for $\Delta_j > 10$, the models are significantly different[@KennethBurnham2004]. In context of SINDy, the AIC was applied successfully for model selection[@ManganModelSelection.2017].

Concept for identification of hysteresis-controlled systems {#sec:Concept}
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This section provides a technical concept for system identification of hysteretic systems using SINDy, named SINDyHybrid. Based upon theoretical considerations of SINDy and hybrid systems, we developed an approach to incorporate tailored basis functions into the function library. To assess the results, we define criteria for model quality. SINDy has been used by [@BruntonSINDy.2016] to model the dynamics of systems with a bifurcation parameter $\mu$, which changes the qualitative behavior of the system’s dynamics. In order to incorporate $\mu$ into the observables, the equation $\dot{x} = f(x)$ was extended by $\mu$, such that $$\begin{aligned}
    \dot{x} &= f(x, \mu), \label{eq:sindyBifurcation}\\
    \dot{\mu} &= 0. \label{eq:MuDerivative}\end{aligned}$$ The function library $\Theta$ then consists of functions of $x$ and $\mu$. The value of $\mu$ however remains unchanged, as indicated by Eq. . A more general formulation has been used in [@BRUNTON_SINDYc], where feedback control signals were included by choosing $$\dot{\mu} = g(x). \label{eq:MuDerivativeSindyC}$$

Expanding the theory for hybrid systems
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We now further generalize the approach of Eq. by choosing $$\dot{\mu} = g(x, \mu), \label{eq:MuDerivativeGeneral}$$ which 