## 1 Introduction  
Transmission line circuits, switched capacitor circuits, neuron model circuits, and constrained circuits are notable as the nonlinear circuits of which behavior may be described by one-dimensional (1-D) maps [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11]. Furthermore, in [12], by using real imperfect integrated devices induced by unavoidable manufacturing imperfections that plausibly have so called hidden dynamics with parasitic effects and nonidealities, the possibility of designing imperfect electronic circuits generating megahertz chaotic oscillations, without the use of additional capacitors or inductors, has been explored and discussed. By using one-dimensional (1-D) map methods, some lossless transmission line circuits with a short at one side terminal [5, 6, 7, 8, 9, 10, 11] (or, with a dc bias voltage source in series with a load resistor at one side terminal [7]) have been actively studied. On the other hand, many weak or strong definitions such that a 1-D map is mathematically _chaotic_ are still being studied [13, 14, 15, 16, 17, 18]. In this paper, we use the following definition that is well known as most traditional and most definite since [PERSON] [19]. Definition 1: A 1-D discrete dynamical system or 1-D map \(\varphi\) on \(M\) is said to be _chaotic_ if there exists an invariant subset \(\Lambda\subset M\) on which some iterates of \(\varphi\) is _topologically conjugate_ to the shift dynamics \((\sigma,\Sigma_{m})\) with \(m\) symbols. The shift dynamics \((\sigma,\Sigma_{m})\) with m symbols has the following properties [13; 15; 16; 17]: (1) There are countably many periodic points with different periods in \(\Sigma_{m}\). (2) The set of all periodic points in dense in \(\Sigma_{m}\). (3) \(\sigma\) is topologically transitive. Namely, there exists a point \(c\in\Sigma_{m}\) where its orbit is dense in \(\Sigma_{m}\). (4) \(\sigma\) is topologically expansive. (5) \(\Sigma_{m}\) is homeomorphic to a Cantor set. (6) The topological entropy of \(\sigma\) is \(h(\sigma)=\log_{\epsilon}m\). For a 1-D discrete dynamical system \(\varphi\), it is well known that \"\(\varphi\)_has a homoclinic point_\" implies Definition 1, and they are often equivalent [15; 20; 21]. Definition 1 is not based on the theoretical measure but is based on the topological viewpoint, and it is called _formal chaos_ or _topological chaos_, which may be not observable generally [13]. Thus, we call the chaos in Definition 1_formal chaos_. Binary sequences based on _observable_ chaotic behavior produced by 1-D maps coming from the nonlinear circuits are reported and well known to have good statistical properties useful for applications relative to several digital communication systems [4; 13]. However, the sufficient conditions for the existence of _observable chaos_ are so strong that most 1-D maps coming from general nonlinear circuits cannot be rigorously proved to possess _observable_ chaos. Indeed, for general 1-D maps such that the absolute slops of them are either greater or less than unity, it is very difficult to rigourously prove the existence of _observable chaos_, except for 1-D maps such as the logistic map family or 1-D piecewise linear maps such that all their absolute slopes of are greater than unity. Once the existence of _formal chaos_ is guaranteed, the degree of the observability of the chaos can be checked by the existence of maximal positive Lyapunov exponent [14] through computer simulations from the viewpoint of engineering. Hence, we pay attention to the sufficient condition that _formal chaos_ in Definition 1 exists in a 1-D map. In [5; 6], a lossless transmission line circuit with a short at one side terminal and with a three-segment piecewise linear (3 SPWL) resistor (so called Nagumo's 3 SPWL resistor) at another side terminal is considered. The \(v-i\) characteristic of the (3 SPWL) resistor is not only in point symmetry at the origin but also in all four quadrants of a \((v,i)\) coordinate system.