## 3 Numerical Analysis and Localization of Chaotic Behavior
By considering normalized values, that is those obtained using optimization routine, we can significantly simplify set of the ordinary differential equations of discovered third-order autonomous deterministic dynamical system Equation (1), namely:  
\[ \begin{array}{l}\frac{d}{dt}v_{1}\ =\ -2.3\cdot v_{1}\ +\ 2.5\cdot v_{2}\ -\ v_{1}^{3}\ +\ (v_{1}\ -\ v_{2})^{3},\\ \frac{d}{dt}v_{2}\ =\ 1.2\cdot v_{1}-v_{2}\ -\ i_{L}\ +\ v_{1}^{3},\ \frac{d}{dt}i_{L}\ =\ v_{2}, \end{array}  \tag{11}\]  
where the state vector is \(\mathbf{x}\) = (\(v_{1}\), \(v_{2}\), \(i_{\mathrm{L}}\))\({}^{\mathrm{T}}\) and parameter group Equation (7) is considered. Circuit realization of this system is provided in Figure 10. For the second set of parameters in Equation (8), this mathematical model slightly changes into:  
\[ \begin{array}{l}\frac{d}{dt}v_{1}\ =\ -2.7\cdot v_{1}\ +\ 2.6\cdot v_{2}\ -\ v_{1}^{3}\ +\ (v_{1}\ -\ v_{2})^{3},\\ \frac{d}{dt}v_{2}\ =\ v_{1}\ -\ 0.9\cdot v_{2}\ -\ i_{L}\ +\ v_{1}^{3},\ \frac{d}{dt}i_{L}\ =\ v_{2}, \end{array}  \tag{12}\]  
Pseudo-component IC (Initial Condition) serve to inject specific initial conditions into circuit, that is, define nonzero voltage in specific node at time instance 0 s. Note that polynomial forward trans-conductances work as nonlinear current sources controlled by voltages. Highlighted parts represent the nonlinear function \(v_{1}\)\({}^{3}\) (green) and \((v_{1}-v_{2})\)\({}^{3}\), respectively. Results of associated time-domain analysis are in Figure 11.  
Figure 10: Maximally idealized third-order dynamical system, circuit realization using pseudo-components MULT (multiplication of two input voltages) and voltage-controlled current-sources G.  
Fourth-order chaotic system can be designed and verified using simulation software on the same level of abstraction, i.e., with ideal controlled sources and multipliers. After substitution of numerical values obtained using a searching procedure mentioned above, we can get following chaotic dynamical system:  
\[ \begin{array}{l}\frac{d}{dt}v_{1}\ =\ -0.6\cdot v_{1}\ +\ 2\cdot v_{2}\ -\ 0.1\cdot v_{3}\ -\ 3\cdot v_{2}^{3},\\ \frac{d}{dt}v_{2}\ =\ -0.3\cdot v_{1}\ -\ 0.1\cdot v_{2}\ -\ 7.9\cdot v_{3}\ +\ 3\cdot v_{1}^{3},\\ \frac{d}{dt}v_{3}\ =\ -0.1\cdot v_{1}\ +\ 0.1\cdot v_{2}\ -\ 0.1\cdot v_{3}\ -\ i_{L},\ \frac{d}{dt}i_{L}\ =\ v_{3}, \end{array}  \tag{13}\]  
where the state vector extends to \(\mathbf{x}\) = (\(v_{1}\), \(v_{2}\), \(v_{3}\), \(i_{L}\))T and the corresponding circuitry can be found in Figure 12. Few selected time domain simulation results are provided in Figure 13.  
Figure 11: Maximally idealized third-order dynamical system, selected Orcad Pspice simulation results: (**a**) plane projections \(v_{1}\) vs. \(i_{L}\) and \(v_{2}\) vs. \(i_{L}\), (**b**) baseband frequency spectrum of the generated signal \(v_{1}\) and \(v_{2}\), (**c**) chaotic signals \(v_{1}\) and \(v_{2}\) plotted with respect to time, (**d–f**) plane projections of interesting strange attractors (slightly modified nominal set of system parameters). However, practical implementation of the real-valued circuitry is of more importance. Note that fitness function utilized inside multi-criteria optimization procedure for chaos localization can be arranged such that both generalized bipolar transistors have the same equivalent circuit, see principal two-port in Figure 14a. Of course, in such a situation, also real circuitry implementation is exactly the same, as evident from schematic given in Figure 14b. The only difference is in numerical values of admittance parameters. In this schematic, upper index \(k=1\), 2 represents \(k\)-th transistor and \(\xi\), is the impedance norm chosen in the beginning stage of design of the chaotic oscillator. Nonlinear forward transconductance has the odd-symmetrical saturation-type shape and can be approximated using piecewise-linear function without losing chance to observe chaotic motion. Chaotic oscillator can be implemented easily if describing differential equations are known.