# New Chaotic Dynamical System with a Conic-Shaped Equilibrium Located on the Plane Structure
## 3 Systematic Design of Chaotic Oscillator and Its Experimental Verification  
Circuit synthesis, based on the prescribed mathematical model, is a straightforward process [30]. In fact, the concept of voltage-mode analog computers can be directly utilized. That is, the operational amplifier based inverting integrators, inverting summing amplifiers and two-ports, with prescribed nonlinear transfer characteristics, are the only building blocks required for the design. Deficiency of the commercially available active devices makes discrete circuit implementation of a current-mode chaotic oscillator much more complicated. However, it is still possible by using a three-terminal  
Figure 4: A two-dimensional surface contour plot of the largest LE as a function of the internal system parameter \(a\) and \(b\), color scale of magnified area: unbounded solution (**white**); strong chaos (**green**); weak chaos (**blue**); color scale of lower image: LE = 0.15 light green, LE = 0.11 dark green, LE = 0.07 light blue and LE = 0.03 dark blue. Figure 5: Basin of attraction: topographically colored surface contour plot of the log-scaled final system energy, original dynamical system with hyperbolic equilibria (upper set of pictures) and system with parallel lines as equilibria (lower set of pictures). second-generation current conveyor AD844 (positive type having input and output current flowing towards active element, supplemented by an output voltage buffer connected to an additional pin), EL2082 (negative type with opposite orientation of input and output current) and a four-quadrant analog current multiplier EL4083. It is still believed that low values of the node resistances and small values of the parasitic capacitances allow designed chaotic oscillators to work properly in higher frequency bands, such that fundamental harmonics can be located above 1 MHz. Both designed variants of oscillators have been verified; first via laboratory experiments and second, through Orcad Pspice circuit simulations. A key design factor in the oscillator realization process is the size of the strange attractor, that is, the volume of state space occupied by the attractor. In practice, this cube is limited by dynamical ranges (for required linear operation) and, consequently, supply voltages of the used active elements. Of course, both the stretching and folding mechanism of the vector field needs to be preserved; these phenomena are responsible for chaos evolution. Thus, we always try to keep final network topology simple with nonlinear transfer functions as close to the original as possible, without the unwanted offsets and curve-shape deformations. Frequency limitations due to parasitic features of active devices are not a problem, since the frequency normalization factor can be arbitrary. We can imagine this constant as a number that multiplies each differential equation. In both practical realizations, \(\tau=1\) ms has been chosen. Thus, chaotic signals are within the audio frequency band and chaos became audible. Circuitry implementation of the dynamical system (1) working in voltage- and current-mode is given in Figures 6 and 7 respectively. Let us discuss the first realization in more detail. Note that the inevitable price for the complex mathematical model is many active elements; one TL084 with four standard voltage-feedback operational amplifiers in a single package and three four-quadrant analog multipliers AD633 with transfer function \(V_{W}=K(V_{X1}-V_{X2})(V_{Y1}-V_{Y2})+V_{Z}\) are required. Both types of the active elements are fed by a symmetrical \(\pm 15\) V voltage supply. This creates sufficient dynamical range for bounded types of the state space attractors. Chaotic signals can be directly measured at the outputs of inverting integrators where state variables \(x\), \(y\) and \(-z\) can be found.