# Fluid Net Models: From Behavioral Properties to Structural Objects
## 3 Bifurcations: Operation Mode Properties
### Bifurcations in a Manufacturing TCPN Model  
Let us introduce intuitively the idea of bifurcation. There can exist path losing/gaining marks when considering a subnet associated with an operation mode. For instance, the places of the subnet in Figure 4(a) lose markings if \(t_{2}\) is fired, while their marking is increased by firing \(t_{5}\), \(t_{4}\), and \(t_{3}\), in that order. As mentioned in Section 2, every operation mode of a TCPN is a linear system \(\hat{\mathbf{m}}=\mathbf{C}\mathbf{\Lambda}\mathbf{\Pi}_{\mathcal{C}}\mathbf{m}\). Depending on the existence of path losing/gaining marks, the firing rates of transitions may be changed to obtain a stable/unstable linear system. For example, considering the net depicted in Figure 2(b), one of its configurations, its associated subnet, and its dynamic matrix are depicted in Figure 4.  
The dynamic matrix of the operation mode associated with the configuration is \(\mathbf{A}_{\mathcal{C}}=\mathbf{C}\mathbf{\Lambda}\mathbf{\Pi}_{\mathcal{C}}\) and it is depicted in Figure 4(b), where \(\mathbf{\Lambda}(t_{2})=\lambda_{2}\) is considered as a parameter. Notice that in the subnet shown in Figure 4(a), for example, the total marking is decremented by the firing of \(t_{2}\) and increased by the firing of \(t_{5}\). In the following lines, we explain how this phenomenon is captured by \(\mathbf{A}_{\mathcal{C}}\). Matrix \(A_{\mathcal{C}}\) has six eigenvalues associated with a basis of P-flows of the net (in this particular case, they are P-semiflows of the net). Of course, all these are independent from \(\lambda_{2}\) and are equal to zero. A similarity transformation \(\boldsymbol{T}\) to obtain a transformed matrix \(\boldsymbol{TA_{\mathcal{C}}T^{-1}}\) for the analysis of the remaining eigenvalues is the following. Note that the last six rows are (transposed) minimal P-semiflows of the net. \[\boldsymbol{T}= \begin{bmatrix}\boldsymbol{I_{6}}&\boldsymbol{0}\\ \boldsymbol{B}_{y}^{T}\end{bmatrix} = \begin{bmatrix}1&0&0&0&0&0&0&0&0&0&0&0\\ 0&1&0&0&0&0&0&0&0&0&0\\ 0&0&1&0&0&0&0&0&0&0&0\\ 0&0&0&1&0&0&0&0&0&0&0\\ 0&0&0&0&1&0&0&0&0&0&0\\ 0&0&0&0&0&1&0&0&0&0&0\\ 0&1&0&0&0&0&0&1&0&0&0\\ 0&0&0&0&0&1&0&0&0&0&1\\ 1&1&1&1&1&1&0&0&0&0&0\end{bmatrix} \]  
\[\boldsymbol{TA_{\mathcal{C}}T^{-1}}= \begin{bmatrix}-2&-1&-1&-1&-1&0&0&0&0&0&1 \\ 1&0&\lambda_{2}&0&0&\lambda_{2}&0&0&0&-\lambda_{2}&0&0\\ 0&0&-\lambda_{2}&1&0&-\lambda_{2}&0&-1&0&\lambda_{2}&0&0\\ 0&0&0&-1&1&0&0&1&-1&0&0\\ 0&0&1&0&-1&1&0&0&1&-1&0&0\\ 0&0&-1&0&0&-2&0&0&0&1&0&0\\ -&-&-&-&-&-&-&-&-&-&-&-\\ 0&0&0&0&0&0&0&0&0&0&0\\ 0&0&0&0&0&0&0&0&0&0&0\\ 0&0&0&0&0&0&0&0&0&0&0\\ 0&0&0&0&0&0&0&0&0&0&0\\ \end{bmatrix} \]  
Figure 5: Subnet of configuration \(\mathcal{C}_{1,81,10,11,86,67}\) of the TCPN model of Figure 1b and the dynamic matrix of its associated operation mode. (**a**) Subnet. (**b**) Dynamic Matrix. In order to obtain the eigenvalues of the transformed matrix, the characteristic polynomial of \(\mathbf{TA_{C}T^{-1}}\) is obtained:  
\[P(s)=s^{6}(s^{6}+(\lambda_{2}+6)s^{5}+(5\lambda_{2}+14)s^{4}+(10\lambda_{2}+15)s ^{3}+(10\lambda_{2}+6)s^{2}+(5\lambda_{2}-1)s+(\lambda_{2}-1))\]  
The eigenvalues of the transformed matrix are the roots of \(P(s)\). Obviously, the transformed matrix has six eigenvalues equal to zero (\(s=0\)), as expected, and they do not depend on \(\lambda_{2}\). Moreover, \(P(s)\) can be factorized as follows:  
\[P(s)=s^{6}(s+1)^{3}(s^{3}+(\lambda_{2}+3)s^{2}+(2\lambda_{2}+2)s+\lambda_{2}-1).\]  
Hence, there are three other eigenvalues that do not depend on \(\lambda_{2}\), and they have the same value \(s=-1\). One of these eigenvalues is due to the firing rate of transition \(t_{1}\) (\(\lambda_{1}=1\)), which determines the velocity of the marking evolution of \(p_{1}\). In the same way, the firing rate of transition \(t_{7}\) (\(\lambda_{7}=1\)) gives another eigenvalue in \(s=-\lambda_{7}=-1\). The other three non-zero eigenvalues depend on the parameter \(\lambda_{2}\) (the speed of retrieving a part). In Figure 6, the variation of these three last eigenvalues is depicted.