## 3 Numerical Results  
Discovered Novel Chaotic System (1) with Parameters (7) has been numerically analyzed using Mathcad and MATLAB. For numerical integration, the fourth-order Runge-Kutta method has been adopted. Figure 2 demonstrates the typical strange attractor generated by a bipolar transistor cell under bias point conditions mentioned above. The final time is \(10^{4}\), time step is \(10^{-2}\), and sensitivity to changes in initial conditions is also visualized. The group of \(10^{4}\) initial conditions were randomly generated around point \(\mathbf{x}_{0}=\) (1, 0, 0)\({}^{\text{T}}\) using normal distribution with deviation of \(10^{-2}\) (red dots); short time evolutions after 1 s are plotted (blue), and then states after 100 s of the integration process are depicted (orange dots), and the final state after 1000 s is provided (green dots). Note that neighborhood state trajectories diverge slowly, and this should be respected when setting input parameters of the upcoming analysis. Figure 3 shows dynamic energy calculated in the state space volume where the strange attractor evolves. In this plot, a rainbow scale is used: red denotes high energy, green stands for average value, and blue marks regions with very low dynamic energy. Figure 4 illustrates basins of attraction for the typical chaotic attractor generated by a transistor-based system. Here, each plot contains a linear grid having \(101\times 101\) points, and the color scale is as follows: the limit cycle is marked by green, chaotic attractor by red, and fixed-point solution is blue. Ranges of horizontal and vertical axes are the same for each plot: \(v_{1}\)\(\in\)(-5, 5) and \(v_{1}\)\(\in\)(-5, 5), respectively.  
Figure 2: Numerical integration outputs: (**a**) typical double-trumpet chaotic attractor, (**b**) sensitivity to the changes in initial conditions, (**c**) zoomed short time evolution, (**d**) zoomed average time evolution, and (**e**) initial conditions close to point \(\mathbf{x}_{0}=\) (0, 0, 0)\({}^{\text{T}}\). Note the different scales of the axis system for individual plots.  
Figure 5 provides us with an idea about potential robustness of the designed chaotic oscillator. This idea is simplified and plotted in two dimensions, namely as a function of cubic terms \(a_{2}\) and \(b_{2}\). This figure also illustrates zoomed plots A and C, where chaotic areas guarantee the generation of structurally stable strange attractors. However, careful circuit realization is still required for future laboratory success. The LLE was calculated using a final time of \(10^{4}\), time step of \(10^{-2}\) s, and Gram-Schmidt orthogonalization. For each line in these three-dimensional plots, a bifurcation diagram can be calculated. Two interesting examples denoted by letters D and E are demonstrated. Note that evolution of trumpet attractors is not subject to a traditional period-doubling route to the chaos scenario. During the calculation of Figure 5, the maximal value of LLE is 0.082, and the Kaplan-Yorke dimension of the associated chaotic attractor is about 2.2.  
Figure 3: Kinetic energy calculated in horizontal slices of state space (from left to right, up to down) and return maps (black dots): \(z=-13\), \(z=-10\), \(z=-6\), \(z=-3\), \(z=-2\), \(z=-1\), \(z=0\), \(z=1\), \(z=2\), \(z=3\), \(z=4\), and \(z=5\).  
Figure 4: Basins of attraction calculated as horizontal slices of a state space: (**a**) \(i_{L}=0\), (**b**) \(i_{L}=0.1\), (**c**) \(i_{L}=0.2\), (**d**) \(i_{L}=0.5\), (**e**) \(i_{L}=0.7\), (**f**) \(i_{L}=0.8\), (**g**) \(i_{L}=1\), (**h**) \(i_{L}=1.5\), (**i**) \(i_{L}=2.0\), (**j**) \(i_{L}=2.5\), (**k**) \(i_{L}=3.0\), (**l**) \(i_{L}=3.5\), (**m**) \(i_{L}=4\), (**n**) \(i_{L}=4.5\), (**e**) \(i_{L}=5\), (**p**) \(i_{L}=6\), (**q**) \(i_{L}=8\), (**r**) \(i_{L}=10\), (**s**) \(i_{L}=11\), and (**t**) \(i_{L}=12\).