## 2 Lossless Transmission Circuit Equations with Terminal Conditions
### Transmission Line Circuit Equation  
The relation of the voltage \(v\) and the current \(i\) in the line in Figure 6 becomes the following:  
\[\frac{\partial v}{\partial\tau}=-\frac{\partial i}{\partial x^{\prime}},\ \frac{ \partial i}{\partial\tau}=-\frac{\partial v}{\partial x}, \tag{1}\]  
where \(Z=\sqrt{\frac{t}{\xi}}\) is the characteristic impedance of the line; \(s=\frac{1}{\sqrt{LC}}\) is the propagation velocity of waves in the line; \(T=\frac{2l}{s}\), \(v=v_{C}/V_{max}\), \(i=i_{L}/I_{max}\), \(x=x_{l}/2l\), \(\tau=t/T\), \(L\) are the series of inductance per unit length of the line; \(C\) is the parallel capacitance per unit length of the line; \(l\) is line length; \(v_{C}\) is the voltage in the line; \(i_{L}\) is the current in the line; \(x_{l}\) is the coordinate in the line; \(v\) is the non-dimensional voltage in the line; \(i\) is the non-dimensional current in the line; \(x\) is the non-dimensional coordinate in the line; and \(\tau\) is non-dimensional time. Accordingly, by eliminating \(i\) (or \(v\)), we have the following. \[\frac{\partial^{2}v(\tau,x)}{\partial\tau^{2}}=\frac{\partial^{2}v(\tau,x)}{ \partial x^{2}},\ \bigg{(}\frac{\partial^{2}i(\tau,x)}{\partial\tau^{2}}=\frac{ \partial^{2}i(\tau,x)}{\partial x^{2}}\bigg{)}. \tag{2}\]  
The following Equations (3) and (4) provide the boundary conditions of Figure 6. Equations (3) and (4) also provide the boundary conditions of Figure 5 as follows. At one side terminal, we have the following: (i) a dc bias voltage source in series with 3 SPWL resistor (N) or (T): \(\xi\
eq 0\) and (ii) 3 SPWL resistor (N) or (T): \(\xi=0\). At another side terminal, we have the following: (1) short: \(E=0\), \(R=0\); (2) open: \(E=0\), \(R\rightarrow\infty\); (3) a load resistor: \(E=0\), \(R\
eq 0\); (4) a dc bias voltage source: \(E\
eq 0\), \(R=0\) ; or (5) a dc bias voltage source in series with a load resistor: \(E\
eq 0\), \(R\
eq 0\). \[v(\tau,0) = E-Ri(\tau,0), \tag{3}\] \[i(\tau,\frac{1}{2}) = f_{k}(v(\tau,\frac{1}{2}),\zeta,\eta,\xi),\ (k=N,T). \tag{4}\]\[f_{k}(v,\zeta,\eta,\zeta)=I_{max}F_{k}(V_{max}(v-E_{0}))=\]  
(N) Nagumo's 3 SPWL Resistor \(f_{N}\):  
\[\left\{ \begin{array}{ll}\frac{v+(1-\zeta)b_{p}}{r}+\frac{\zeta b_{p}}{r}&,v<-( 1-\zeta)b_{p},\\ -\frac{\zeta b_{p}}{r}+\frac{\zeta b_{p}}{r}&,-(1-\zeta)b_{p}\leq v\\ &\leq(\eta+\zeta)b_{p},\\ \frac{(v-(\eta+\zeta)b_{p})}{r}-\frac{\zeta tb_{p}}{r}&,(\eta+\zeta)b_{p}<v, \end{array} \right. \tag{5}\]  
Here, \(\zeta>0\), \(\eta>-1\), \(r=\frac{R_{1}}{R_{max}}(>0)\), \(b_{p}=\frac{E_{p}^{-}}{V_{max}}(>0)\), \(\eta b_{p}=\frac{E_{p}^{+}}{V_{max}}(>0>-b_{p})\). (T) TD 3 SPWL Resistor \(f_{T}\):  
\[\left\{ \begin{array}{ll}\frac{v}{r}-\frac{\zeta b_{p}}{r}&,v<(1+\zeta)b_{p}, \\ -\frac{\zeta((v-(1+\zeta)b_{p})}{r}+\frac{b_{p}}{r}&,(1+\zeta)b_{p}\leq v\\ &\leq(\eta+\zeta)b_{p},\\ \frac{2(v-(\eta+\zeta)b_{p})}{r}-\frac{\zeta(\eta-1)b_{p}}{r}+\frac{b_{p}}{r}&,(\eta+\zeta)b_{p}<v,\end{array} \right. \tag{6}\]  
Here, \(\zeta>0\), \(\eta>1\), \(r=\frac{R_{1}}{R_{max}}(>0)\),\(b_{p}=\frac{E_{p}}{V_{max}}(>0)\), \(\eta b_{p}=\frac{E_{p}}{V_{max}}(>b_{p})\). In addition in (N) or (T), \(\zeta b_{p}=E_{0}\), \(R=R_{L}/R_{max}\), \(E_{0}=E_{B0}/V_{max}\), \(E=E_{B}/V_{max}\), \(E_{B0}\), and \(E_{B}\) are the dc bias voltage sources; \(R_{L}\) is the load resistor; and \(f_{k}(v,\zeta,\eta,\zeta)\) is the non-dimensional current of 3 SPWL resistor, \((k=N,T)\). **3. 1-D Map Describing the Behavior of the Transmission Line Circuit**  
_3.1. Derivation of 1-D Map_  
Here, we discuss how to derive the 1-D map or the difference equation of which the dynamics completely describes the dynamics of Equations (2)-(6). The 1-D wave equation of Equation (2) has the d'Alembert's solution, which is described as the following. \[v(\tau,x) = \psi(\tau-x)-\phi(\tau+x),\] \[i(\tau,x) = \frac{\psi(\tau-x)+\phi(\tau+x)}{Z}. \tag{7}\]  
Equation (7) can be solved with regard to the scattering variables, that is,\(\psi\) and \(\phi\). Then, we have the following description. \[\left( \begin{array}{c}\psi\\ \phi\end{array}