Max Snijders, max.snijders@gmail.com
December 7^{th}, 2018
Prof. Dr. Ulrich Bauer  Prof. Dr. Oliver Junge  Prof. Dr. Konstantin Mischaikow  Max Snijders 
Technical University of Munich  Technical University of Munich  Rutgers University  Technical University of Munich & LudwigMaximilians University Munich 
\[ H^1(X; \mathbb{Z}) \simeq \lbrack X, S^1 \rbrack \]
\[ \frac{\partial}{\partial t} \begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} \sigma (y  x) \\ x (\rho  z)  y \\ x y  \beta z \end{pmatrix} \]
We use: \( \beta = \nicefrac{8}{3}, \sigma = 10, \rho = 28 \)
This work relies on a large body of preexisting work, including:
Paper  Persistent Homology 
Comparison against competition  Realworld data 
Thesis refs.: 
Backup slides: 
Normal slides: 
Def.  For a given point metric space \( (X, d) \) we denote by \( \mathrm{Rips}_r(X,d) \) the Rips complex of radius \(r \in \mathbb{R}\) with vertex set \(X\) and as simplices all subsets of \(X\) whose constituent vertices are pairwise within distance \(r\). 
Starting from a cocycle \( \eta \) which represents a cohomology class, which can be seen as a map on the complex' edges, we perform the following optimisation programme: \[ \begin{align} \bar{f} &= \mathrm{argmin}_{f \in C^0(\bar{X}, \Phi; \mathbb{R})} \eta + d_0 f^2_2 \\ \eta_2^2 &= \sum_{ab \in (\bar{X}, \Phi)}  \eta(ab) ^2 \end{align} \]
Backup Index
\( \eta_{\alpha},  \eta_{\alpha} ^2_2 = 1 \)

\( \eta_{\beta},  \eta_{\beta} ^2_2 = \nicefrac{1}{2} \)

\( \eta_{\gamma},  \eta_{\gamma} ^2_2 = \nicefrac{1}{7} \)

Coordinate \( \alpha \) s.t. \( \eta_{\gamma} = \eta_{\alpha} + d_0 \alpha\)

Cohomology is defined by dualizing the chains and the boundary maps between them by applying the \( \mathrm{Hom}\left( \_, G \right) \) functor as follows:
Intuitively, whereas an \(n\)chain is a sum of \(n\)simplices, an \(n\)cochain is a function taking values on each \(n\)simplex.
Backup Index
Def. 
A simplicial complex \( (K, \Phi) \) is a totally ordered vertex set \( K \) together with a family of nonempty subsets \(\Psi\) of \( K \) called the simplices such that the following hold:

Def.  We define the \(n\)chains of a simplicial complex \( (K, \Phi) \), \(C_n(K,\Phi)\) to be the group generated by the \(n\)simplices of \((K,\Phi)\) under formal summation. 
Def.  For a given \(n \in \mathbb{Z}\) we define the \(n\)^{th} boundary operator \( \partial_n: C_n(K, \Phi) \rightarrow C_{n1}(K,\Phi)\) on individual \(n\)simplices: \[ \partial_n: \{ v_0, ..., v_N \} \mapsto \sum_{i=0}^n (1)^i \{ v_0, ..., v_{i1}, v_{i+1}, ..., v_N \} \] and extend it linearly w.r.t. the summation on simplices. 
Def. 
For a given simplicial complex \( K, \Phi \) and integer \(n \in \mathbb{Z} \) we denote two special subgroups of \(C_n(K,\Phi)\), namely the \(n\)cycles \( Z_n(K, \Phi) = \mathrm{ker}\left( \partial_n: C_n(K, \Phi) \rightarrow C_{n1}(K, \Phi)\right)\) and the \(n\)boundaries \( B_n(K, \Phi) = \mathrm{im}\left( \partial_{n+1}: C_{n+1}(K, \Phi) \rightarrow C_{n}(K, \Phi)\right)\). We note that \( B_n(K, \Phi) \subseteq Z_n(K, \Phi) \) as \( \partial_{n} \circ \partial_{n+1} = 0 \) and so the following group is welldefined: \[ H_n(K, \Phi) = \nicefrac{Z_n(K, \Phi)}{B_n(K, \Phi)} \] we call this group the \(n\)^{th} homology group for \((K, \Phi)\). 
We will only need the first two homology groups, \( H_0 \) and \( H_1 \).
Backup Index
Def.  A lifted coordinate function \( \theta: \{ t_1, ..., t_N \} \rightarrow \mathbb{R} \) is said to be \(\epsilon\)increasing on \( J \subseteq \{ t_1, ..., t_N \} \) iff \( \frac{\theta(t')  \theta(t)}{t'  t} \ge \epsilon \) for all \(t, t' \in J\) whenever \(t' > t\). 
Def.  A lifted coordinate function \( \theta: \{ t_1, ..., t_N \} \rightarrow \mathbb{R} \) is said to be \(\epsilon\)cycling on an interval \( (a, b) \subseteq \mathbb{R} \) for \( a,b \in \{ t_1, ..., t_N \} \) if \( \theta \) is \(\epsilon\)increasing on \( \{t_1, ..., t_N\} \big_{(a,b)} \) and \( \theta(b)  \theta(a) \ge 1\). 
Def.  A cube \( \xi \) in the cover is said to be \(\epsilon\)cycling w.r.t. the function \(f: \bar{X} \rightarrow \mathbb{R}\) and the boxed trajectory \( \gamma: \{t_1, ..., t_N\} \rightarrow \bar{X} \) if there is some interval \( (a,b) \subseteq \mathbb{R} \) s.t. \( \gamma((a,b)) \ni \xi \) where \( f \circ \gamma: \{ t_1, ..., t_N \} \rightarrow \mathbb{R} \) is \( \epsilon \)cycling on the interval \( (a, b) \). 
Def.  A cube \( \xi \) in the cover is said to be \(\epsilon\)transient w.r.t. the function \(f: \bar{X} \rightarrow \mathbb{R}\) and the boxed trajectory \( \gamma: \{t_1, ..., t_N\} \rightarrow \bar{X} \) if for a maximal interval \( (a,b) \subseteq \mathbb{R} \) s.t. \( \gamma((a,b)) \ni \xi \) and \( f \circ \gamma: \{ t_1, ..., t_N \} \rightarrow \mathbb{R} \) is \( \epsilon \)cycling on the interval \( (a, b) \) we have that \( f(\xi)  f(\gamma(b)) \le 1 \), i.e. the box is in the last cycle of the interval . 
Def.  A dynamical system is some system with timedependent state, i.e. some system \( S \) with a transition map \(f: S \rightarrow S\) or a continuoustime evolution map \(f: \mathbb{R} \times S \rightarrow S\). 
In our case, our statespace consists of \( \mathbb{R}^n \) and our timeevolution map is provided by an ODE.
We want to decompose the statespace of a given dynamical system into regions where we can understand behaviour and then study how the system transitions between these regions.
Def.  Let \(f, g: X \rightarrow Y\) be two maps between topological spaces. \(f\) and \(g\) are said to be homotopyequivalent or homotopic if there exists a continuous map \(H: X \times \lbrack 0, 1 \rbrack \rightarrow Y\) such that \(H \big_{X \times \{0\}} = f \) and \( H \big_{X \times \{1\}} = g \). 
Def.  Two topological spaces \(X\) and \(Y\) are said to be homotopyequivalent or homotopic if there exist continuous maps \(f : X \rightarrow Y \) and \(g: Y \rightarrow X\) such that \(f \circ g: Y \rightarrow Y\) is homotopic to the identity map on \(Y\) and \(g \circ f: X \rightarrow X\) is homotopic to the identity map on \(X\). 
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