This work focuses on the preservation of attractors and saddle points of ordinary differential equations under discretisation. In the 1980s, key results for autonomous ordinary differential equations were obtained - by Beyn for saddle points and by Kloeden & Lorenz for attractors. One-step numerical schemes with a constant step size were considered, so the resulting discrete time dynamical system was also autonomous. One of the aims of this book is to present new findings on the discretisation of dissipative nonautonomous dynamical systems that have been obtained in recent years, and in particular to examine the properties of nonautonomous omega limit sets and their approximations by numerical schemes - results that are also of importance for autonomous systems approximated by a numerical scheme with variable time steps, thus by a discrete time nonautonomous dynamical system.
Xiaoying Han obtained her PhD from the University at Buffalo, USA in 2007 and is currently a professor at Auburn University, USA. Her main research interests are in random and nonautonomous dynamical systems and their applications. In addition to mathematical analysis of dynamical systems, she is also interested in the modelling and simulation of applied dynamical systems in biology, chemical engineering, ecology, material sciences, etc. Professor Han is a co-author of the books "Applied Nonautonomous and Random Dynamical Systems" (with T. Caraballo) published in the SpringerBrief series and "Random Ordinary Differential Equations and Their Numerical Solutions" (with P. E. Kloeden) published by Springer. Peter E. Kloeden completed his PhD and DSc at the University of Queensland, Australia in 1975 and 1995. He is currently a Professor of Mathematics at Huazhong University of Science & Technology in China, and an affiliated professor at Auburn University, USA. He has wide interests in the applications of mathematical analysis, numerical analysis, stochastic analysis and dynamical systems. Professor Kloeden is a co-author of several influential books on nonautonomous dynamical systems, metric spaces of fuzzy sets, and in particular "Numerical Solutions of Stochastic Differential equations" (with E. Platen) published by Springer in 1992. He is a Fellow of the Society of Industrial and Applied Mathematics and was awarded the W.T. & Idalia Reid Prize in 2006. His current interests focus on nonautonomous and random dynamical systems and their applications in the biological sciences.
Part I Dynamical systems and numerical schemes.- 1 Lyapunov stability and dynamical systems.- 2 One step numerical schemes.- Part II Steady states under discretization.- 3 Linear systems.- 4 Lyapunov functions.- 5 Dissipative systems with steady states.- 6 Saddle points under discretisation . Part III Autonomous attractors under discretization.- 7 Dissipative systems with attractors.- 8 Lyapunov functions for attractors.- 9 Discretisation of an attractor. Part IV Nonautonomous limit sets under discretization.- 10 Dissipative nonautonomous systems .- 11 Discretisation of nonautonomous limit sets.- 12 Variable step size.- 13 Discretisation of a uniform pullback attractor.- Notes.- References.