Applied Nonautonomous and Random Dynamical Systems (eBook)

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.

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