set of points within the $\epsilon$-range is reached. An exception occurs, when the actual threshold $\alpha$ or $\beta$ is reached. In this case, the switch is done regardless of whether this is the extremum within this set.

![Switches of the proximity hysteron[]{data-label="fig:proximity_hysteron"}](proximityHysteronSVG){width="\linewidth"}

Due to the introduction of the proximity hysteron, state transitions can even be localized under harsh conditions – if the threshold points do not occur in the measurement data. The connection between signal, threshold values, indicator functions and hysterons is visualized in Fig.\[fig:buildingOfHysterons\].

![Interaction between signal, threshold values, indicator functions and hysterons[]{data-label="fig:buildingOfHysterons"}](simSignal "fig:"){width="\linewidth"}\
![Interaction between signal, threshold values, indicator functions and hysterons[]{data-label="fig:buildingOfHysterons"}](simIndicatorsSVG "fig:"){width="\linewidth"}\
![Interaction between signal, threshold values, indicator functions and hysterons[]{data-label="fig:buildingOfHysterons"}](simHysteronSVG.pdf "fig:"){width="\linewidth"}

System Identification with SINDy
--------------------------------

Once the hysterons are built and other preprocessing measures have been taken, the SINDy algorithm can start. The schematics are shown in \[fig:sindyFlow\]. Creating the data matrices includes choosing the right state vector $X$. The past values $H(k-1)_i$ for $i = 1, ..., m$ (see Eq. ) of all $m$ hysterons have to be included, holding information about the system’s current discrete state. Past measurement values $x(k-1), ..., x(k-q) $ up to a defined horizon $q$ can also be considered, 