0 $} \\
\end{cases} \label{eq:indicator1} \\
\bar{I}(x) &= \begin{cases} 
1,  & \text{if $x < 0 $} \\
0,  & \text{if $x \geq 0 $} \\
\end{cases}
\end{split}\end{aligned}$$

A hysteron can be formed out of two indicator functions, which are not both true at the same time. For all matching indicator pairs $I_\alpha, I_\beta$, the hysteron $H_{\alpha, \beta}$ is built. This way, $I(\tilde{x}_2)$ and $\bar{I}(\tilde{x}_1)$ would also be combined. For each hysteron, a complementary hysteron $\bar{H}_{\alpha, \beta}$ will also be built. The evaluation is done according to

$$\label{eq:hysteron}
H(k)_{\alpha, \beta} = \begin{cases}
1, & \text{if $I_\beta = 1$.} \\
0, & \text{if $I_\alpha = 1$.} \\
H(k-1), & \text{for $I_\beta = I_\alpha = 0$.}
\end{cases}$$

for each step $k>1$.

This is equivalent to the formulation of the relay hysteron in Eq. . Presuming alternating switches between the states, $H(1)_{\alpha, \beta}$ can be initialized with the opposite state of the first occuring switch. This presumption has to be reviewed depending on the use case.

Robust handling of transitions – the proximity hysteron {#sec:proximity}
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A problem in the design of hysterons is the fact, that the critical threshold values $\alpha$ and $\beta$ from Eq. are only reached for a short moment. Due 