\[ \frac{\partial}{\partial t} \begin{pmatrix}
x \\ y \\ z
\end{pmatrix} = \begin{pmatrix}
\sigma (y - x) \\
x (\rho - z) - y \\
x y - \beta z
\end{pmatrix} \]
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:
\[ \begin{align}
\bar{f} &= \mathrm{argmin}_{f \in C^0(\bar{X}, \Phi; \mathbb{R})} ||\eta + d_0 f||^2_2 \\
||\eta||_2^2 &= \sum_{ab \in (\bar{X}, \Phi)} || \eta(ab) ||^2
\end{align} \]
Cohomology is defined by dualizing the chains and the boundary maps between them by applying the \( \mathrm{Hom}\left( \_, G \right) \) functor as follows:
We dualize the chains by replacing them with maps from the chains to our coefficient group \(G\) giving us the cochains \( C^n(K, \Phi; G) = \mathrm{Hom}\left( C_n(K, \Phi \right), G) \).
Furthermore, we dualize the boundary operator to get the coboundary operator , which behaves as follows:
\[
\begin{align}
d_n : C^n(K, \Phi; G) &\rightarrow C^{n+1}(K, \Phi; G) \\
c &\mapsto c \circ \partial_{n+1}
\end{align}
\]
Intuitively, whereas an \(n\)-chain is a sum of \(n\)-simplices, an \(n\)-cochain is a function taking values on each \(n\)-simplex.
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:
\( k \in K \implies \{ k \} \in \Phi \)
\( a, \in \Phi, b \subseteq a \implies b \in \Phi \)
We call a simplex \(a \in \Phi\) an \(n\)-simplex if \(|a| = n+1\).
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 \( \partial_n: C_n(K, \Phi) \rightarrow C_{n-1}(K,\Phi)\) on individual \(n\)-simplices:
\[
\partial_n: \{ v_0, ..., v_N \} \mapsto \sum_{i=0}^n (-1)^i \{ v_0, ..., v_{i-1}, v_{i+1}, ..., v_N \}
\]
and extend it linearly w.r.t. the summation on simplices.
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 \( Z_n(K, \Phi) = \mathrm{ker}\left( \partial_n: C_n(K, \Phi) \rightarrow C_{n-1}(K, \Phi)\right)\) 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 well-defined:
\[
H_n(K, \Phi) = \nicefrac{Z_n(K, \Phi)}{B_n(K, \Phi)}
\]
we call this group the \(n\)th homology group for \((K, \Phi)\).
We will only need the first two homology groups, \( H_0 \) and \( H_1 \).
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 .
A dynamical system is some system with time-dependent state, i.e. some system \( S \) with a transition map \(f: S \rightarrow S\) or a continuous-time evolution map \(f: \mathbb{R} \times S \rightarrow S\).
In our case, our state-space consists of \( \mathbb{R}^n \) and our time-evolution map is provided by an ODE.
We want to decompose the state-space of a given dynamical system into regions where we can understand behaviour and then study how the system transitions between these regions.
Let \(f, g: X \rightarrow Y\) be two maps between topological spaces. \(f\) and \(g\) are said to be homotopy-equivalent 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 homotopy-equivalent 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 \(X\).
