Attractors under discretisation

Description

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. Read more... Abstract: 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 Front Matter ....Pages i-xi Front Matter ....Pages 1-1 Lyapunov Stability and Dynamical Systems (Xiaoying Han, Peter Kloeden)....Pages 3-10 One Step Numerical Schemes (Xiaoying Han, Peter Kloeden)....Pages 11-31 Front Matter ....Pages 33-33 Linear Systems (Xiaoying Han, Peter Kloeden)....Pages 35-39 Lyapunov Functions (Xiaoying Han, Peter Kloeden)....Pages 41-48 Dissipative Systems with Steady States (Xiaoying Han, Peter Kloeden)....Pages 49-53 Saddle Points Under Discretisation (Xiaoying Han, Peter Kloeden)....Pages 55-66 Front Matter ....Pages 67-67 Dissipative Systems with Attractors (Xiaoying Han, Peter Kloeden)....Pages 69-75 Lyapunov Functions for Attractors (Xiaoying Han, Peter Kloeden)....Pages 77-81 Discretisation of an Attractor: General Case (Xiaoying Han, Peter Kloeden)....Pages 83-88 Front Matter ....Pages 89-89 Dissipative Nonautonomous Systems (Xiaoying Han, Peter Kloeden)....Pages 91-97 Discretisation of Nonautonomous Limit Sets (Xiaoying Han, Peter Kloeden)....Pages 99-104 Variable Step Size Discretisation of Autonomous Attractors (Xiaoying Han, Peter Kloeden)....Pages 105-109 Discretisation of a Uniform Pullback Attractor (Xiaoying Han, Peter Kloeden)....Pages 111-117 Back Matter ....Pages 119-122