on the importance of the theory in the Applied Sciences. It starts by discussing the basic concepts from the theory of autonomous dynamical systems, which are easier to understand and can be used as the motivation for the non-autonomous and stochastic situations. The book subsequently establishes a framework for non-autonomous dynamical systems, and in particular describes the various approaches currently available for analysing the long-term behaviour of non-autonomous problems. Here, the major focus is on the novel theory of pullback attractors, which is still under development. In turn, the third part represents the main body of the book, introducing the theory of random dynamical systems and random attractors and revealing how it may be a suitable candidate for handling realistic models with stochasticity. A discussion of future research directions serves to round out the coverage.

Recenzje

"This brief monograph gives an introduction to the theory of non-autonomous, random and  stochastic dynamical systems. The book is accessible for a wide audience and gives some citations to more recent research. ... The main emphasis is on the study of the long-term behavior of solutions and pullback attractors. A discussion of future research directions supplements the topics presented." (Henri Schurz, Mathematical Reviews, June, 2017)

Spis treści

1 Introduction.- 2 Autonomous dynamical systems.- 2.1 Basic Stability Theory.- 2.2 Attractors. - 2.3 Applications. - 2.3.1 Application to ecology: a chemostat model.- 2.3.2 Application to epidemiology: the SIR model.- 2.3.3 Application to climate change: the Lorenz-84 model.- 3 Nonautonomous dynamical systems.- 3.1 Formulations of nonautonomous dynamical systems.- 3.1.1 Process formulation.- 3.1.2 Skew product flow formulation.- 3.2 Nonautonomous Attractors.- 3.2.1 Nonautonomous attractors for processes.- 3.2.2 Nonautonomous attractors for skew product flows.- 3.3 Applications.- 3.3.1 Nonautonomous chemostat model.- 3.3.2 Nonautonomous SIR model.- 3.3.3 Nonautonomous Lorenz-84 model.- 4 Random dynamical systems.- 4.1 Noise is present almost everywhere.- 4.2 Formulation of Random Dynamical System and Random Attracto.- 4.2.1 Some properties of the random attractor.- 4.2.2 Generation of random dynamical systems.- 4.2.3 A brief introduction to stochastic differential equations.- 4.2.4 Global asymptotic behavior of SDEs: conjugation of RDS.- 4.3 Applications.- 4.3.1 Random chemostat.- 4.3.2 Random and stochastic SIR.- 4.3.3 Stochastic Lorenz models.- 4.4 Stabilization of dynamical systems. References 

O autorze

Tomás Caraballo graduated with a degree in Mathematical Sciences in 1984, and received his Ph.D.  in the same subject at the Universidad de Sevilla (Spain) in 1988. He is currently a Catedrático de Universidad (Full Professor) at the Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla. He has participated in or been Principal Investigator for more than 20 research projects. He has authored or co-authored more than 180 papers, has supervised 10 doctoral theses, and has been associate or guest editor of numerous scientific journals. His current research interests are non-autonomous and random dynamical systems, both in finite and infinite dimensions, stochastic ordinary and partial differential equations with memory, delay, impulses and their applications to real models from the applied sciences.

Xiaoying Han received her Ph.D. in mathematics from the State University of New York at Buffalo in 2007. She is an associate professor of mathematics at Auburn University’s Department of Mathematics and Statistics, Alabama, USA. Her main research focuses are nonautonomous and random dynamical systems and their applications, and she is also interested in modeling, analysis and simulation of stochastic systems arising in applied sciences and engineering.