Preface; Contents; About the Authors; Part I Dynamical Systems and Numerical Schemes; 1 Lyapunov Stability and Dynamical Systems; 1.1 Lyapunov Stability; 1.2 Autonomous Dynamical Systems; 1.3 Invariant Sets; 1.4 Limit Sets; 1.5 Attractors; 2 One Step Numerical Schemes; 2.1 Discretisation Error; 2.2 General One Step Schemes; 2.2.1 Taylor Schemes; 2.2.2 Schemes Derived by Integral Approximations; 2.3 Orders of Local and Global Convergence; 2.4 Consistency; 2.5 Numerical Instability; 2.6 Steady States of Numerical Schemes; Part II Steady States Under Discretisation; 3 Linear Systems

3.1 Linear ODE in mathbbR13.2 Linear ODE in mathbbC1; 3.3 The General Linear Case; 4 Lyapunov Functions; 4.1 Linear Systems Revisited; 4.2 Application: The Linear Euler Scheme; 4.3 Application: The Nonlinear Euler Scheme; 5 Dissipative Systems with Steady States; 6 Saddle Points Under Discretisation; 6.1 Saddle Points and the Euler Scheme; 6.1.1 A Nonlinear Example; 6.1.2 Shadowing; 6.2 General Case: Beyn's Theorem; Part III Autonomous Attractors Under Discretisation; 7 Dissipative Systems with Attractors; 7.1 Euler Scheme Dynamics; 7.2 Convergence of the Numerical Attractors

8 Lyapunov Functions for Attractors8.1 Lyapunov Stability of Sets; 8.2 Yoshizawa's Theorem; 9 Discretisation of an Attractor: General Case; Part IV Nonautonomous Limit Sets Under Discretisation; 10 Dissipative Nonautonomous Systems; 10.1 Nonautonomous Omega Limit Sets; 10.2 Asymptotic Invariance; 10.2.1 Asymptotic Positive Invariance; 10.2.2 Asymptotic Negative Invariance; 11 Discretisation of Nonautonomous Limit Sets ; 11.1 The Implicit Euler Scheme; 11.2 Upper Semi Continuous Converence of the Numerical Omega Limit Sets; 12 Variable Step Size Discretisation of Autonomous Attractors

12.1 Variable Time Step Limit Sets12.2 Upper Semi Continuous Convergence of the Numerical Omega Limit Sets; 13 Discretisation of a Uniform Pullback Attractor; Notes; References