Symbolic CTQ-analysis
Denote a discrete dynamical system in the form of a mapping: \begin{aligned} \mathbf{s}_{k+1}=\mathbf{f}\left(\mathbf{s}_{k},\,\mathbf{p}\right), \end{aligned} with the properties: \begin{aligned} \mathbf{s}\in\mathrm{S}\subset\mathbb{R}^N,\quad \mathbf{p}\in\mathrm{P}\subset\mathbb{R}^L,\quad k\in\mathrm{K}\subseteq\mathbb{Z},\, n=\overline{1,\,N},\, l=\overline{1,\,L}, \end{aligned} where \(\mathbf{s}\) is a state variable of the system and \(\mathbf{p}\) is a vector of parameters.
With the mapping, we associate its trajectory in space \(\mathrm{S}\times\mathrm{K}\), which has the form of a semisequence~\(\{\mathbf{s}_{k}\}^K_{k=1}\), \(k=\overline{1,\,K}\).
Define the initial mapping, which encodes (in terms of the final T-alphabet) the shape of the \(n\)-th component of a sequence \(\{\mathbf{s}_{k}\}^K_{k=1}\): \begin{aligned} \left\lbrace\mathbf{s}^{(n)}_{k-1},\,\mathbf{s}^{(n)}_{k},\,\mathbf{s}^{(n)}_{k+1}\right\rbrace \Rightarrow T^{\alpha\varphi}_k|_n,\quad T^{\alpha\varphi}_k = \left[T^{\alpha\varphi}_k|_1,\,\ldots,\,T^{\alpha\varphi}_k|_N\right]. \end{aligned} Thus, the T-alphabet includes the following set of symbols: \begin{aligned} \mathrm{T}^{\alpha\varphi}_o=\{\mathtt{T0}, \,\mathtt{T1}, \,\mathtt{T2}, \,\mathtt{T3N}, \,\mathtt{T3P}, \,\mathtt{T4N}, \,\mathtt{T4P}, \,\mathtt{T5N}, \,\mathtt{T5P}, \,\mathtt{T6S}, \,\mathtt{T6}, \,\mathtt{T6L}, \,\mathtt{T7S}, \,\mathtt{T7}, \,\mathtt{T7L}, \,\mathtt{T8N}, \,\mathtt{T8P}\}. \end{aligned} The graphic diagrams illustrating the geometry of the symbols \(T^{\alpha\varphi}|_n\) for the \(k\)-th sample and the \(n\)-th phase variable are shown in Figure.

As follows from the figure, we can distinguish the subset of symbols: \begin{aligned} \mathrm{T}^{\alpha\varphi}_c=\{\mathtt{T3N}, \,\mathtt{T3P}, \,\mathtt{T5N}, \,\mathtt{T5P}, \,\mathtt{T6S}, \,\mathtt{T6L}, \,\mathtt{T7S}, \,\mathtt{T7L}\}. \end{aligned}
In addition to the symbols \(T^{\alpha\varphi}|_n\), we introduce the symbols \(Q^{\alpha\varphi}|_n\): \begin{aligned} Q^{\alpha\varphi}_k|_n\equiv T^{\alpha\varphi}_k|_n\rightarrow T^{\alpha\varphi}_{k+1}|_n,\quad Q^{\alpha\varphi}_k = \left[Q^{\alpha\varphi}_k|_1,\,\ldots,\,Q^{\alpha\varphi}_k|_N\right]. \end{aligned} All admissible transitions constitute a set of symbols of the alphabet\(\mathrm{Q}^{\alpha\varphi}_o\ni Q^{\alpha\varphi}|_n\). These transitions are shown in Figure.

The subset \(\mathrm{Q}^{\alpha\varphi}_o\) which is correspond \(\mathrm{T}^{\alpha\varphi}_c\) alphabet we denote through \(\mathrm{Q}^{\alpha\varphi}_c\).
The interactive diagram (Wolfram CDF-technology) is demonstrated principle of encoding initial sequence \(\{\mathbf{s}_{k}\}^K_{k=1}\) to T- and Q-alphabets.
Let us introduce a finite graph: \begin{aligned} \Gamma^{TQ}|_n=\left\langle\mathrm{V^\Gamma}|_n,\,\mathrm{E^\Gamma}|_n\right\rangle,\quad \mathrm{V^\Gamma}|_n\subseteq\mathrm{T}^{\alpha\varphi}_o,\, \mathrm{E^\Gamma}|_n\subseteq\mathrm{Q}^{\alpha\varphi}_o, \end{aligned} where \(\mathrm{V^\Gamma}|_n\) is the vertex set and \(\mathrm{E^\Gamma}|_n\) is the edge set. According to its topology, the graph \(\Gamma^{TQ}|_n\) is a connected directed graph without multiple arcs but with loops. Let us denote the graph \(\Gamma^{TQ}|_n\) corresponding to the full alphabets \(\mathrm{T}^{\alpha\varphi}_o\) and \(\mathrm{Q}^{\alpha\varphi}_o\) by \(\Gamma^{TQ}_o\).
The graph \(\Gamma^{TQ}|_n\) is a particular symbolic TQ-image of the dynamical system with respect to its \(n\)-th phase variable. The set of graphs \begin{aligned} \Gamma^{TQ}=\left[\Gamma^{TQ}|_1,\,\ldots,\,\Gamma^{TQ}|_N\right], \end{aligned} is a complete symbolic TQ-image of the dynamical system.
In formalism of the symbolic CTQ-analysis we are studying several constructive extensions:
TQ-bifurcation;
TQ-complexity;
T-synchronization;
Q-predict;
Q-control.
In addition: The papers is on theme the project.