main.toc (4344B)
1 \babel@toc {english}{}\relax 2 \contentsline {section}{\numberline {0}Notations and conventions}{2}{}% 3 \contentsline {section}{\numberline {1}Part 1}{2}{}% 4 \contentsline {subsection}{\numberline {1.1}Modelling}{2}{}% 5 \contentsline {subsection}{\numberline {1.2}Autonomous ODE in $\mathbb {R}^n$}{3}{}% 6 \contentsline {subsection}{\numberline {1.3}Invariant subspaces}{8}{}% 7 \contentsline {subsection}{\numberline {1.4}Stability of equilibria}{14}{}% 8 \contentsline {subsection}{\numberline {1.5}Polar coordinates}{19}{}% 9 \contentsline {subsection}{\numberline {1.6}Asymptotic behavior}{21}{}% 10 \contentsline {paragraph}{Chlorine dioxide-Iodine-Malonic-Acid reaction: $(X = I, Y = ClO_2^-)$}{23}{}% 11 \contentsline {paragraph}{Hilbert's 16th problem:}{25}{}% 12 \contentsline {subsection}{\numberline {1.7}LaSalle's invariance principle}{25}{}% 13 \contentsline {subsection}{\numberline {1.8}Hamiltonian systems in 2D}{27}{}% 14 \contentsline {subsection}{\numberline {1.9}Special Hamiltonian systems: Newtonian systems}{28}{}% 15 \contentsline {subsection}{\numberline {1.10}Gradient systems in ${\mathbb {R}^n}$}{29}{}% 16 \contentsline {subsection}{\numberline {1.11}First integral (or constant of motion)}{30}{}% 17 \contentsline {subsection}{\numberline {1.12}How to find centers?}{33}{}% 18 \contentsline {subsubsection}{\numberline {1.12.1}Planar S-systems}{33}{}% 19 \contentsline {subsubsection}{\numberline {1.12.2}Reversible systems}{33}{}% 20 \contentsline {paragraph}{Reversibility with respect to $x = y$ line:}{34}{}% 21 \contentsline {subsection}{\numberline {1.13}Stable and unstable manifolds}{34}{}% 22 \contentsline {subsection}{\numberline {1.14}Center manifold}{36}{}% 23 \contentsline {paragraph}{Bad news:}{37}{}% 24 \contentsline {paragraph}{Good news:}{37}{}% 25 \contentsline {subsection}{\numberline {1.15}Andronov--Hopf bifurcation}{38}{}% 26 \contentsline {paragraph}{Back to 2d:}{42}{}% 27 \contentsline {section}{\numberline {2}Part 2}{42}{}% 28 \contentsline {subsection}{\numberline {2.1}Ideas from the General theory of dynamical systems}{42}{}% 29 \contentsline {subsubsection}{\numberline {2.1.1}Continuous vs. discrete time DS}{42}{}% 30 \contentsline {subsubsection}{\numberline {2.1.2}Continuous time systems can also give you discrete systems: $(\phi _t)_{t\geq 0}$ semiflows of $X$}{43}{}% 31 \contentsline {subsubsection}{\numberline {2.1.3}Relations between systems}{43}{}% 32 \contentsline {subsubsection}{\numberline {2.1.4}Example: Mathematical Billiards "table" $Q \subseteq \mathbb {R}^2$, open}{44}{}% 33 \contentsline {paragraph}{Poincare sections:}{44}{}% 34 \contentsline {subsubsection}{\numberline {2.1.5}Questions and structure}{44}{}% 35 \contentsline {paragraph}{Coarse structure:}{44}{}% 36 \contentsline {paragraph}{Topological dynamics:}{45}{}% 37 \contentsline {subsection}{\numberline {2.2}Circle rotations}{45}{}% 38 \contentsline {subsubsection}{\numberline {2.2.1}Rational rotation}{45}{}% 39 \contentsline {subsubsection}{\numberline {2.2.2}Irrational Rotations}{45}{}% 40 \contentsline {subsubsection}{\numberline {2.2.3}Linear flows on the $2$-torus $\mathbb {T}^2$}{45}{}% 41 \contentsline {subsubsection}{\numberline {2.2.4}Some notions of topological dynamics}{46}{}% 42 \contentsline {subsubsection}{\numberline {2.2.5}Distribution of orbits}{46}{}% 43 \contentsline {paragraph}{An application:}{48}{}% 44 \contentsline {subsubsection}{\numberline {2.2.6}More general circle maps}{49}{}% 45 \contentsline {subsubsection}{\numberline {2.2.7}Circle homeomorphisms with periodic points}{54}{}% 46 \contentsline {subsection}{\numberline {2.3}Maps with complicated orbit structure}{56}{}% 47 \contentsline {subsubsection}{\numberline {2.3.1}Warmup}{56}{}% 48 \contentsline {subsubsection}{\numberline {2.3.2}Basic properties}{56}{}% 49 \contentsline {paragraph}{Periodic orbits:}{56}{}% 50 \contentsline {paragraph}{Question:}{56}{}% 51 \contentsline {subsubsection}{\numberline {2.3.3}Symbolic dynamics and coding}{57}{}% 52 \contentsline {paragraph}{Application:}{57}{}% 53 \contentsline {paragraph}{Metric on $\Omega _2$:}{58}{}% 54 \contentsline {subsubsection}{\numberline {2.3.4}The general uniformly expanding circle maps $T:\mathbb {T}\to \mathbb {T}$ of degree 2}{58}{}% 55 \contentsline {subsection}{\numberline {2.4}Outlook: Coding for other systems}{59}{}% 56 \contentsline {paragraph}{Warning:}{60}{}% 57 \contentsline {subsubsection}{\numberline {2.4.1}Outlook: Measurable dynamics (Ergotic theory)}{60}{}%