\contentsline {section}{\numberline {1.}Introduction}{2}
\contentsline {subsection}{\numberline {1.1.}Chaos --- the dynamical systems approach}{2}
\contentsline {subsection}{\numberline {1.2.}Conservative dynamical systems}{4}
\contentsline {subsection}{\numberline {1.3.}Dissipative dynamical systems}{4}
\contentsline {subsection}{\numberline {1.4.}Spatio-temporal chaos}{7}
\contentsline {subsection}{\numberline {1.5.}I want dynamics --- Why study numerics?}{7}
\contentsline {section}{\numberline {2.}Dynamical systems}{8}
\contentsline {subsection}{\numberline {2.1.}A one-dimensional map}{9}
\contentsline {subsection}{\numberline {2.2.}A one-dimensional diffeomorphism}{10}
\contentsline {subsection}{\numberline {2.3.}A dissipative two-dimensional diffeomorphism}{10}
\contentsline {subsection}{\numberline {2.4.}An area-preserving diffeomorphism}{10}
\contentsline {subsection}{\numberline {2.5.}A two-dimensional continuous-time dynamical system}{11}
\contentsline {subsection}{\numberline {2.6.}An infinite-dimensional continuous-time dynamical system}{12}
\contentsline {subsection}{\numberline {2.7.}One- and $1\frac {1}{2}$-degree of freedom Hamiltonian systems}{12}
\contentsline {subsection}{\numberline {2.8.}$N$-body Hamiltonian systems}{13}
\contentsline {section}{\numberline {3.}Forward vs.\ backward error}{13}
\contentsline {subsection}{\numberline {3.1.}Forward \& backward error and dynamical systems}{14}
\contentsline {section}{\numberline {4.}Attractors and dimensions}{15}
\contentsline {subsection}{\numberline {4.1.}Approximation of attractors}{15}
\contentsline {subsection}{\numberline {4.2.}Fractals and approximating dimensions}{17}
\contentsline {subsection}{\numberline {4.3.}Other estimates of dimension \& their relationships}{18}
\contentsline {subsection}{\numberline {4.4.}Amount of data}{19}
\contentsline {subsection}{\numberline {4.5.}Time-series and embeddings}{19}
\contentsline {subsection}{\numberline {4.6.}Random or chaotic?}{23}
\contentsline {section}{\numberline {5.}Stable, unstable \& centre subspaces \& manifolds}{24}
\contentsline {subsection}{\numberline {5.1.}Computing the stable and unstable subspaces}{25}
\contentsline {subsection}{\numberline {5.2.}Equations for the stable manifold}{26}
\contentsline {subsection}{\numberline {5.3.}Stable and unstable manifolds for maps and diffeomorphisms}{27}
\contentsline {subsection}{\numberline {5.4.}Stable and unstable manifolds for periodic orbits}{27}
\contentsline {subsection}{\numberline {5.5.}Hyperbolicity for attractors}{28}
\contentsline {subsection}{\numberline {5.6.}Non-hyperbolic fixed points --- centre manifolds}{29}
\contentsline {section}{\numberline {6.}Lyapunov exponents}{31}
\contentsline {subsection}{\numberline {6.1.}Computing Lyapunov exponents}{32}
\contentsline {section}{\numberline {7.}Continuation and bifurcation}{34}
\contentsline {subsection}{\numberline {7.1.}Bifurcation jargon}{34}
\contentsline {subsection}{\numberline {7.2.}Some basic bifurcations}{35}
\contentsline {subsection}{\numberline {7.3.}Continuation}{37}
\contentsline {subsection}{\numberline {7.4.}Applications of continuation}{38}
\contentsline {subsection}{\numberline {7.5.}Continuation techniques}{39}
\contentsline {subsection}{\numberline {7.6.}When a path meets a bifurcation\dots }{40}
\contentsline {section}{\numberline {8.}Inertial manifold theory}{40}
\contentsline {subsection}{\numberline {8.1.}Technical ideas}{42}
\contentsline {subsection}{\numberline {8.2.}Approximate inertial manifolds}{43}
\contentsline {section}{\numberline {9.}Finding shadows}{43}
\contentsline {subsection}{\numberline {9.1.}Why shadowing can be done}{44}
\contentsline {subsection}{\numberline {9.2.}Numerical methods for shadows}{46}
\contentsline {subsection}{\numberline {9.3.}Shadowing autonomous ODEs}{46}
\contentsline {section}{\numberline {10.}Concluding remarks}{46}
\contentsline {section}{References}{49}
\contentsline {section}{Index}{54}