Titlebook: Attractors Under Discretisation; Xiaoying Han,Peter Kloeden Book 2017 The Author(s) 2017 One step numerical schemes.Autonomous dynamicl sy

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Linear SystemsStability of linear systems by eigenvalue conditions is introduced. Stability conditions for one and two dimensional, as well as general linear systems, are established.

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Lyapunov FunctionsLyapunov functions are defined and used to investigate the stability of the zero solution to Euler schemes for linear and nonlinear ODEs.

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Dissipative Systems with Steady StatesThe preservation or stability of the zero solution to Euler schemes for dissipative systems is established using Lyapunov functions.

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Saddle Points Under DiscretisationSaddle points for Euler schemes for ODEs are discussed. Numerical stable and unstable manifolds are illustrated through a set of examples, and compared to the stable and unstable manifolds of the ODEs. The shadowing phenomenon is briefly illustrated. Finally, Beyn’s Theorem is presented.

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Dissipative Systems with AttractorsEuler schemes for dissipative ODE systems with attractors are presented and shown to possess numerical attractors that converge to the ODE attractors upper semi continuously. A counterexample shows that the numerical attractor need not convergence lower semi continuously.

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Discretisation of an Attractor: General CaseKloeden and Lorenz’s Theorem on the existence of a maximal numerical attractor of one step numerical schemes for general autonomous ODEs with a global attractor is stated and proved.

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Variable Step Size Discretisation of Autonomous AttractorsDiscretising autonomous ODEs with variable step size results in discrete nonautonomous semi-dynamical systems. Numerical omega limit sets for such dynamical systems are constructed and shown to converge to the attractor for the ODEs upper semi continuously.