## 3 Numerical Analysis and Localization of Chaotic Behavior
Namely, these planes are assumed as \(z=0\), \(z=1\), \(z=2\), \(z=3\), \(z=4\), \(z=5\) for Figure 5c and \(z=0\), \(z=0.5\), \(z=1\), \(z=1.5\), \(z=2\), \(z=2.5\), \(z=3\), \(z=3.5\), \(z=4\), \(z=4.5\), \(z=5\), \(z=6\), and \(z=7\) for Figure 5d. Basins for the negative values of \(z\) coordinate need not be visualized due to the vector field symmetry. Figure 6 represents contour graphs of first (lower plots) and second (upper plots) LE if these are sorted from the largest to the smallest value, again with respect to the cubic polynomial terms. Color scale for second largest LE is the same as it is for Figure 5. However, positive value of both LEs means that system starts to exhibit the so-called hyperchaotic behavior. In fact, this kind of a very complex motion where the vector field stretches two-dimensional subspace occurs very rarely and is marked by dark red color in upper plots. These  
Figure 4: Typical strange attractors generated by analyzed fourth order dynamical system and visualized using 3D subspaces, different internal parameters of second generalized bipolar transistors, namely: (**a**) \(\widetilde{b}\ =\ 1.4\), (**b**) \(\widetilde{b}\ =\ 1\), (**c**) \(\widetilde{y}_{11}\ =\ 0.22\). Ranges for individual state variables are \(v_{1}{\in}(-1,1)\) V, \(v_{2}{\in}(-1,1)\) V, \(v_{3}{\in}(-100,100)\) mV, and \(i_{L}{\in}(-100,100)\) mA. are especially visible in parameter space defined by Figure 6a,b,g,h,m,n. Nevertheless, note that areas where analyzed dynamical system exhibit \"simple\" chaos are wide and continuous. Figure 5: Selected numerical results associated with third order chaotic system: (**a,b**) rainbow-scaled contour plots of largest Lyapunov exponent (LLE) as two-dimensional functions of the internal parameters of chaotic oscillator, parameter step 0.001, graph contains 10,201 points. Colored basins of attraction for ranges of state variables \(x\)\(\in\)(\(-\)5, 5) and \(y\)\(\in\)(\(-\)5, 5), chaotic solution (red), unbounded solution (red), fixed point (green): (**c**) dynamical system Equation (1) with parameter group Equation (8), (**d**) dynamical system Equation (1) having parameter set Equation (8). Figure 6: Rainbow-scaled contour plots of LLE as two-dimensional functions of parameters of generalized transistors for fourth order dynamical system: (**a**) second LE and (**b**) first LE for parameter range \(\widetilde{a}\in(-2,\ -1),\widetilde{a}\in(-2,\ -1)\), (**c**) second LE and (**d**) first LE for parameter range \(\widetilde{a}\in(-3,\ -2),\widetilde{a}\in(-2,\ -1)\), (**e**) second LE and (**f**) first LE for parameter range \(\widetilde{a}\in(-4,\ -3),\widetilde{a}\in(-2,\ -1)\), (**g**) second LE and (**h**) first LE for parameter range \(\widetilde{a}\in(-2,\ -1),\widetilde{a}\in(-3,\ -2)\), (**i**) second LE and (**j**) first LE for parameter range \(\widetilde{a}\in(-3,\ -2),\widetilde{a}\in(-3,\ -2)\), (**k**) second LE and (**l**) first LE for parameter range \(\widetilde{a}\in(-4,\ -3),\widetilde{a}\in(-3,\ -2)\), (**m**) second LE and (**n**) first LE for parameter range \(\widetilde{a}\in(-2,\ -1),\widetilde{a}\in(-4,\ -3)\), (**o**) second LE and (**r**) first LE for parameter range \(\widetilde{a}\in(-3,\ -2),\widetilde{a}\in(-4,\ -3)\), (**o**) second LE and (**r**) first LE for parameter range \(\widetilde{a}\in(-4,\ -3),\widetilde{a}\in(-4,\ -3)\). Individual images are plotted for different shapes of forward polynomial trans-conductances. and robustness of strange attractor. This proposition should be considered also from the viewpoint of time constant of constructed chaotic oscillator. The first step toward circuit realization of chaotic oscillator should be computer-aided design and simulation. Of course, software such as Orcad Pspice can treat with both real and normalized numerical values of the circuit components.