# Fluid Net Models: From Behavioral Properties to Structural Objects
## 4 Discontinuities and Non-Monotonicities
### A \"Dual\" Perspective: The Initial Marking Variation Case  
In this case, formal definitions for discontinuities and non-monotonicities are given taking now the initial marking as a parameter. More precisely, these equilibrium throughput behaviors are defined when \(\mathbf{\lambda}\) is fixed and \(\mathbf{m}_{0}\) varies.  
**Definition 7**.: _Let \(\langle N,\mathbf{\lambda},\mathbf{m}_{0}\rangle\) be an MTS TCPN system with \(\mathbf{\lambda}\) fixed. Assume that the initial marking \(\mathbf{m}_{0}\) is arbitrary. Its equilibrium throughput is:_  
* Monotonic _with respect to the_ initial marking_, denoted as_ \(M(\mathbf{m}_{0})\)_, if_ \(\forall(\mathbf{m}_{0},\mathbf{m}_{0}^{\prime})\)_,_ \(\mathbf{m}_{0}\leq\mathbf{m}_{0}^{\prime}\implies\alpha(\mathbf{m}_{0})\leq\alpha(\mathbf{m}_{ 0}^{\prime})\)_;_
* Deadlock-free monotonic _with respect to the_ initial marking_, denoted as_ \(DFM(\mathbf{m}_{0})\)_, if_ \(\forall(\mathbf{m}_{0},\mathbf{m}_{0}^{\prime})\) _with_ \(\mathbf{m}_{0}\leq\mathbf{m}_{0}^{\prime}\)_,_ \(\alpha(\mathbf{m}_{0})>\mathbf{0}\implies\alpha(\mathbf{m}_{0}^{\prime})>\mathbf{0}\)_;_
* Continuous _with respect to the_ initial marking_, denoted as_ \(C(\mathbf{m}_{0})\)_, if for every_ \(\mathbf{m}_{0}\)_, given_ \(\epsilon>0\)_, there is_ \(\delta>0\) _such that_ \(\|\mathbf{m}_{0}-\mathbf{m}_{0}^{\prime}\|_{2}<\delta\implies\|\alpha(\mathbf{m}_{0})- \alpha(\mathbf{m}_{0}^{\prime})\|_{2}<\epsilon\)_, where_ \(\|\cdot\|_{2}\) _is the euclidean norm._  
A discontinuity of the equilibrium throughput due to smooth variations of the initial marking can occur, as can be seen in Figure 2a. A similar procedure to the previous one, for the analysis of this behavior with respect to the initial marking variations, can be carried out using the homothecy property to obtain the following result (arrow (c) in Figure 10).  
Figure 10: Relationships between configurations and behavioral properties. Continuity, monotonicity, and deadlock-free monotonicity w.r.t. the firing rates are abbreviated as \(C(\lambda)\), \(M(\lambda)\), and \(DFM(\lambda)\), respectively. In the same way, continuity, monotonicity, and deadlock-free monotonicity w.r.t. the initial marking are abbreviated as \(C(\mathbf{m}_{0})\), \(M(\mathbf{m}_{0})\), and \(DFM(\mathbf{m}_{0})\), respectively. Abbreviations \(nB\) and \(B\) stand for nB-Problematic systems and B-Problematic systems, respectively.  
**Theorem 3** ([15]).: _Let \(\langle N,\mathbf{\lambda},\mathbf{m}_{0}\rangle\) be an MTS TCPN system with \(\mathbf{\lambda}\) fixed and \(\mathbf{m}_{0}\) arbitrary. If the equilibrium throughput is monotonic with respect to \(\mathbf{m}_{0}\), then it is continuous with respect to \(\mathbf{m}_{0}\)._  
Furthermore, directly derived from Definition 7, a monotonic system never decreases its system throughput when the initial markings are increased, thus its throughput cannot be zero. Hence, the system is also deadlock-free monotonic (arrow (d) in Figure 10).  
**Proposition 5**.: _Let \(\langle N,\mathbf{\lambda},\mathbf{m}_{0}\rangle\) be an MTS TCPN system with \(\mathbf{\lambda}\) fixed and \(\mathbf{m}_{0}\) arbitrary. If the equilibrium throughput is monotonic with respect to \(\mathbf{m}_{0}\), then it is deadlock-free monotonic with respect to \(\mathbf{m}_{0}\)._  
Proof.: It derives directly from Definition 7.  
The lower part of the diagram of Figure 10 shows these results, arrows (c) and (d). Our running example shows that, when initial marking variations are considered, a non-monotonic systems does not imply a discontinuous system, arrow (e). The implication represented by arrow (f) is discussed in Section 5. The relationships between these behaviors and the bifurcations of the equilibrium marking, give us an insight that the configurations of the net could provide some information about it. The structural properties that configurations provide are related to the behavioral properties observed in the equilibrium throughput of the TCPN system. Section 5 is devoted to clarify this fact.