# Automobile Technological Transition Scenarios Based on Environmental Drivers
## 2 Materials and Methods
In particular, we conducted the empirical structure confirmation test and empirical parameters confirmation test. The theoretical structure confirmation test was performed through the literature review mentioned above to explain the causal rules construction. The theoretical parameters confirmation was carried out during the construction of the model by defining the application ranges, as shown above. The dimensional consistency test appears together with the equations explanation, verifying the units proposed. We also carried out extreme value tests allowing us to recognize the limits and scope of the model. When possible, the bifurcation analysis is better than the system sensitivity tests because the bifurcation analysis can identify all the probable system behaviors, while the sensitivity analysis only allows visualizing the transitory state under the parameter variation in a limited range, using the Monte Carlo method. For this reason, our bifurcation analysis shown below supports the validity of our proposed model. Finally, we carried out phase relationship tests to compare the behavior between the state variables and verify the behavior consistency through phase portraits, as shown in Section 3. We did not conduct behavior validation with real time series, because it is still too early to have enough national data that could be comparable with the behaviors shown by our prospective mathematical model. The mathematical model of Equation (4) is a continuous dynamical system. For this reason, we analyzed the equilibrium points and their stability and determined that the system has bifurcations in codimension two. A dynamical system (see [60; 61]) is a triple \((X,T,\varphi)\), where \(X\) is the state space, \(T\) is an ordered set representing time, and  
\[\varphi:T\times X\longrightarrow X \tag{5}\] \[(t,x)\longmapsto\varphi_{t}(x)\]  
is a parameterized family of evolution operators that satisfies the following:* The system state does not change spontaneously \[\varphi_{0}(x)=x\quad\text{for all }x\in X\] (6)
* The evolution law of the system does not change in time \[CAI=a\cdot x\cdot(1-CEA)\qquad CAO=b\cdot x \tag{2}\]0 (7)  
In the model of Equation (4), the state space is defined by the state variables of conventional automobiles \(x\) and electric automobiles \(y\); that is, \(X=\{(x,y)\in\mathds{R}\}\). The set of time is a subset of the real numbers, so the dynamical system will be said to be continuous; that is, \(T=\{t\in\mathds{R}:t\geq 0\}\). Finally, the evolution law is given by the solution of the differential system of Equation (4). The mathematical perspective from dynamical systems is relevant in a trend behavior analysis because it allows characterizing the qualitative structure of the system's behavior beyond just understanding the behavior for a specific time, which could be misleading. Roughly speaking, the dynamical systems identify invariant sets and characterize their stability. Invariant sets are states that do not change as the system evolves, i.e., state sets that remain in time. Examples include equilibrium points, periodic orbits, and strange attractors. The stability can characterize the system trajectories as attracting, repelling, fixing (centers), or combining these three options. Since the equilibrium points occur when the system does not change, it must be satisfied that \(\dot{x}\) and \(\dot{y}\) are zero [60; 61], making the differential problem an algebraic issue, as follows:  
\[ \begin{array}{c}\dot{x}=0\\ \implies\\ \dot{y}=0\end{array} \quad x(\alpha-\delta y)=0\\ y(\beta-\lambda x)=0 \tag{8}\]  
The stability of the equilibrium points is calculated with the eigenvalues of the evolution matrix of the differential system around the equilibrium points [60; 61], i.e., consider Equation (4) around the equilibrium points in the form \(\dot{X}=JX\), where \(X=(x,y)^{T}\) is the state vector, and \(J\) is the Jacobian defined as:  
\[J(x,y)=\left(