# An Algebraic Approach to Identifiability
## 4 Examples  
### The Van der Pol Oscillator  
The Van der Pol oscillator is considered with an increasing number of parameters. In its simplest form, the equations of motion read  
\[\dot{x}_{1} =x_{2}\] \[\dot{x}_{2} =-x_{1}-\epsilon x_{2}\left(x_{1}^{2}-1\right)\] \[y =x_{1}\]  
where \(\epsilon\in\Theta=\mathbb{R}\) is a parameter. The chain of ideals defined by Equation (5) starting with \(\langle x_{1}-\tilde{x}_{1}\rangle\) generated by the extended output map stabilizes after the fifth Lie derivate with respect to the parametric vector field. This stabilized ideal is already real and reads  
\[\mathfrak{I}=\langle x_{1}-\tilde{x}_{1},x_{2}-\tilde{x}_{2},x_{1}(\epsilon- \bar{\epsilon}),x_{2}(\epsilon-\bar{\epsilon})\rangle\]  
and can be written as an intersection  
\[\langle x_{1},x_{2},\tilde{x}_{1},\tilde{x}_{2}\rangle\cap\langle x_{1}- \tilde{x}_{1},x_{2}-\tilde{x}_{2},\epsilon-\bar{\epsilon}\rangle\]  
of prime ideals. The latter of these ideals corresponds to equal states and parameters, i.e., the ideal \(\mathfrak{J}\). The first one describes the not identifiable part, which consists of the (sole) equilibrium \((x_{1},x_{2})=(0,0)\) for each system copy. Thus, the system is locally identifiable except at the equilibrium, where the output trajectory is identically zero for all parameters \(\epsilon\).  
The system is modified to include two parameters and reads  
\[\dot{x}_{1} =x_{2}\] \[\dot{x}_{2} =-x_{1}-x_{2}\Big{(}ax_{1}^{2}-b\Big{)}\]  
further on, while the output map is preserved. The chain of ideals generated by Lie derivatives of the extended output map stabilizes after the seventh Lie derivative has been added. This ideal is not yet radical. However, the radical takes a similar form  
\[\mathfrak{I}=\langle x_{1},x_{2},\tilde{x}_{1},\tilde{x}_{2}\rangle\cap\langle x _{1}-\tilde{x}_{1},x_{2}-\tilde{x}_{2},a-\bar{a},b-\bar{b}\rangle\]  
as for the previous system. Again, the parameters are structurally identifiable, and the equilibrium is the only exceptional point.  
If an additional parameter \(c\) is included such that  
\[\dot{x}_{1} =x_{2}\] \[\dot{x}_{2} =-cx_{1}-x_{2}\Big{(}ax_{1}^{2}-b\Big{)},\]  
Lie derivatives up to order nine are required until the chain of ideals stabilizes. The radical of the stabilized Lie derivative can be written as an intersection of five irreducible components:\[\mathfrak{J} =\langle x_{1},x_{2},\bar{x}_{1},\bar{x}_{2}\rangle\] \[\cap\big{\langle}x_{1}-\bar{x}_{1},x_{2}-\bar{x}_{2},a-\bar{a},b- \bar{b},c-\bar{c}\big{\rangle}\] \[\cap\big{\langle}x_{1}-\bar{x}_{1},x_{2}-\bar{x}_{2},a,\bar{a},x_ {1}(c-\bar{c})-x_{2}(b-\bar{b}),x_{1}x_{2}\bar{b}-x_{1}^{2}\bar{c}-x_{2}^{2},\] \[x_{2}(b\bar{b}-\bar{b}^{2}-c+\bar{c})-x_{1}\bar{c}(b-\bar{b}),bc \bar{b}-c\bar{b}^{2}-b^{2}\bar{c}+b\bar{b}\bar{c}-c^{2}+2c\bar{c}-\bar{c}^{2} \big{\rangle}\] \[\cap\big{\langle}x_{1}-\bar{x}_{1},x_{2}-\bar{x}_{2},b,\bar{b}, ax_{2}^{2}+c,\bar{a}x_{2}^{2}+\bar{c},a\bar{c}-\bar{a}c\big{\rangle}\] \[\cap\big{\langle}x_{1}-\bar{x}_{1},x_{2}-\bar{x}_{2},c,\bar{c} \big{\rangle}\]  
The first two components are similar to the previous cases. As the ideal \(\mathfrak{J}\) occurs in the second prime component, the system is locally identifiable again, except at the equilibrium. The remaining three components describe other constellations for that local identifiability fails. As can be seen, these correspond to degenerate parameter constellations \(a=0\), \(b=0\), or \(c=0\), respectively.