## 6 Conclusions  
1. We have described an implicit 1-D map of the incident and reflected waves that is derived from a lossless transmission line circuit with a dc bias voltage source in series with a load resistor at one side terminal and with a three-segment piecewise linear resistor at another side terminal.
2. We have provided Theorem 1 establishing a 1-D map such as an incident wave: \(\psi(\tau-1)\to\) a reflected wave: \(\phi(\tau)\); Corollary 1 establishing 1-D map such as an incident wave: \(\psi(\tau-1)\to\) a reflected wave: \(\phi(\tau)\); and Corollary 2 establishing 1-D map such as a voltage: \(v(\tau-\frac{1}{2},\frac{1}{2})\to\) a voltage: \(v(\tau+\frac{1}{2},\frac{1}{2})\) in the case of Nagumo's 3 SPWL resistor or TD 3 SPWL Resistor at the side terminal: \(x=\frac{1}{2}\).
3. We have provided Theorem 3 such as an easy-to-use sufficient condition implying _formal chaos_ existence in Definition 1.
4. We have provided Theorem 2 to guarantee that for any initial point, every orbit by the iteration of the 1-D map ultimately penetrates in an invariant interval.
5. We have provided Theorems 2 and 5 to guarantee that the dynamics of the second iterate map of the 1-D map on an invariant subset of the invariant interval has _formal chaos_.
6. We have found that _formal chaos_ exists under the condition such that \(V_{max}=10\) [V], \(Z=300\) [\(\Omega\)], \(b_{p}=\frac{3}{50}\), \(r=\frac{3}{25}\), \(\eta=\frac{20}{3}\), \(\zeta=\frac{1}{10}\), \(E=\frac{143}{320}\), and \(R=\frac{7}{16}\).
7. We have obtained the codimension of one type of four various bifurcation diagrams including the found _formal chaos_, with one of the bifurcation parameters: \(\zeta\), \(\eta\), \(\xi\), or \(E\). From each bifurcation diagram with the Lyapunov exponent, the obsavability of the found _formal chaos_ is considered to be rather high.
8. As with the case of [12] such that the hidden dynamics of the circuit based on the integrated device has been unveiled on the basis of an analogy with the well-known Colpitts oscillator with the chaotic oscillations, we think that the hidden dynamics of imperfect transmission lines with parasitic effects and nonidealities inside realintegrated devices can be unveiled on the basis of an analogy with the transmission line circuit of Equations (2)-(6) with _formal chaos_ and that the parameters of the hidden dynamics can also be estimated by synchronizing a transmission line circuit with the chaotic oscillations acquired from the experimental circuit.
* We will report bifurcation processes with each of bifurcation parameters, \(Z\) or \(R\), because nonlinear phenomena such as _intermittency of chaos_ or _blue sky bifurcation_ are observed in simulations.
* We will establish a method to find the bifurcation parameter ranges such that _formal chaos_ exists for applications relative to several digital communication systems.  
[PERSON] supervised this research. [PERSON] and [PERSON] initiated the research, and contributed of a lossless transmission line circuit conceptualization and investigation of the transmission line circuit. [PERSON] and [PERSON] managed all of the research. [PERSON] provided theorems, and proofs. [PERSON] and [PERSON] contributed development of MAPLE simulation algorithm. [PERSON] also carried out the simulations. [PERSON] contributed validation of the methodology. [PERSON], [PERSON] and [PERSON] contributed the all of analysis result. The original manuscript was written by [PERSON], [PERSON] and [PERSON]. And all of authors reviewed and edited the manuscript. All authors have read and agreed to the published version of the manuscript.  
This research was supported by the research budget of the Graduate School of Electrical and Computer Engineering, Shonan Institute of Technology.  
Not Applicable.  
Data Availability StatementData is contained within the article.  
The authors declare no conflict of interest.