## 2 Mathematical Description of Third Order and Fourth Order Model  
The lambda diode is denotation commonly used for a two-terminal circuit element that is composed by two transistors (usually two field-effect transistors) with opposite conductivity. As depicted in Figure 1a, configuration is specific. Emitters of both transistors are connected as well as base-collectors. The lambda diode exhibits an area of differential negative resistance in the ampere-voltage characteristics, similarly, as a tunnel diode. Typically, this area is relatively narrow (few volts) with negative conductance slope about hundreds of micro-siemens. Of course, each mentioned quantity depends on the type of transistors. Generally, a lambda diode is biased to work, as negative resistance operates with the larger voltages such that it cannot replace the tunnel diode directly. Intuitive application of the negative resistor is inside harmonic oscillators where it compensates frequency-independent loses of series or parallel inductor-capacitor tanks. The upcoming section investigates dynamic behavior of such an oscillator if two complementary bipolar transistors in the equivalent circuit model presented in the recent paper [27] is assumed. Therefore, it contributes to research where class of chaotic systems with several transistors is presented, like in paper [28]. In this work, the brute force method combining random search and the heuristic approach was successfully utilized to find up to forty-nine unique circuits generating chaotic signals. The generalized bipolar transistor proposed in paper [27] is based on the full admittance matrix description of the non-unilateral transistor understood as a two-port. Based on the equivalent circuit for negative resistance oscillator with lambda diode provided in Figure 1b, the following system of ordinary differential equations can be derived:  
\[ \begin{array}{c}C_{be}\frac{d}{dt}v_{1}\,=\,(\overline{y}_{22}\,+\,\widetilde {y}_{11}\,-\,\overline{y}_{12})\cdot v_{1}\,+\,(\overline{y}_{11}\,+\, \widetilde{y}_{22}\,-\,\widetilde{y}_{12})\cdot(v_{2}\,-\,v_{1})\,+\\ \overline{y}_{21}(v_{2}\,-\,v_{1})\,-\,\widetilde{y}_{21}(-v_{1}),\\ C\frac{d}{dt}v_{2}\,=\,\,(\overline{y}_{11}\,+\,\widetilde{y}_{22})\cdot(v_ {1}\,-\,v_{2})\,+\,\overline{y}_{12}\cdot v_{1}\,+\,\widetilde{y}_{21}(-v_{1} )\,-\,i_{L},\\ L\frac{d}{dt}i_{L}\,=\,v_{2},\end{array}  \tag{1}\]  
Figure 1: Negative resistance oscillator with the lambda diode: (**a**) principal schematic with generalized bipolar transistors, (**b**) equivalent small-signal model with nonlinear voltage controlled current sources. where scalar nonlinear trans-conductances can be expressed as a cubic polynomial with zero quadratic and absolute term, that is:  
\[ \begin{array}{l}\overline{y}_{21}(v_{2}\,-\,v_{1})\,=\,\overline{a}\cdot(v_{2 }\,-\,v_{1})^{3}\,+\,\overline{b}\cdot(v_{2}\,-\,v_{1}),\\ \widetilde{y}_{21}(-v_{1})\,=\,\widetilde{a}\cdot(-v_{1})^{3}\,+\,\widetilde{b }\cdot(-v_{1}),\end{array}  \tag{2}\]  
where \(\overline{a}\), \(\overline{b}\), \(\widetilde{a}\), \(\widetilde{b}\) are real numbers. Without loss of generality (and chance to reach chaos), unity normalized values of accumulation elements can be considered, i.e., \(C_{be}=C=1\) F and \(L=1\) H. Results presented in paper [27] suppose generalized transistors that behave like ideal current sources at output terminal, that is parameter \(y_{22}\) is assumed zero. In addition, the two-port possess significant normalized value of backward trans-conductance, i.e., parameter \(y_{12}\) is a subject of optimization (see clarification below). Linear analysis of this system can start with determination of the fixed points, in our case, the following solutions:  
\[