## 1. Introduction

_{21}(v

_{1}), while work [20] assumes a cubic polynomial for both functions y

_{12}(v

_{2}), y

_{21}(v

_{1}). Therefore, all seven distinct chaotic cases revealed in this manuscript are algebraically simpler than the single example proposed in paper [20]. In contrast, paper [21] deals with smooth nonlinear function y

_{21}(v

_{1}) and presents strange attractors discovered for only two shapes of forward trans-conductance. On the other hand, in the upcoming analysis, smooth polynomial nonlinearity up to the fifth order that describes backward trans-conductance could be found within the set of the ordinary differential equations. Of course, the different mathematical model considered here leads to completely different numerical results as well as much simpler circuitry implementation. From the practical perspective, the third-order deterministic chaotic systems provided in this paper generate waveforms with different features. Readers can pick and use our dynamical system that fits a specific application. In general, the upcoming sections represent more comprehensive analysis of the class C amplifier than case study [20]. Simple circuits with one or two transistors are analyzed in paper [22], again from the viewpoint of the evolution of chaotic behavior. In these networks, the parasitic properties of transistors are not considered for the numerical investigations.

## 2. Single Transistor Stage

_{jk}are admittance parameters of a bipolar transistor considered as two-port in a common emitter configuration and the state vector is $x={\left({v}_{1},{v}_{2},{i}_{\ell}\right)}^{\mathrm{T}}$. Symbol c

_{1}represents a parasitic base-emitter capacitance and c

_{2}is a sum of parasitic collector-emitter capacitance and working capacitance of the parallel resonant tank. Resistor R

_{2}given in the schematic could contain both the output admittance of transistor y

_{22}and inductance resistive losses. In practice, entries of 2 × 2 transistor´s admittance matrix could be complex numbers, especially if high-frequency applications are addressed. It is necessary to realize that the word “high” is relative—it could be tens of MHz depending on the type of bipolar transistor. Coefficients of the transistor´s admittance matrix could also be nonlinear functions, specifically in the case of power amplifiers or if signals with high amplitudes are processed.

_{21}will be scalar constant as well, which is a linear function of input voltage v

_{1}without offset. Therefore, a characteristic polynomial associated with the isolated system (1) and evaluated at the fixed point ${x}_{e}={\left({v}_{1}^{0},{v}_{2}^{0},{i}_{\ell}^{0}\right)}^{\mathrm{T}}$ is:

**E**is the unity matrix. A partial derivative of backward trans-conductance of a bipolar transistor is evaluated at equilibrium structure d

**x**/dt =

**0**, where

**0**is a vector of zeroes. Note that, at this moment, accumulation elements are normalized with respect to time and impedance. This fact is emphasized by utilization of the small letters ${c}_{1},{c}_{2},\ell $ throughout this manuscript.

#### 2.1. Local Polynomial Backward Trans-Conductance

_{1}, c

_{2}, $\ell $ and positive system dissipation, i.e., y

_{11}> 0 and y

_{22}> 0.

_{11}= y

_{22}= 0, formulas for the eigenvalues (6), (7) and (8) significantly simplify into the following relation:

#### 2.2. Local Piecewise-Linear Backward Trans-Conductance

_{12}(v

_{2}) by two- or three-segment piecewise linear (PWL) curves. A set of three ordinary differential equations that describe the class C amplifier is:

_{2}, −ϕ

_{1}, ϕ

_{1}, ϕ

_{2}} can be expressed as:

_{0}is slope of segment around zero, ρ

_{1}is slope of segment between breakpoint ϕ

_{1}and ϕ

_{2}(ϕ

_{1}< ϕ

_{2}), and ρ

_{2}is slope of segment in the outer regions of the vector field, i.e., x > ϕ

_{2}. For even-symmetry of the vector field, PWL function could possess three breakpoints {−ϕ, 0, ϕ} and be characterized by the following simple relation:

_{0}for 0 < x < ϕ, slope −ρ

_{0}for −ϕ < x< 0, slope −ρ

_{1}for x < −ϕ, and finally slope ρ

_{1}for x > ϕ. Both PWL functions generate a single equilibrium point located at the origin and the entire vector field is separated into five and four affine segments for odd (11) and (12) even-symmetrical PWL function, respectively. Note that the existence of other fixed points is not conditioned by the shape of PWL functions. In other words, for scalar PWL functions, there is one equilibrium point located at ${\left(-{y}_{12}\left(0\right)/{y}_{11},0,{y}_{21}\xb7{y}_{12}\left(0\right)/{y}_{11}\right)}^{T}$. Assume constant term ρ

_{const}in PWL function (11) or (12). Then, ρ

_{const}can be used to move the equilibrium point to a new position in the state space along a line.

#### 2.3. Alternative Mathematical Models of Class C Amplifier

**Y**; neither from the viewpoint of linear analysis nor circuitry realization. For a bipolar transistor modeled by impedance matrix

**Z**=

**Y**

^{−1}we can obtain the following algebraic relations:

_{22}= 0 and a normalized forward trans-conductance y

_{21}= 0. Analogically, for a bipolar transistor described by the hybrid matrix, we can obtain:

_{22}= 0 and y

_{21}= 1 provides simplification again. Obviously, a chaotic system based on a class C amplifier can be constructed using (13) and (14), or by introducing a linear transformation of coordinates applied on system (3) or (10).

#### 2.4. Searching for Chaotic Case

_{22}→0 and forward trans-conductance y

_{21}= 1 S. Normalized eigenvalues associated with the origin will be γ

_{1}= −y

_{11}, γ

_{2,3}= ±j, i.e., neighborhood trajectories are attracted to an eigenplane where limit cycle is evolved. This is a quite unusual situation in chaos theory. Therefore, the sixth-dimensional hyperspace of the internal parameters of a dynamical system (1) with the edges

**Ψ**∈{y

_{11}, a, b, c, d, e} undergoes deep investigation. The last five parameters shape nonlinear feedback function (3).

**Ψ**with a reasonable (from the viewpoint of potential practical applications involving experimental construction of the chaotic circuit) seven-sided volume have been found. More details can be found in Table 1 for polynomial vector field and Table 2 for PWL case. The provided cases represent differently shaped (in the geometric sense) strange attractors and this list is by no means complete.

_{12}is a nonlinear scalar function of variable v

_{2}. A single higher-order differential equation has a simple circuit representation: cascade connection of integrators with two-port feedback branches and an input summation/differentiation stage.

**Ψ**

_{1}up to

**Ψ**

_{7}represents a new chaotic system that cannot be transformed into some known third-order system via a linear change of coordinates.

## 3. Numerical Results

**x**

_{0}= (−1, 0, 0)

^{T}for parameter set

**Ψ**

_{1,2,3,5,6}and

**x**

_{0}= (2, 0, 0)

^{T}for parameter set

**Ψ**

_{4}. Individual cases of parameters

**Ψ**

_{1–6}are associated with Table 1.

**Ψ**

_{1}case of the chaotified class C amplifier (see Table 1), but similar results can be obtained for the rest of the system cases, both polynomial and PWL. In these graphs, red dots represent 10

^{4}initial conditions with normal distribution, standard deviation 0.01 and nominal value

**x**

_{0}= (1, 0, 0)

^{T}. Other colors have the following meanings: final state is stored after 1 s (green points), ending state after 10 s (blue dots) and final state after 100 s (black dots). Note that neighborhood trajectories diverge slowly, and after 10 s fiducial points are still closely spaced. There is one exception: system case

**Ψ**

_{2}possesses the higher degree of long-time unpredictability.

**Ψ**

_{1–7}(see Table 1). Here, red color denotes high kinetic energy, green marks average local energy and magenta indicates a very low local energy. Numerical values associated with these rainbow scaled plots normalized to unity time intervals are also provided. Initial conditions and time step were kept the same as for the analysis given in Figure 2; both with integration input parameters and a set of the initial conditions. Firstly, note that strange attractors occupy different sized volumes in the state space. Therefore, each plotted high-resolution plane has different axis ranges but uniform step size 0.01; concrete boundaries can be found within descriptors of the individual figures. For system cases

**Ψ**

_{1}and

**Ψ**

_{4}, it is obvious that average dynamic energy rises with the absolute value of state variable z. Using visualized plots, geometrical similarity between system cases

**Ψ**

_{1}and

**Ψ**

_{4}can be observed. Finally, strange attractors primarily do not evolve within areas with very high or low normalized energy. Along with kinetic energy distributions, Poincaré return maps for the horizontal slices of the state space (z = const) are visualized. Obviously, geometrical shapes of generated strange attractors are distinct and attractors are dense in the state space, outside regions of a high local differential growth. If indicated in the plot, vector field symmetry causes the strange attractor to be mirrored with respect to the zero plane (z = 0) and Poincaré sections could only be provided for upper or lower half state space (system cases

**Ψ**

_{2,3,6}).

**Ψ**

_{1}up to

**Ψ**

_{7}. The first flow quantifier is the largest Lyapunov exponent (LLE) calculated using the mathematical model, see [30,31] for an overall description and algorithm explanation. Based on the spectrum of one-dimensional Lyapunov exponents (real numbers calculated with transient behavior omitted), the so-called Kaplan–Yorke dimension (KYD) of a generated strange attractor is established [32,33]. Capacity dimension (CD) of the state space attractor established by using the box counting method [34] is also provided. Mentioned flow quantifiers adapted for third-order dynamical systems can be calculated as follows:

**Z**(t), LE

_{1}> LE

_{2}> LE

_{3}are one-dimensional Lyapunov exponents arranged in a decreasing order, and N(ε) is the number of cubes with edge ε required to fully cover the inspected state attractor.

**Ψ**. Here, a data sequence with a length of 1000 samples, embedding dimension 3 and time delay 1, was adopted. Table 3 can be roughly evaluated as follows: second case

**Ψ**

_{2}can be considered as the most unpredictable system with the most complex geometric structure of the strange attractor, while cases

**Ψ**

_{1},

**Ψ**

_{5}and

**Ψ**

_{7}produce chaotic waveforms with the most significant entropic properties. These are probably a good choice for applications in secure communications, chaos-based modulation/masking techniques, etc.

**Ψ**are wide enough such that the geometry of desired strange attractors will be structurally stable and experimentally observable—consult paper [37] for details. This is important since real values of the circuit components fluctuate with time, ambient temperature and heating, and they are inaccurate due to the fabrication tolerances, etc. Moreover, these effects neither compensate each other nor have mutual correlations. Color scale (legends with values of LLE are provided directly within individual plots) corresponds to the solution of a dynamical system (1) as follows: red denotes unbounded solution, yellow and green represent strong and weak chaos, respectively, blue color marks areas where the ω-limit set is periodic solution, and magenta highlights areas where trajectory is slowly attracted to the fixed point. In this case, final time was extended to 5000 s and the data sequence for calculation was stored after 500 s to remove short as well as long transients. To obtain sufficient accuracy (high resolution) of all plots, the parameter step was decreased to 0.01 such that each plot contained 101 × 101 = 10,201 points. Vertical axis is provided using a linear scale starting with zero. Note that system case

**Ψ**

_{3}has a very narrow parameter subspace that leads to the geometrically stable predefined chaotic attractor. On the other hand, nominal values of internal parameters of system case

**Ψ**

_{1}can be adjusted such that the prescribed strange attractor is very robust and cannot be violated by various imperfections during circuit construction.

_{11}. However, corresponding patterns are different for the particular cases

**Ψ**

_{1–6}. To demonstrate this property, dynamical flow was quantified and divided into the following classes: unbounded solution (white), strong chaos (red), weak chaos (yellow), limit cycle (green) and fixed point solution (blue). The parameter step for all plots is chosen uniformly as 0.05, axis scale for case

**Ψ**

_{1}is b∈(2, 3), d∈(−2, −1), for second system

**Ψ**

_{2}it is c∈(2, 3), e∈(−2, −1), third case

**Ψ**

_{3}has axis ranges a∈(4, 5), c∈(−2, −1), fourth case

**Ψ**

_{4}is characterized in ranges b∈(2, 3), d∈(−3, −2), results for fifth case

**Ψ**

_{5}are given in ranges c∈(2, 3), e∈(−3, −2), and for the case

**Ψ**

_{6}it is a∈(1, 2), e∈(−1.5, −0.5). For this kind of analysis, low resolution plots with 21 × 21 = 441 points have been calculated.

**Ψ**

_{1}transistor cell. Plots are calculated for ranges x

_{0}∈(−1, 1), y

_{0}∈(−1, 1) and each plot contains 201 × 201 = 40,401 sets of initial conditions. Note that neighborhood of equilibrium is not a part of the basin of attraction for the chaotic attractor. Basins of attraction are colored as follows: unbounded solution (red), strange attractor (green), limit cycle (light blue) and fixed point solution (dark blue).

## 4. Design of Flow-Equivalent Chaotic Oscillator

**Ψ**

_{1}and

**Ψ**

_{3}systems with impedance norm 10

^{3}and frequency norm 10

^{6}. Of course, having ideal voltage-controlled current-sources (G) and ideal voltage multiplication blocks (MULT), both norms can be arbitrary; only simulation profile setup needs to be adjusted accordingly. In our case, final time was set to 10 ms (to visualize the robust strange attractor), maximum allowed time step was reduced to 1 μs (to demonstrate the density of the strange attractor) and pseudo-components IC = −1V (IC1) served to set nonzero initial conditions into the circuit. In real circuitry, the injection of certain initial conditions is a much more complicated task.

**x**= (v

_{1}, v

_{2}, v

_{3})

^{T}and K = 0.1 is the internally trimmed transfer constant of AD633. Note that this chaotic system models the behavior of function (3) with nonzero values a and c, whereby other terms are zero.

_{2}/(R

_{1}+ R

_{2}) = 1 holds. The fundamental time constant of this circuit is τ = R⋅C = 10

^{4}⋅10

^{−8}= 100 μs, but the main frequency components can be shifted toward the GHz band easily by appropriate frequency rescaling. Considering the normalized numerical values provided in Table 1, impedance rescaling 10

^{4}and frequency norm 10

^{8}circuit components for (18) with parameter set

**Ψ**

_{3}are: C = 10 nF, R = 10 kΩ, R

_{1}= 33 kΩ, R

_{2}= 50 Ω, and R

_{3}=2 kΩ. Analogically, circuit components for (19) with parameter set

**Ψ**

_{2}(

**Ψ**

_{4}) are the following: C = 10 nF, R = 10 kΩ, R

_{1}= 1 kΩ, R

_{2}= 9 kΩ, R

_{3}= 18 kΩ (25 kΩ), R

_{4}= 4.8 kΩ (3.7 kΩ), and R

_{5}= 91 Ω (50 Ω). Figure 20a shows the PCB (Printed Circuit Board) of two uncoupled two-ports modeled by the adjustable admittance parameters. While input and output admittance is linear and represented by a variable resistor, trans-admittance y

_{12}and y

_{21}are polynomials up to the fourth order. The unoccupied socket is dedicated for integrated circuit TL084 (four operational amplifiers in a single package), or its empty pins can be used to connect PCB with the breadboard. PCB is designed such that a user can use switches to change signs of all coefficients of the polynomial trans-conductance y

_{12}(v

_{2}) and/or y

_{21}(v

_{1}). Figure 20b demonstrates the simplicity of the designed chaotic system.

## 5. Experimental Verification

**Ψ**

_{3}and

**Ψ**

_{1}, respectively—values are given in Table 1. A uniform 100 mV grid is used for numerical integration results. For the latter case, numerical mirrors of the visualized strange attractors are not provided. During measurement, strong sensitivity of type of the steady state to the initial conditions imposed into the chaotic oscillator has been confirmed. Nevertheless, goodish correspondence between theory and practical experiment was achieved. The route-to-chaos scenario can be traced via the shaping of nonlinear y

_{12}(v

_{2}) function, namely by variable resistors R

_{4}and R

_{5}in Figure 19b. To obtain classes of the chaotic system characterized by sets

**Ψ**

_{2},

**Ψ**

_{5},

**Ψ**

_{6}, and

**Ψ**

_{7}, additional AD633 is necessary. However, a major part of the proposed oscillator remains unchanged.

_{21}is realized by a single-input single-output operational trans-conductance amplifier. Nonlinear transfer function is implemented by couple (third-order polynomial for

**Ψ**

_{3}, fourth-order polynomial for

**Ψ**

_{1}and

**Ψ**

_{4}) or three (fifth-order polynomial to reach sets of parameters

**Ψ**

_{2},

**Ψ**

_{5}and

**Ψ**

_{6}) AD633.

#### Fractional-Order Chaotified Class C Amplifier

_{c}for very low frequencies and reaches to infinity for high frequencies. A set of ordinary differential equations that describes circuitry given in Figure 24b with an FO inductor L

_{x}with seven sections is as follows:

_{0}= 1/(2π) Hz is 1 s

^{1−α}/F, where α represents math order.

## 6. Discussion

- General mathematical models analyzed in this paper (3) and (10) contain normalized values of all accumulation elements. After optimization, to observe strange attractors, resulting parasitic capacitance as well as capacitance and inductance located within the LC resonant tank are of comparable orders. Therefore, parasitic accumulation elements turn into functional. This fact increases the intrinsic number of degrees of freedom and forces a naturally non-chaotic analogue building block to behave chaotically. Because of the internal structure of bipolar transistors commonly used in class C amplifiers, this kind of motion is possible only for assumed high-frequency operation. In practice, generated chaotic waveform can be easily misinterpreted as noise.
- The second condition for chaos evolution is the presence of a specific local nonlinear feedback. In the mathematical model of the analyzed dynamical system, either polynomial or PWL scalar function is the only nonlinearity.
- The third specific property of a bipolar transistor is linear backward trans-conductance. Its value is non-zero and relatively large.

## 7. Conclusions

- Parasitic capacitors are working ones,
- Nonlinearity is typical for a large signal model of a bipolar transistor,
- An additional degree of freedom is presented because driving force (processed signal) changes the operational point of an analyzed circuit.

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**General circuit concepts analyzed in this paper: (

**a**) fundamental cell of class C amplifier, (

**b**) equivalent schematic of class C amplifier for useful small-amplitude AC (Alternating Current) signals.

**Figure 2.**Plane projection v

_{1}vs. v

_{2}(black) and rainbow colored three-dimensional perspective views on the typical strange attractors generated by: (

**a**) parameter set

**Ψ**

_{1}substituted into the expression (1), (

**b**) parameter set

**Ψ**

_{1}substituted into system (6), (

**c**) parameter set

**Ψ**

_{2}substituted into differential Equation (1), (

**d**) parameter set

**Ψ**

_{2}substituted into jerk dynamics (6), (

**e**) parameter set

**Ψ**

_{3}numerically integrated using Equation (1), (

**f**) integration of system (1) with parameter set

**Ψ**

_{4}, (

**g**) parameter set

**Ψ**

_{5}substituted into Equation (1), and (

**h**) parameter set

**Ψ**

_{6}substituted into system (1) and integrated.

**Figure 3.**Sensitivity to tiny changes of initial condition demonstrated for first case of chaotic system: starting situation (red points), short time evolution (green points), average time evolution (blue dots) and long time separation (black dots). Nominal starting position is chosen as follows: (

**a**)

**x**

_{0}= (1, 0, 0)

^{T}, (

**b**)

**x**

_{0}= (−1, 0, 0)

^{T}, (

**c**)

**x**

_{0}= (0, −1, 0)

^{T}and (

**d**)

**x**

_{0}= (0, 1, 0)

^{T}. Magnified areas showing states are demonstrated.

**Figure 4.**Horizontal state space slices given by z = const. showing kinetic energy distribution of typical chaotic attractors of case

**Ψ**

_{1}system, associated Poincaré sections (black dots). Figures sorted from left to right and up to down: z = −0.9, z = −0.7, z = −0.5, z = −0.2, z = 0, z = 0.3, z = 1, z = 1.5, z = 2, z = 2.4, and z = 2.47.

**Figure 5.**Horizontal state space slices defined by the plane z = const. and providing dynamical energy distribution of typical chaotic attractors of case

**Ψ**

_{2}system (white curve), associated Poincaré sections (black dots). Figures sorted from left to right and up to down: z = −3.3, z = −3, z = −2.5, z = −2, z = −1.5, z = −1, z = −0.8, z = −0.6, z = −0.4, z = −0.2, and z = 0.

**Figure 6.**Horizontal state space slices given by plane z = const. providing rainbow scaled dynamical energy distribution of the typical chaotic attractors of case

**Ψ**

_{3}system, associated Poincaré sections (black dots). Individual figures are sorted from left to right and up to down with respect to the planes: z = −9.4, z = −9, z = −8.5, z = −8, z = −7, z = −6.5, z = −6, z = −4, z = −2, z = −1, and z = 0.

**Figure 7.**Horizontal state space slices given by plane z = const. providing rainbow scaled dynamical energy distribution of the typical chaotic attractors of case

**Ψ**

_{4}system (white trajectory), and associated Poincaré sections (black dots). Figures sorted from left to right and up to down: z = −0.4, z = −0.2, z = 0, z = 0.2, z = 0.3, z = 0.5, z = 0.8, z = 1.2, z = 1.6, z = 2, and z = 2.5.

**Figure 8.**Horizontal state space slices given by plane z = const. providing rainbow scaled dynamical energy distribution of the typical chaotic attractors of case

**Ψ**

_{5}system (white trajectory), and associated Poincaré sections (black dots). Individual figures are sorted from left to right and up to down: z = −1, z = −0.8, z = −0.6, z = −0.2, z = 0.4, z = 0.6, z = 1, z = 1.5, z = 2, z = 2.5, and z = 2.9.

**Figure 9.**Horizontal state space slices defined by plane z = const. providing rainbow scaled dynamical energy distribution of typical chaotic attractors of case

**Ψ**

_{6}system (white state trajectory), associated Poincaré sections (black dots). Figures sorted from left to right and up to down are given by: z =−3.3, z = −2.7, z = −2, z = −1.2, z = −0.7, z = 0, z = 0.4, z = 1, z = 1.7, z = 2.3, and z = 3.3.

**Figure 10.**Horizontal state space slices given by plane z = const. providing rainbow scaled dynamical energy distribution of the typical chaotic attractors of case

**Ψ**

_{7}system (white trajectory), and associated Poincaré sections (black dots). Individual figures sorted from left to right and up to down are given by: z = −1.4, z = −1, z = −0.5, z = −0.2, z = 0, z = 0.2, z = 0.5, z = 1, z = 1.5, z = 1.8, z = 2.2, and z = 2.7.

**Figure 11.**Rainbow scaled surface-contour plot of the largest Lyapunov exponent as two-dimensional function of nonlinear feedback, calculated for

**Ψ**

_{1}case of chaotic circuit and total range of parameters is a∈(0, 3) and b∈(−3, 0).

**Figure 12.**Rainbow scaled surface-contour plot of the largest Lyapunov exponent as two-dimensional function of nonlinear feedback, area covering both

**Ψ**

_{2},

**Ψ**

_{5}, and

**Ψ**

_{7}with dissipation coefficient y

_{11}= 0.4, total range of parameters is c∈(2, 5) and e∈(−3, 0).

**Figure 13.**Rainbow scaled surface-contour plot of the largest Lyapunov exponent as two-dimensional function of nonlinear feedback, area covering case

**Ψ**

_{3}and total range of parameters is a∈(3, 6) along with b∈(−3, 0).

**Figure 14.**Rainbow scaled surface-contour plot of the largest Lyapunov exponent as two-dimensional function of nonlinear feedback, area covering case

**Ψ**

_{4}and total range of parameters is b∈(0, 3) along with d∈(−3, 0).

**Figure 15.**Rainbow scaled surface-contour plot of the largest Lyapunov exponent as two-dimensional function of nonlinear feedback, area covering case

**Ψ**

_{6}and total range of parameters is a∈(1, 4) along with e∈(−3, 0).

**Figure 16.**Rainbow scaled plot showing flow quantification for the individual cases

**Ψ**

_{1–6}(rows 1 to 6) of chaotic class C amplifier and increased value of system dissipation y

_{11}= 0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9 (columns from left to right), see text for better clarification.

**Figure 17.**Colored basins of attraction, individual slices are the horizontal planes: (

**a**) z0 = −3, (

**b**) z0 = −2.5, (

**c**) z0 = −2, (

**d**) z0 = −1.5, (

**e**) z0 = −1.25, (

**f**) z0 = −1, (

**g**) z0 = −0.75, (

**h**) z0 = −0.5, (

**i**) z0 = −0.3, (

**j**) z0 = −0.1, (

**k**) z0 = 0.0, (

**l**) z0 = 0.1, (

**m**) z0 = 0.2, (

**n**) z0 = 0.3, (

**o**) z0 = 0.4, (

**p**) z0 = 0.5, (

**q**) z0 = 0.75, (

**r**) z0 = 1.0, (

**s**) z0 = 1.5, and (

**t**) z0 = 2.

**Figure 18.**Idealized circuit realization of a chaotic system with the emulated bipolar transistor stage: (

**a**) case

**Ψ**

_{1}system with parameters taken from Table 1, (

**b**) case

**Ψ**

_{3}system with parameter set taken from Table 1, (

**c**) Monge projections v

_{y}

_{1}vs. v

_{x}

_{1}(blue) and i

_{L}

_{1}vs. v

_{x}

_{1}(red), (

**d**) frequency spectrum of generated signal v

_{x}

_{1}(blue) and v

_{y}

_{1}(red). Red areas represent polynomial feedback transfer functions.

**Figure 20.**Photos captured during experimental investigation: (

**a**) PCB showing two two-ports where transconductances y

_{12}and y

_{21}are polynomials up to the fourth-order, (

**b**) two views onto breadboard with designed chaotic oscillator based on generalized class C amplifier.

**Figure 21.**Dynamical system (1) with (3) and values

**Ψ**

_{3}from Table 1, Comparison between numerical integration process (blue) and laboratory experiment (green): (

**a**,

**b**) v

_{1}vs. v

_{3}plane, (

**c**,

**d**) v

_{2}vs. v

_{3}plane, (

**e**,

**f**) v

_{1}vs. v

_{2}plane.

**Figure 22.**Dynamical system (1) with (3) and values

**Ψ**

_{1}from Table 1, comparison between numerical integration process (blue) and laboratory experiment (green): (

**a**,

**b**) v

_{1}vs. v

_{3}plane, (

**c**,

**d**) v

_{1}vs. v

_{2}plane, (

**e**,

**f**) v

_{2}vs. v

_{3}plane.

**Figure 23.**Different Monge projections of strange attractors not mutually connected with numerical analysis of generalized chaotic class C amplifier.

**Figure 24.**Two alternative lumped circuitry implementations of class C potentially chaotic amplifier: (

**a**) principal schematic of dynamical system with passive approximated fractional-order inductor, (

**b**) realization based directly on the state model (20).

**Table 1.**Numerical values of internal parameters of system (3) with mathematical orders α = β = γ = 1 that result in robust chaotic motion.

Case | y_{11} | a | b | c | d | e |
---|---|---|---|---|---|---|

Ψ_{1} | 0.56 | 0 | 2.1 | 0 | −1.1 | 0 |

Ψ_{2} | 0.50 | 0 | 0 | 3 | 0 | −1.5 |

Ψ_{3} | 0.30 | 5 | 0 | −2 | 0 | 0 |

Ψ_{4} | 0.40 | 0 | 2.7 | 0 | −2 | 0 |

Ψ_{5} | 0.30 | 0 | 0 | 3 | 0 | −2 |

Ψ_{6} | 0.50 | 2 | 0 | 0 | 0 | −0.5 |

Ψ_{7} | 0.40 | 0 | 0 | 2 | 0 | –1 |

**Table 2.**Numerical values of internal parameters of system (10) with mathematical orders α = β = γ = 1 and either (11) or (12) that result in structurally stable chaotic motion (NA means Not Available).

Case | y_{11} | ϕ | ϕ_{1} | ϕ_{2} | ρ_{0} | ρ_{1} | ρ_{2} |
---|---|---|---|---|---|---|---|

Ψ_{8} | 0.56 | 1.1 | NA | NA | 1 | −4.3 | NA |

Ψ_{9} | 0.5 | NA | 0.3 | 1.1 | 0.3 | 2 | −7 |

Ψ_{10} | 0.3 | NA | 0.6 | 1.18 | 4.6 | 0.6 | −9.9 |

Ψ_{11} | 0.3 | NA | 0.4 | 1 | 0.2 | 1.5 | −9.5 |

Case | LLE | KYD | CD | ApEn |
---|---|---|---|---|

Ψ_{1} | 0.071 | 2.113 | 2.15 | 0.539 |

Ψ_{2} | 0.156 | 2.239 | 2.24 | 0.558 |

Ψ_{3} | 0.045 | 2.132 | 2.14 | 0.440 |

Ψ_{4} | 0.020 | 2.050 | 2.10 | 0.503 |

Ψ_{5} | 0.069 | 2.186 | 2.20 | 0.564 |

Ψ_{6} | 0.047 | 2.081 | 2.15 | 0.518 |

Ψ_{7} | 0.050 | 2.160 | 2.13 | 0.620 |

**Table 4.**Numerical values of fully passive series-parallel circuit realization of fractional-order (FO) inductor with mathematical order 9/10, i.e., phase shift between voltage and current 81°.

R_{a} | R_{1} | R_{2} | R_{3} | R_{4} | R_{5} | R_{6} | R_{7} |
---|---|---|---|---|---|---|---|

0.6 Ω | 3.3 Ω | 22.7 Ω | 153 Ω | 1031 Ω | 6944 Ω | 46.7 kΩ | 313 kΩ |

L_{a} | L_{1} | L_{2} | L_{3} | L_{4} | L_{5} | L_{6} | L_{7} |

144 mH | 120 mH | 98 mH | 79 mH | 64 mH | 52 mH | 42 mH | 37 mH |

**Table 5.**Numerical values of fully passive series-parallel circuit realization of FO inductor with mathematical order 8/9, i.e., phase shift between voltage and current 80°.

R_{a} | R_{1} | R_{2} | R_{3} | R_{4} | R_{5} | R_{6} | R_{7} |
---|---|---|---|---|---|---|---|

1 Ω | 6.3 Ω | 44.4 Ω | 319 Ω | 2286 Ω | 16.4 kΩ | 118 kΩ | 833 kΩ |

L_{a} | L_{1} | L_{2} | L_{3} | L_{4} | L_{5} | L_{6} | L_{7} |

203 mH | 230 mH | 194 mH | 152 mH | 120 mH | 93 mH | 73 mH | 62 mH |

**Table 6.**Numerical values of fully passive series-parallel circuit realization of FO inductor with mathematical order 4/5, i.e., phase shift between voltage and current 72°.

R_{a} | R_{1} | R_{2} | R_{3} | R_{4} | R_{5} | R_{6} | R_{7} |
---|---|---|---|---|---|---|---|

1.1 Ω | 4.7 Ω | 25.8 Ω | 141 Ω | 769 Ω | 4184 Ω | 22.7 kΩ | 133 kΩ |

L_{a} | L_{1} | L_{2} | L_{3} | L_{4} | L_{5} | L_{6} | L_{7} |

35 mH | 237 mH | 155 mH | 101 mH | 66 mH | 43 mH | 28 mH | 22 mH |

**Table 7.**Numerical values of fully passive series-parallel circuit realization of FO inductor with mathematical order 3/4, i.e., phase shift between voltage and current 67.5°.

R_{a} | R_{1} | R_{2} | R_{3} | R_{4} | R_{5} | R_{6} | R_{7} |
---|---|---|---|---|---|---|---|

1.2 Ω | 4.5 Ω | 22 Ω | 108 Ω | 526 Ω | 2591 Ω | 12.7 kΩ | 55.6 kΩ |

L_{a} | L_{1} | L_{2} | L_{3} | L_{4} | L_{5} | L_{6} | L_{7} |

13 mH | 210 mH | 132 mH | 78 mH | 46 mH | 27 mH | 16 mH | 10 mH |

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