## 2 Method
### A Simple Specification of the Dynamic Process of Route Choice and Signal Control  
In this section, we show how the analysis developed in Section 2.2 can be further specified through the definition of explicit functional forms to provide the necessary elements for the subsequent numerical tests of the dynamical system's behavior. We assume that the travel cost functions of the two links have the following forms:  
\[K_{1}(F,G)=a+\frac{bF}{QG} \tag{41}\]  
\[K_{2}(F,G)=a+\frac{b(1-F)}{Q(1-G)} \tag{42}\]  
where \(a,\ b\) are positive parameters and \(Q\) is the link saturation flow. The latter is a measure of the link's physical capacity without consideration of signal control so that the actual link capacity is obtained as the product of saturation flow and green split. Note that we assume simplified cost functions for the sake of analytical tractability of the model. More complex expressions representing in a realistic way the delay incurred by vehicles at signal-controlled intersections are adopted in the numerical tests presented in Section 4. The share of travel demand assigned to link 1 as a function of the difference between mean perceived costs is specified through a Logit model with parameter \(\theta\) as follows:  
\[S(C_{1}-C_{2})=S(Z)=\frac{1}{1+\exp(\theta Z)} \tag{43}\]  
The green split of link 1 is determined by means of the Logit-like signal control policy introduced in Section 2.3 as follows:  
\[G=H(F)=\frac{\exp\left(\frac{\gamma F}{Q}\right)}{\exp\left(\frac{\gamma F}{Q} \right)+\exp\left[\frac{\gamma(1-F)}{Q}\right]}=\frac{1}{1+\exp\left[\frac{ \gamma(1-2F)}{Q}\right]} \tag{44}\]  
Subtracting (42) from (41) and substituting expression (44) for \(G\), we obtain (see Appendix A for details)  
\[V[F,H(F)]=\frac{b}{Q}\left\{F\left[\exp\left(\frac{\gamma(2F-1)}{Q}\right)+ \exp\left(\frac{\gamma(1-2F)}{Q}\right)+2\right]-\exp\left(\frac{\gamma(2F-1) }{Q}\right)-1\right\} \tag{45}\]  
If we introduce (43) and (45) into (17) and (18), the day-to-day evolution of route choice and signal control is described by the following dynamical system:  
\[Z^{t}=\beta\frac{b}{Q}\Big{\{}F^{t-1}\Big{[}\exp\left(-U\Big{(}F^{t-1}\Big{)} \right)+\exp\left(U\Big{(}F^{t-1}\Big{)}\right)+2\Big{]}-\exp\left(-U\Big{(}F ^{t-1}\Big{)}\right)-1\Big{\}}+(1-\beta)Z^{t-1} \tag{46}\]  
\[F^{t}=\frac{\alpha}{1+\exp[\theta W(F^{t-1},Z^{t-1})]}+(1-\alpha)F^{t-1} \tag{47}\]  
having defined  
\[U(F)=\frac{\gamma(1-2F)}{Q} \tag{48}\]  
\[W(F,Z)=\beta\frac{b}{Q}\left\{F[\exp(-U(F))+\exp(U(F))+2]-\exp(-U(F))-1\right\} +(1-\beta)Z \tag{49}\]  
where (49) can be recognized as being a specific instance of (29). A fixed point of (46) and (47) is a pair \((\widetilde{Z},\widetilde{F})\), such that  
\[\widetilde{Z}=\frac{b}{Q}\bigg{\{}\widetilde{F}\Big{[}\exp\left(-U(\widetilde {F})\right)+\exp\left(U(\widetilde{F})\right)+2\Big{]}-\exp\left(-U(\widetilde {F})\right)-1\bigg{\}} \tag{50}\]  
\[K_{2}(F,G)=a+\frac{b(1-F)}{Q(1-G)} \tag{42}\]0  
In order to analyze fixed point local stability, we compute the entries of the Jacobian matrix (24) applied to (46) and (47)  
\[K_{2}(F,G)=a+\frac{b(1-F)}{Q(1-G)} \tag{42}\]1  
\[K_{2}(F,G)=a+\frac{b(1-F)}{Q(1-G)} \tag{42}\]2  
\[K_{2}(F,G)=a+\frac{b(1-F)}{Q(1-G)} \tag{42}\]3  
\[K_{2}(F,G)=a+\frac{b(1-F)}{Q(1-G)} \tag{42}\]4Next, we focus our analysis on the pair \((\widetilde{Z}=0,\widetilde{F}=0.5)\), which is a solution to (50) and (51), and we derive stability conditions for this fixed point.