## 2 Method
### Day-to-Day Dynamics of Route Choice and Signal Control in a Two-Link Network  
We consider a two-link road network connecting a single Origin-Destination (O-D) pair with a total travel demand of 1 unit. The two links intersect at some point before reaching the destination, as shown in Figure 1. As a result of flow conservation, the flow pattern is uniquely determined by the value of a single variable; in what follows, we denote \(F\) as the flow on link 1 and \((1-F)\) as the flow on link 2. The intersection between the two links is controlled by a two-phase traffic signal operating with a fixed and given cycle length. For the sake of simplicity, lost times occurring during phase transitions are not considered, and minimum/maximum green splits are not enforced. The state of the traffic signal can thus be identified by the value of one of the two green splits; in what follows, we denote \(G\) as the green split assigned to link 1 and \((1-G)\) as the green split assigned to link 2. Figure 1: Two-link network with a signal-controlled intersection. Based on the above assumptions, expressions (1)-(3) can be rewritten in scalar form as follows:  
\[C_{1}^{t}=\beta K_{1}\Big{(}F^{t-1},G^{t-1}\Big{)}+(1-\beta)C_{1}^{t-1} \tag{11}\]  
\[C_{2}^{t}=\beta K_{2}\Big{(}1-F^{t-1},1-G^{t-1}\Big{)}+(1-\beta)C_{2}^{t-1} \tag{12}\]  
\[F^{t}=\alpha S\big{(}C_{1}^{t}-C_{2}^{t}\big{)}+(1-\alpha)F^{t-1} \tag{13}\]  
\[G^{t}=H\big{(}F^{t}\big{)} \tag{14}\]  
where we note that, according to (13), the fraction of \"habit-free\" drivers choosing link 1 on day \(t\) is expressed as a function of the difference between the perceived costs of the two links on day \(t\). Next, we define \(Z=C_{1}-C_{2}\) and \(V(F,G)=K_{1}(F,G)-K_{2}(1-F,1-G)\) and subtract (12) from (11), so that (11)-(13) are reduced to the following expressions:  
\[Z^{t}=\beta V\Big{(}F^{t-1},G^{t-1}\Big{)}+(1-\beta)Z^{t-1} \tag{15}\]  
\[F^{t}=\alpha S\big{(}Z^{t}\big{)}+(1-\alpha)F^{t-1} \tag{16}\]  
Substituting Equation (15) for \(Z^{t}\) into (16) and expressing \(G\) via the signal control policy (14), we obtain the following two-dimensional nonlinear dynamical system:  
\[Z^{t}=\beta V\Big{[}F^{t-1},H\Big{(}F^{t-1}\Big{)}\Big{]}+(1-\beta)Z^{t-1} \tag{17}\]  
\[F^{t}=\alpha S\Big{\{}\beta V\Big{[}F^{t-1},H\Big{(}F^{t-1}\Big{)}\Big{]}+(1- \beta)Z^{t-1}\Big{\}}+(1-\alpha)F^{t-1} \tag{18}\]  
Fixed points \(\widetilde{(Z,\widetilde{F})}\) of (17) and (18) are found by setting  
Figure 1:0  
Figure 1:1  
which results in the following equations:  
Figure 1:2  
Figure 1:3  
with the corresponding fixed-point green split of link 1 being given by  
Figure 1:4  
We now turn to the derivation of conditions for fixed point local stability based on the Jacobian matrix of (17) and (18)  
\[J= \begin{bmatrix}\frac{\partial Z^{t}}{\partial Z^{t-1}}&\frac{\partial Z^{t} }{\partial F^{t-1}}\\ \frac{\partial F^{t}}{\partial Z^{t-1}}&\frac{\partial F^{t}}{\partial F^{t-1 }}\end{bmatrix}  \tag{24}\]  
whose entries are  
\[\frac{\partial Z^{t}}{\partial Z^{t-1}}=1-\beta \tag{25}\]  
\[\frac{\partial Z^{t}}{\partial F^{t-1}}=\beta\bigg{(}\frac{\partial V}{ \partial F}+\frac{\partial V}{\partial H}\frac{dH}{dF}\bigg{)} \tag{26}\]\[\frac{\partial F^{t}}{\partial Z^{t-1}}=\alpha(1-\beta)\frac{dS}{dW} \tag{27}\] \[\frac{\partial F^{t}}{\partial F^{t-1}}=\alpha\frac{dS}{dW}\beta \bigg{(}\frac{\partial V}{\partial F}+\frac{\partial V}{\partial H}\frac{dH}{ dF}\bigg{)}+1-\alpha \tag{28}\]  
where  
\[W(F,Z)=\beta V[F,H(F)]+(1-\beta)Z \tag{29}\]  
As stated in Section 2.1, local stability is ensured if all eigenvalues of \(J\), evaluated at the fixed point, have a modulus smaller than 1.