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Analysis of infinite dimensional dynamical systems by set-oriented numerics / von Adrian Ziessler ; [Gutachter: Prof. Dr. Michael Dellnitz, Prof. Dr. Oliver Junge]. Paderborn, 2018
Content
1 Introduction
2 Classical set-oriented techniques
2.1 Theoretical background
2.2 The subdivision algorithm
2.3 The continuation method
2.4 Computation of invariant measures
2.4.1 Stochastic transition functions and probability measures
2.4.2 The transfer operator
2.4.3 Numerical approximation of invariant measures
2.4.4 Convergence result
3 From finite to infinite dimensional embeddings
3.1 Taken's embedding theorem
3.2 Extension to fractal sets
3.2.1 Prevalence
3.2.2 Box-counting dimension
3.3 Infinite dimensional embedding theory
3.3.1 The thickness exponent
3.3.2 An infinite dimensional embedding result for linear maps
3.3.3 An infinite dimensional delay embedding result
4 The core dynamical system
5 Set-oriented techniques for embedded invariant sets
5.1 Extension to continuous dynamical systems
5.2 Computation of embedded attractors via subdivision
5.3 Continuation for embedded unstable manifolds
6 Applications
6.1 Delay differential equations
6.1.1 Numerical realization of the delay-coordinate map R
6.1.2 Numerical realization of the map E
6.1.3 Examples
6.2 Partial differential equations
6.2.1 Numerical realization of the observation map R
6.2.2 Numerical realization of the map E
6.2.3 Examples
7 Improving the numerical efficiency
7.1 A modified selection step for the subdivision algorithm
7.2 Development of a sequential procedure
7.2.1 Computation of new observations
7.2.2 Creating a new box covering
7.3 Koopman operator based continuation step
7.3.1 The Koopman operator
7.3.2 The trust region
7.3.3 Examples
8 Conclusion and outlook
8.1 The core dynamical system
8.2 Set-oriented techniques for embedded invariant sets
8.3 Improving the numerical efficiency
8.4 Future work
Bibliography
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