## 3 Numerical Analysis and Localization of Chaotic Behavior  
Assume that the symbolic expression for the mathematical model in the form of system of the first order ordinary differential equations is available. Let us start with numerical values associated with generalized transistor presented in paper [27]. We can put suitable dynamical flow quantifier (calculation of Lyapunov exponents) inside a parallel processing loop in Matlab, create a seventh dimensional hyperspace of internal system parameters using full grid with decimal parameter step, and run repeated calculations. This numerical routine finds chaotic behavior for the several distinct sets of parameters. For both chaotic cases mentioned below, we adopted the additional condition \(\vec{a}\ =\ \widetilde{a}\) and this choice reduce parameter space investigated by search algorithm by one. This also shrinks possible cases of third order chaotic dynamical systems to those having a single fixed point located at the origin, namely those with a saddle spiral local geometry with a stability index one. Obviously, the fourth order chaotic system possess more equilibrium points including one located at origin. Firstly, note that all circuit parameters are still defined using normalized values, with respect to both impedance and frequency. Two sets of parameters that lead to the chaotic behavior were discovered for third order dynamical system. The first one is:  
\[ \begin{array}{l}\overline{y}_{11}\ =\ 1.0,\ \overline{y}_{12}\ =\ 2.2,\ \overline{y}_{22}\ =\ 0.0,\ \overline{a}\ =\ -1.0,\ \overline{b}\ =\ 1.5,\\ \widetilde{y}_{11}\ =\ 0.4,\ \widetilde{y}_{12}\ =\ 2.0,\ \widetilde{y}_{22}\ =\ 0.0,\ \overline{a}\ =\ -1.0,\ \overline{b}\ =\ 2.0,\end{array}  \tag{7}\]  
where line and wavelet sign mean parameter associated with upper and lower transistor respectively (see Figure 1a). The second set can be expressed as:  
\[ \begin{array}{l}\overline{y}_{11}\ =\ 0.9,\ \overline{y}_{12}\ =\ 2.1,\ \overline{y}_{22}\ =\ 0.0,\ \overline{a}\ =\ -1.0,\ \overline{b}\ =\ 1.5,\\ \widetilde{y}_{11}\ =\ 0.0,\ \widetilde{y}_{12}\ =\ 2.0,\ \widetilde{y}_{22}\ =\ 0.0,\ \overline{a}\ =\ -1.0,\ \overline{b}\ =\ 2.0,\end{array}  \tag{8}\]  
In the case of fourth-order dynamical system we have one set of internal system parameters that leads to chaotic solution, namely:  
\[ \begin{array}{l}\overline{y}_{11}\ =\ 8.0,\ \overline{y}_{12}\ =\ 1.0,\ \overline{y}_{22}\ =\ 0.0,\ \overline{a}\ =\ -3.0,\ \overline{b}\ =\ 2.0,\\ \widetilde{y}_{11}\ =\ 0.5,\ \widetilde{y}_{12}\ =\ 0.1,\ \widetilde{y}_{22}\ =\ 0.0,\ \overline{a}\ =\ -3.0,\ \widetilde{b}\ =\ 1.4,\end{array}  \tag{9}\]  
Numerically integrated strange attractor obtained by using Mathcad and build-in fourth order Runge-Kutta method and for list of values Equation (7) is demonstrated in  
Figure 2: Negative resistance oscillator with a resistor-biased lambda diode: (**a**) principal schematic considering generalized bipolar transistors, (**b**) equivalent small-signal model with the nonlinear voltage controlled current sources. Figure 3a. Parameters leading to this result are final time 5000 s with fixed time step 0.1 s and initial conditions \(\mathbf{x}_{0}=(1,\,2,\,0)^{\mathrm{T}}\). Figure 3b shows that same state trajectory plotted with respect of time, time range has linear scale, starts with 0 s and ends with 200 s. Chaotic trajectory evolves accordingly to the spiral repellor located at state \((0,\,0,0)^{\mathrm{T}}\). Figure 3c demonstrates that the analyzed mathematical model exhibits extreme sensitivity of system solution to tiny changes of the initial conditions in the time instance from 0 s to 20 s. The second trajectory is integrated with initial state \(\mathbf{x}_{0}=(1.01,\,2,\,0)^{\mathrm{T}}\). Note that both integrated state trajectories start to significantly differ approximately in the middle of provided time interval, that is after 10 s. Figure 3d provides numerical integration of the state trajectory for parameter set Equation (8) and for initial conditions \(\mathbf{x}_{0}=(-1,\,-2,\,0)^{\mathrm{T}}\).