## 4 Formal Chaos Existing Conditions of 1-D Maps
### Formal Chaos Existing Conditions of a 1-D Map Family  
As mentioned precisely in the later subsections, for a lossless transmission 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, the existence of a three-segment piecewise linear 1-D map describing the dynamics of the circuit, the existence of the invariant interval of the 1-D map, and the coordinate transformation from the invariant interval to the normalized interval \(I=[0,1]\) are provided. Therefore, in this subsection, we pay our attention to the normalized 1-D map on the normalized interval \(I\) after applying the coordinate transformation to the three-segment piecewise linear 1-D map. We discuss the _formal chaos_ verification of the normalized three-segment piecewise linear 1-D map \(H\) on \(I\) such that the absolute slopes of the map are either greater or less than unity. In general such map \(H\) satisfies Theorem 3 under very narrow range-limited circuit parameters. Instead of \(H\), focusing on \(H^{2}\) of which the circuit parameters are expected to have the broader range implying that \(H^{2}\) satisfies Theorem 3, we provide Theorem 5 to guarantee that the dynamics of \(H^{2}\) on \(I\) is _topologically conjugate_ with respect to the shift dynamics \((\sigma,\Sigma_{2})\) with two symbols. Since one tends to consider it as routine work for providing the sufficient inequality conditions of Theorem 5, once again we note that,in order to obtain the inequality conditions, one needs an elaborate symbolical manipulation technique. We consider the normalized 1-D map as follows (see in Figure 7):  
\[x_{n+1}=H(x_{n})=\left\{ \begin{array}{ll}H_{1}(x_{n})=\dfrac{(1-n_{1})x_{n}}{k_ {1}}+n_{1}&,x_{n}<k_{1},\\ H_{2}(x_{n})=\dfrac{x_{n}+k_{1}}{k_{2}-k_{1}}+1&,k_{1}\leq x_{n}\leq k_{2},\\ H_{3}(x_{n})=\dfrac{n_{2}(x_{n}-k_{2})}{1-k_{2}}&,k_{2}<x_{n},\end{array} \right. \tag{19}\]  
where \(I=[0,1]\subset\mathcal{R}\), \(x_{n},x_{n+1}\in I\), and \(\mathcal{R}\) is the set of real numbers. **Theorem 4**. : _If \(1-k_{1}(k_{2}-k_{1})>n_{1}>n_{2}>(k_{2}-k_{1})(1-k_{2})\), \(n_{1}>1-n_{2}>k_{2}>n_{2}>k_{1}>1-n_{1}\) and \(0<k_{1},k_{2},n_{1},n_{2}<1\), then the following is the case:_  
\[x_{n+1}=H^{2}(x_{n})=\left\{ \begin{array}{ll}H_{1}(H_{1}(x_{n})),\text{when} \ x_{n}<-\frac{(k_{1}-n_{1})k_{1}}{n_{1}-1},\\ H_{2}(H_{1}(x_{n})),\text{when}-\frac{(k_{1}-n_{1})k_{1}}{n_{1}-1}\leq x_{n}\\ <-\frac{(k_{2}-n_{1})k_{1}}{n_{1}-1},\\ H_{3}(H_{1}(x_{n})),\text{when}-\frac{(k_{2}-n_{1})k_{1}}{n_{1}-1}\leq x_{n}<k _{1},\\ H_{3}(H_{2}(x_{n})),\text{when}\ k_{1}\leq x_{n}<k_{2}-k_{2}(k_{2}-k_{1}),\\ H_{2}(H_{2}(x_{n})),\text{when}\ k_{2}-k_{2}(k_{2}-k_{1})\leq x_{n}<k_{2}(1-k _{1})+k_{1}^{2},\\ H_{1}(H_{2}(x_{n})),\text{when}\ k_{2}(1-k_{1})+k_{1}^{2}\leq x_{n}<k_{2},\\ H_{1}(H_{3}(x_{n})),\text{when}\ k_{2}\leq x_{n}<k_{2}+\frac{k_{1}(1-k_{2})}{n_ {2}},\\ H_{2}(H_{3}(x_{n})),\text{when}\ k_{2}+\frac{k_{1}(1-k_{2})}{n_{2}}\leq x_{n} <k_{2}+\frac{k_{2}(1-k_{2})}{n_{2}},\\ H_{3}(H_{3}(x_{n})),\text{when}\ k_{2}+\frac{k_{2}(1-k_{2})}{n_{2}}<x_{n},\end{array} \right. \tag{20}\]  
\[-\frac{(k_{1}-n_{1})k_{1}}{n_{1}-1} < -\frac{(k_{2}-n_{1})k_{1}}{n_{1}-1}<0, \tag{21}\] \[0<k_{1} < k_{2}-k_{2}(k_{2}-k_{1}),\] (22) \[k_{2}-k_{2}(k_{2}-k_{1}) < k_{2}(1-k_{1})+k_{1}^{2}<k_{2},\] (23) \[k_{2} < k_{2}+\frac{k_{1}(1-k_{2})}{n_{2}}<1,\] (24) \[1 < k_{2}+\frac{k_{2}(1-k_{2})}{n_{2}}, \tag{25}\]  
_such that \(|DH_{1}H_{1}|=\left|\dfrac{(1-n_{1})^{2}}{k_{1}^{2}}\right|<1\), \(|DH_{2}H_{2}|=\left|\dfrac{1}{(k_{2}-k_{1})^{2}}\right|>1\), \(|DH_{3}H_{3}|=\left|\dfrac{n_{2}^{2}}{(1-k_{2})^{2}}\right|<1\), \(|DH_{3}H_{1}|=|DH_{1}H_{3}|=\left|\dfrac{n_{2}(1-n_{1})}{k_{1}(1-k_{2})}\right|<1\), \(|DH_{3}H_{2}|=|DH_{2}H_{3}|=\left|\dfrac{-n_{2}}{(k_{2}-k_{1})(1-k_{2})}\right|>1\), \(|DH_{1}H_{2}|=|DH_{2}H_{1}|=\left|\dfrac{-(1-n_{1})}{k_{1}(k_{2}-k_{1})}\right|>1\), where \(DH_{i}H_{j}\) is the slope of \(H_{i}(H_{j}(x_{n}))\), \(1\leq i,j\leq 3\)._  
Figure 7: Normalized 1-D map.