## 3 Numerical Analysis and Localization of Chaotic Behavior
Colored three dimensional views are rotated along vertical axis \(z\) as indicated by arrow, i.e., along line \(x=y=0\). Fundamental property of chaos, that is the extreme sensitivity of system solution to the tiny uncertainties of the initial states, is visualized in Figure 3e,f, respectively. Explanation of these plots is following: red dots represent \(10^{4}\) initial states generated with normal distribution (standard deviation 0.01) around point \(\mathbf{x}_{0}=(-1,\,-2,\,0)^{\mathrm{T}}\). After a short time of the system evolution (10 s), blue dots are stored, after additional 10 s green dots are placed and finally, after 100 s of system evolution, black dots are placed. Numerical integration was performed using a decreased time step 0.01 s to ensure smooth trajectory evolution and better accuracy. Figure 3: Numerical analysis of third order dynamical system: (**a**) 3D plot of the typical strange attractor generated by (1) with parameter set (7), (**b**) chaotic waveforms in time domain, (**c**) sensitivity of system solution to small changes of the initial conditions plotted with respect to time, (**d**) 3D plot of the typical strange attractor generated by (1) with parameter group (8), (**e,f**) sensitivity of system solution visualized in the state space, see text for details. Ranges for individual state variables in plot (**a**) are \(v_{1}\in(-4,\,4)\) V, \(v_{2}\in(-2,\,2)\) V, and \(i_{L}\in(-2,\,2)\) A, axis ranges for plot (**d**) are \(v_{1}\in(-1,\,1)\) V, \(v_{2}\in(-2,\,2)\) V, and \(i_{L}\in(-2,\,2)\) A. Figure 4 shows colored projections of the strange attractors generated by fourth order analyzed system into three-dimensional state space subspaces. Three differently shaped dense attractors have been discovered for the same mathematical model, with an exception that one internal parameter has different numerical value. Note that attractors generated by both third order and fourth order dynamical systems exhibit symmetry with respect to horizontal plane \(z=0\). In addition, dynamical flows are invariant with respect to the coordinate inversion, that is linear transformation \((x,y,z)^{\mathrm{T}}{\rightarrow}(-x,-y,-z)^{\mathrm{T}}\). The meaning of the graph of largest Lyapunov exponent (LLE) as function of system parameter is to mark regions (locations and compactness) where system behavior is truly chaotic. Thus, this graph is especially important if parameter space is associated to circuit realization of chaotic oscillator. \[\frac{d}{dt}\left( \begin{array}{c}v_{1}\\ v_{2}\\ i_{L}\end{array} \right)\ =\ \left( \begin{array}{ccc}a_{11}&a_{12}&a_{13}\\ a_{21}&a_{22}&a_{23}\\ a_{31}&a_{32}&a_{33}\end{array} \right).\left( \begin{array}{c}v_{1}\\ v_{2}\\ i_{L}\end{array} \right)\ +\ \left( \begin{array}{c}f(v_{1},\ v_{2})\\ g(v_{1},\ v_{2})\\ 0\end{array} \right), \tag{10}\]  
where functions \(f(. )\) and \(g(. )\) comprises all nonlinear terms. Figure 5a,b represents visualization of LLE as a two-dimensional surface-contour plot with respect to coefficients \(a_{21}\) and \(a_{12}\) in the sense of matrix Equation (10). The type of dynamical motion corresponds to the colored legend provided therein for the largest LE: a value close to zero is limit cycle, positive value indicates chaos and areas with negative value marks fixed point solution. As will be clarified later, both swept coefficients correspond to the variable resistors in the practical circuit realization. During this analysis, final time used for LLE calculation was increased to \(10^{4}\) s with time step 10 ms. Numerical values of system parameters should be adjusted precisely to observe desired strange attractor. Figure 5c,d provide rough specification of basins of attraction for dynamical system Equation (1) with set Equation (7), individual plots represent the horizontal slices of the state space, i.e., planes \(z=\mathrm{const}\).