# New Nonlinear Active Element Dedicated to Modeling Chaotic Dynamics with Complex Polynomial Vector Fields

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Evolution of Differential Voltage Trans-Conductance Multiplier (DV-TC-M)

^{2}for a bipolar multiplier and k = 1.3 mA/V

^{2}for a CMOS multiplier. Note that such transfer function is versatile especially in the case where individual ordinary differential equations are implemented by directly following Kirchhoff’s first law; that is, the sum of the currents associated with a single node equals zero. Summation/subtraction building block performs a useful operation, returning the sum and subtraction of three input voltages in accordance to the following equation:

## 3. Numerical Analysis of Chaotic Systems Dedicated for Circuit Synthesis

#### 3.1. Numerical Investigation of Dynamical System with Cyclically Symmetrical Vector Field

**x**

_{0}= (0, 0, 0)

^{T}and

**x**

_{0}= (b/a, b/a, b/a)

^{T}. Phase portrait of a typical chaotic attractor together with the measure of its sensitivity to tiny changes in the initial conditions is provided as in Figure 9. Sensitivity graphs are plotted for 10

^{4}randomly generated initial conditions that are uniformly distributed in all three state space dimensions to create a filled cube with edge size 0.01 (red dots). Final states are marked by a green color. Note that the dynamical system is clearly extremely sensitive to tiny changes in initial conditions. The desired strange attractor occupies a state space cube with edges −0.2 up to 0.3 in all directions, i.e., for all state variables. Thus, the essential condition for a dynamical system to be implementable by using DV-TC-M active elements is satisfied. However, the larger state attractor can be also modeled by circuitry with DV-TC-M, as long as suitable linear transformation of the coordinates is applied to the original system that makes the state attractor smaller.

**x**

_{0}= (0.1, 0, 0)

^{T}. The static system energy has a direct connection to the power dissipation of chaotic oscillator while the dynamic energy is bounded to a slew rate and measured frequency responses (pole frequencies, GBP in particular) of used active elements. Figure 11 shows a plot of LLE associated with a mathematical model (3) as two-dimensional function of the internal system parameters. Here, various limit cycles correspond to magenta while light blue stands for weak chaos, green represents strong chaos, and red means unbounded solution. During calculations, the final time was set to 1000 s with variable time step and the initial conditions

**x**

_{0}= (0.1, 0, 0)

^{T}. In the optimal circumstances, a dynamical system can possess the following promising set of three Lyapunov exponents 0.106, 0, −0.967 and corresponding fractal Kaplan–Yorke dimension of about 2.11. Note that, in a given two-dimensional space of system parameters, the region of chaos is wide enough for practical construction of a robust generator of chaotic waveform and surrounded by a limit cycle solution.

#### 3.2. Numerical Analysis of Fourth-Order and Fifth-Order Jerky Dynamics

**x**

_{e}= (±1, 0, 0, 0)

^{T}. To observe structurally stable strange attractors, the following values need to be kept: a = 1, b = 5.2, c = 2.7, and d = 4.5. For these numerical values, a typical single-scroll strange attractor provided in Figure 12 can be observed. Here, a numerical FFT calculation for the real circuit is also provided.

**x**

_{e}= (±1, 0, 0, 0, 0)

^{T}. Graphical visualization of the typical chaotic ω-limit set associated with dynamical system (5) is given in Figure 13.

#### 3.3. Numerical Analysis of Hyperchaotic Dynamics

**T**= diag(x

_{r}, y

_{r}, z

_{r}, w

_{r}) = (50, 50, 100, 150) represents a fundamental transformation matrix used for strange attractor compression and a = 10, b = 28, c = 2.7, and d = 5 are system parameters that lead to hyperchaotic motion. By considering mathematical model (6) and the numerical values mentioned above, we can obtain the strange attractor provided in the different 3D projections by means of Figure 16. During calculations, the final time was set to 1000 s with a fixed time step of 10 ms and initial conditions

**x**

_{0}= (0, 0, 0, 0.1)

^{T}. By utilizing the proposed linear change of the state coordinates, the desired state attractor shrinks from unfeasible cube (±20, ±30, +50, ±50) to much smaller cube (±0.4, ±0.5, +0.5, ±0.5). This process is graphically visualized by means of Figure 17. It turns out that even the greater compression of the strange attractor is possible. However, large values of system constants can cause practical problems, both from the viewpoint of trajectory convergence and basin of attraction.

## 4. New Circuitry Realizations of Chaotic Dynamical Systems Employing DV-TC-M

#### 4.1. Analog Functional Blocks Based on DV-TC-M Active Element

_{12}= y

_{21}= y

_{22}= 0, y

_{11}= −1/R), unilateral resistor (with current flowing in single direction, two-port admittance matrix y

_{11}= y

_{12}= y

_{22}= 0, y

_{21}= 1/R), negative bilateral trans-admittance circuit (two-port admittance matrix y

_{11}= y

_{22}= 0, y

_{12}= −1/R

_{1}, y

_{21}= −1/R

_{2}), differential voltage current source I

_{out}= (V

_{Y}

_{1}− V

_{Y}

_{2})/R, etc. Each mentioned building block is depicted in Figure 19, and can significantly simplify the final network that creates the linear part of the vector field. If speaking about nonlinear elements of chaotic oscillators, the trans-conductance nature of multiplier allows easy implementation of a polynomial resistor by simple connection of input and output terminal. Besides common nonlinear operations such as multiplication, division, squaring, and square rooting (corresponding circuitries can be found in datasheets of almost any commercially available multiplier) developed DV-TC-M allow the user to construct relatively simple resistors with sine-type and cosine-type ampere–voltage characteristics in both signum variants. Change of sign can be realized by reconnection of one input terminal of multiplier; there is no need to change circuit topology. An example of sine-type structure is provided in Figure 20. It is a two-terminal device having input current

_{2}/R

_{1}. Numerical values of all resistors can be calculated by comparing individual terms of (7) with a power series expansion of sine function, that is, sin(x) = x − x

^{3}/6 + x

^{5}/120. Of course, because this series is finite, approximation is valid only in in the close neighborhood of origin. This is a limitation of the dynamical range.

#### 4.2. Chaotic Oscillator with Cyclically Symmetrical Vector Field

**v**= (v

_{1}, v

_{2}, v

_{3})

^{T}is composed of voltages across grounded capacitors taken from left to right, with k

_{j}being the trans-conductance of j-th multiplier. The simultaneous change of all resistors (for example, via a tandem potentiometer) varies with system parameter b while parameter a stays fixed on a constant value given by the impedance norm of this circuit. For experimental verification, capacitors have a uniform value of 100 nF. Experiments show that the time constant of this chaotic oscillator can be lowered; the smallest value of applicable capacitors is 33 pF together with nominal resistance 1 kΩ. The measured power dissipation of this chaotic oscillator lies between 55 and 65 mW; the concrete value depends on the operational regime (chaos vs. limit cycles).

#### 4.3. Circuitry Implementation of Fourth-Order Hyperjerk Function

_{out}is output voltage of passive filter. Since both the energy source and nonlinearity are created by a single active element (DV-TC-M), the corresponding chaotic oscillator cannot be simplified further. The proposed circuit exhibits a chaotic motion for C

_{1}= 1 nF, C

_{2}= 470 pF, C

_{3}= 820 pF, L = 1.8 mH, and R = 1.4 kΩ. The state description of this oscillator is

**x**= (v

_{1}, v

_{2}, v

_{3}, i

_{L})

^{T}and k is internally trimmed constant of DV-TC-M element, namely k = 1.3 mA/V

^{2}. Structurally stable strange attractor can be observed for following list of the circuit parameters: C

_{1}= 3.9 nF, C

_{2}= 1.7 nF, C

_{3}= 4.2 nF, L

_{1}= 1.8 mH, R

_{1}= 200 Ω, R

_{2}= 2500 Ω (both resistances are potentiometers), and R

_{3}= 750 Ω. External voltages V

_{a}and V

_{b}can be considered as natural bifurcation parameters. Similar concept will be used to design fifth-order chaotic oscillator. In this case, Equation (3) can be rewritten providing more details as

_{out}is output voltage of fifth-order low-pass passive filter. This is the right place to remark that it is possible to substitute trans-impedance RLC passive ladder filter with active low-pass filter with the same, i.e., a trans-impedance type, filter. This holds under specific circumstances, namely in case of the same positions of the transfer function poles in the complex plane. State description of this oscillator can be expressed as

**x**= (v

_{1}, v

_{2}, v

_{3}, i

_{L}

_{1}, i

_{L}

_{2})

^{T}and k is a trans-conductance constant of DV-TC-M. In order to generate a structurally stable dense strange attractor, a list of the circuit parameters should be the following: C

_{4}= 330 pF, C

_{5}= 390 pF, C

_{6}= 470 pF, L

_{2}= L

_{3}= 1.8 mH, R

_{4}= 200 Ω, R

_{5}= 2500 Ω (both resistances are potentiometers). This set is not unique. During experimentation, few other value configurations turn out to be reasonable for the evolution of robust chaos. The time constant of this circuit can be lowered if each capacitor and inductor is divided by the same real number. The analogical change of time constant can be applied in the case of the fourth-order jerky system.

#### 4.4. Hyperchaotic System

_{j}is trans-conductance of j-th DV-TC-M multiplier and V

_{1}, V

_{2}are external DC voltages. This mathematical expression directly corresponds to a circuitry provided in Figure 23. Note that only three DV-TC-M elements are required for design of this chaotic oscillator. By considering unified values of the trans-conductances k

_{1,2,3,4,5}= 1.3 mA/V

^{2}, the desired chaotic attractor can be observed if the values of the remaining circuit components are C

_{1}= 10 nF, C

_{2}= 5.2 nF, C

_{3}= C

_{4}= 100 nF, R

_{1}= 1 kΩ, R

_{2}= 600 Ω, R

_{3}= 333 Ω, R

_{4}= 4.9 kΩ, R

_{5}= 50 kΩ, and external voltages V

_{1}= 290 mV, V

_{2}= 2 V. The time constant of this circuit is τ = 100 µs, and can be changed by dividing the value of each capacitor by the same number. However, the smallest allowed value of time constant for which the system generated an undistorted shape of a strange attractor is 200 ns. This leads to the set of working capacitors C

_{1}= 3.2 pF, C

_{2}= 1.7 pF, C

_{3}= C

_{4}= 32 pF, while resistors remain unchanged, and the nominal value equals 1 kΩ.

_{j}= 0.1 is a non-changeable internal constant of j-th fourth-quadrant analog multiplier AD633. Corresponding simulated strange attractor visualized as plane projections are provided by means of Figure 25; numerical values of passive circuit components and the location of symmetrical supply voltage ±15 V are included therein. The time constant of this circuit can be changed by boosting or lowering each capacitor by the same value. However, the smallest value that leads to an undistorted shape of the strange attractor is about 100 pF.

## 5. Experimental Verification

## 6. Discussion

^{2}is reported while three DV-TC-M devices alone will occupy an area of 0.242 mm

^{2}. The authors of [72] announced an operational power consumption of 99.5 mW (simulation), and this is value is close to the measured power dissipation for the proposed hyperchaotic circuit. The hyperchaotic oscillator implemented on a chip is the subject of [73]. The total chip area occupied by this circuit is 0.69 mm × 0.84 mm, and this value is still less than the area potentially required by our network topology. However, power consumption is significantly high; up to 475 mW was reported. Interestingly, on-chip realization of a multigrid spiral attractor generator is presented in [74]. The design process is nicely described though the oscillators are based on a conventional electronically programmable second-generation current conveyors, and the vector field is piecewise linear. The authors report excellent frequency responses, while the chip area needed is a little bit larger, and static power consumption (for the simplest configuration of spirals) is about 3.7 mW. This value can be compared to the power dissipation of our generator of a single-scroll attractor, that is, fourth-order jerky dynamics provided in Figure 22. Here, the measured value is approximately four times greater. Another successful realization of multiscroll attractor generator can be found in paper [75]. Of course, a list of the available publications dealing with integrated versions of chaotic systems (the most often fabricated using CMOS technology) is by no means complete. A few others can be found in [76,77], where complete simulations can be found, including a post-layout and process–voltage–temperature variations. However, these results cannot be directly compared and discussed in the context of individual circuitry realizations presented in this paper. Roughly speaking, a fully integrated fashion of implementation of simple as well as complex chaotic oscillators is preferred if portable systems are required for applications.

## 7. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

^{2}(bipolar multiplier), 761 µm × 203 µm = 0.155 mm

^{2}(CMOS multiplier) and 430 µm × 203 µm = 0.087 mm

^{2}for voltage summation/subtraction cell. Bias power consumption is: 10 mW (bipolar multiplier), 8 mW (CMOS multiplier), and 9 mW (voltage summation/subtraction cell).

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**Figure 1.**Symbols of active cells employed in designed chaotic oscillators: multiplier with internal voltage-to-current conversion (

**left**) and summation/subtraction (

**right**) unit.

**Figure 2.**Internal topologies of the active cells fabricated in a single integrated circuit (IC) package.

**Figure 3.**Comparison between measured and simulated responses of bipolar core-based multiplier: DC transfer responses X

_{1}→Z for V

_{Y1}controlled by DC voltage (

**upper plot**), DC transfer responses Y

_{1}→Z for V

_{X1}controlled by DC voltage (

**middle plot**), and dependence of g

_{m}on V

_{Y1}/V

_{X1}(

**lower plot**).

**Figure 4.**Magnitude of AC transfer responses X

_{1}→Z for V

_{Y1}controlled by DC voltage (

**upper left plot**), magnitude of AC transfer responses Y

_{1}→Z for V

_{X1}controlled by DC voltage (

**upper right graph**), magnitude vs. frequency plot of input impedance at X

_{1}terminal (

**middle left plot**), magnitude vs. frequency plot of input impedance at Y

_{1}terminal (

**middle right plot**), and magnitude vs. frequency plot of output impedance at Z terminal, example for V

_{X,Y}= −0.5 V (

**lower plot**).

**Figure 5.**Comparison between measured and simulated responses of CMOS core-based multiplier: DC transfer responses X

_{1}→Z for V

_{Y1}controlled by DC voltage (

**upper image**), DC transfer responses Y

_{1}→Z for V

_{X1}controlled by DC voltage (

**middle plot**), and dependence of g

_{m}on V

_{Y1,X1}(

**lower image**).

**Figure 6.**Measured magnitude of AC transfer X

_{1}→Z responses for V

_{Y1}controlled by DC voltage (

**upper left plot**), magnitude of AC transfer responses Y

_{1}→Z for V

_{X1}controlled by DC voltage (

**upper right plot**), magnitude vs. frequency plot of input impedance at X

_{1}node (

**middle left image**), magnitude vs. frequency plot of input impedance at Y

_{1}node (

**middle right plot**), and magnitude vs. frequency plot of output impedance at Z terminal, example for V

_{X1}= −0.5 V (

**lower graph**).

**Figure 7.**Comparison between measured and simulated responses of a summation/subtraction unit (see text for evaluation): DC transfer responses between terminals Y

_{1,2,3}→W.

**Figure 8.**Summation/subtraction unit and magnitude AC transfer responses between terminals Y

_{1,2,3}→W (

**upper left plot**), magnitude of the input impedances Y

_{1,2,3}vs. frequency (

**upper right graph**), and magnitude vs. frequency plot of output impedance measured at W terminal (

**lower image**).

**Figure 9.**Three-dimensional perspective view on the typical strange attractor associated with the dynamical system (3) (

**black color**) and corresponding plane projections: x vs. y (

**green**), x vs. z (

**blue**) and y vs. z (

**orange**) for initial conditions

**x**

_{0}= (0.1, 0, 0)

^{T}. Sensitivity of the dynamical system (3) to initial conditions for different final time of integration; see text for clarification.

**Figure 10.**Energy distribution over strange attractor (see text): static (

**blue**) and dynamic (

**orange**).

**Figure 11.**2D and 3D view of the rainbow color-scaled surface contour plot of the largest Lyapunov exponent (LLE) as a function of the internal system parameters a ∈ (1, 2) with step size 0.01 and b ∈ (0.3, 0.5) with step size 0.001.

**Figure 12.**Perspective view on a typical strange attractor associated with system (4) is demonstrated in the left plot and circuit-oriented equivalent flow (10) is provided on the middle plot. The next pictures give generated chaotic waveforms in time and frequency domain for normalized and real circuit.

**Figure 13.**Three-dimensional perspective view (black) of the typical strange attractor associated with system (5) is provided in the left plot: x vs. dx/dt (green), x vs. dx

^{2}/dt

^{2}(blue) and x vs. dx

^{3}/dt

^{3}(orange). Remaining images represent the individual plane projections of a fifth-dimensional strange attractor: (

**a**) x vs. dx/dt, (

**b**) x vs. d

^{2}x/dt

^{2}, (

**c**) x vs. dx

^{3}/dt

^{3}, (

**d**) x vs. dx

^{4}/dt

^{4}, (

**e**) dx/dt vs. dx

^{2}/dt

^{2}, (

**f**) dx/dt vs. dx

^{3}/dt

^{3}, (

**g**) dx/dt vs. dx

^{4}/dt

^{4}, (

**h**) dx

^{2}/dt

^{2}vs. dx

^{3}/dt

^{3}, (

**i**) dx

^{2}/dt

^{2}vs. dx

^{4}/dt

^{4}, (

**j**) dx

^{3}/dt

^{3}vs. dx

^{4}/dt

^{4}.

**Figure 14.**Basin of attraction calculated for dynamical system (4), state space degraded into cubes defined by x = 0 and plotted as planes d

^{4}x/dt

^{4}(from left to right), upper row is set {−2, −1, 0, 1, 2, 3, 4}, lower row is continuation {5, 6, 7, 8, 9, 10, 11}.

**Figure 15.**Kaplan–Yorke dimension calculated in close neighborhood of nominal values of internal system parameters, low-resolution one-dimensional bifurcation diagram calculated with respect to the external voltage V

_{a}∈ (0.4, 0.5)V with a step of 1 mV (red) and V

_{b}∈ (−0.45, −0.2)V with a step of 1 mV (green).

**Figure 16.**3D perspective plots of typical strange attractor generated by 4D hyperchaotic system (6). Upper left plot: y vs. z plane (

**orange**), x vs. z plane (

**blue**) and x vs. y plane (

**green**). Upper middle plot: z vs. w plane (

**orange**), y vs. w plane (

**blue**) and y vs. z plane (

**green**). Upper right plot: x vs. w plane (purple). Lower row represents 3D cross sections defined by planes x = {−0.25, −0.2, −0.1, 0, 0.1, 0.25}.

**Figure 17.**Three-dimensional views of a typical strange attractor associated with 4D hyperchaotic system (6) for increased zoom: original dynamical system (

**blue**) and transformed system (

**red orbit**). Size of cube side of individual plane projection, from left to right: 50, 10, 4, 0.4.

**Figure 18.**Rainbow color-scaled plots of two LLE as function of internal parameters of hyperchaotic system (6). Colored contours represent maximal LLE while line contours second largest LLE; see text for clarification. Overlapped maxima mark dynamical behavior expanding in two directions.

**Figure 19.**Four examples of differential voltage trans-conductance multiplier (DV-TC-M) subcircuits that are interesting in terms of synthesis of the linear part of the vector field: (

**a**) negative grounded resistor, (

**b**) unilateral resistor, (

**c**) negative trans-admittance two-port, and (

**d**) voltage controlled current source.

**Figure 20.**DV-TC-M based sine-type resistor that considers power series approximation with three terms, cosine- type resistor can be implemented by a cascade of three squarers, negative resistor, and constant current source (each can be implemented by single proposed summation/subtraction block).

**Figure 21.**Circuit realization of fully analog chaotic oscillator based on mathematical model having cyclically symmetrical vector field and three pieces of DV-TC-M.

**Figure 22.**Circuitry implementation of fourth-order (

**left schematic**) and fifth-order (

**right picture**) jerky dynamical system, both with only a single DV-TC-M active element.

**Figure 24.**Realization of the hyperchaotic system based on the integrator block schematic ready for OrCAD PSpice circuit simulation, five active devices needed, nodes with the same label are connected, individual capacitors can be pre-charged using pseudo-component IC1 (not mandatory).

**Figure 25.**Circuit simulation of hyperchaotic system based on the integrator block schematic. Upper plot: v

_{x}vs. v

_{y}plane (

**blue**), v

_{x}vs. v

_{z}plane (

**red**). Lower plot: v

_{w}vs. v

_{y}plane (

**blue**), v

_{w}vs. v

_{z}plane (

**red**).

**Figure 26.**Fabricated printed circuit board (PCB) dedicated to experimental verification of DV-TC-M-based applications: fourth-order jerky function as a robust generator of the chaotic waveforms (

**left photo**) and chaotic oscillator with cyclically symmetrical vector field (

**right photo**).

**Figure 27.**Experimental verification of chaotic circuit given in Figure 21: selected plane projections.

**Figure 28.**Experimental verification of chaotic circuit given in Figure 21: generated chaotic signals.

**Figure 29.**Outputs coming from the numerical integration process (

**upper row of plots**) compared to true experimental results captured as the oscilloscope screenshots: v

_{1}vs. v

_{2}plane projections of the generated state attractors (including chaotic orbits), different values of the external voltage V

_{a}and fixed voltage V

_{b}= 350 mV.

**Figure 30.**Experimental results captured as the oscilloscope screenshots: v

_{1}vs. v

_{3}plane projections of generated state attractors (including chaotic orbits), different values of external voltage V

_{a}and fixed voltage V

_{b}= 350 mV.

**Figure 31.**Oscilloscope screenshots captured during measurement of fourth-order jerky dynamics, individual rows represent different plane projection of the limit cycles and typical chaotic attractors.

**Figure 32.**Oscilloscope screenshots captured during verification of a fifth-order jerky chaotic circuit, plane projections of different limit cycles (left column) and chaotic motion (the rest of photos).

**Figure 33.**Oscilloscope screenshots associated with measurement of a hyperchaotic circuit, different plane projections of the compressed strange attractor.

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Petrzela, J.; Sotner, R.
New Nonlinear Active Element Dedicated to Modeling Chaotic Dynamics with Complex Polynomial Vector Fields. *Entropy* **2019**, *21*, 871.
https://doi.org/10.3390/e21090871

**AMA Style**

Petrzela J, Sotner R.
New Nonlinear Active Element Dedicated to Modeling Chaotic Dynamics with Complex Polynomial Vector Fields. *Entropy*. 2019; 21(9):871.
https://doi.org/10.3390/e21090871

**Chicago/Turabian Style**

Petrzela, Jiri, and Roman Sotner.
2019. "New Nonlinear Active Element Dedicated to Modeling Chaotic Dynamics with Complex Polynomial Vector Fields" *Entropy* 21, no. 9: 871.
https://doi.org/10.3390/e21090871